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| Line 6: |
Line 6: |
| | The theory is described in [[Kagemoto and Yue 1986]] and in | | The theory is described in [[Kagemoto and Yue 1986]] and in |
| | [[Peter and Meylan 2004]]. | | [[Peter and Meylan 2004]]. |
| − |
| |
| − |
| |
| − |
| |
| − |
| |
| − |
| |
| − |
| |
| − | We extend the finite depth interaction theory of [[kagemoto86]] to
| |
| − | water of infinite depth and bodies of arbitrary geometry. The sum
| |
| − | over the discrete roots of the dispersion equation in the finite depth
| |
| − | theory becomes
| |
| − | an integral in the infinite depth theory. This means that the infinite
| |
| − | dimensional diffraction
| |
| − | transfer matrix
| |
| − | in the finite depth theory must be replaced by an integral
| |
| − | operator. In the numerical solution of the equations, this
| |
| − | integral operator is approximated by a sum and a linear system
| |
| − | of equations is obtained. We also show how the calculations
| |
| − | of the diffraction transfer matrix for bodies of arbitrary
| |
| − | geometry developed by [[goo90]] can be extended to
| |
| − | infinite depth, and how the diffraction transfer matrix for rotated bodies can
| |
| − | be easily calculated. This interaction theory is applied to the wave forcing
| |
| − | of multiple ice floes and a method to solve
| |
| − | the full diffraction problem in this case is presented. Convergence
| |
| − | studies comparing the interaction method with the full diffraction
| |
| − | calculations and the finite and infinite depth interaction methods are
| |
| − | carried out.
| |
| − |
| |
| − |
| |
| − |
| |
| − | ==Introduction==
| |
| − | The scattering of water waves by floating or submerged
| |
| − | bodies is of wide practical importance.
| |
| − | Although the problem is non-linear, if the
| |
| − | wave amplitude is sufficiently small, the
| |
| − | problem can be linearised. The linear problem is still the basis of
| |
| − | the engineering design of most off-shore structures and is the standard
| |
| − | model of geophysical phenomena such as the wave forcing of ice floes.
| |
| − | While analytic solutions have been found for simplified problems
| |
| − | (especially for simple geometries or in two dimensions) the full
| |
| − | three-dimensional linear diffraction problem can only be solved by
| |
| − | numerical methods involving the discretisation of the body's surface.
| |
| − | The resulting linear system of equations has a dimension equal to the
| |
| − | number of unknowns used in the discretisation of the body.
| |
| − |
| |
| − | If more than one body is present, all bodies scatter the
| |
| − | incoming waves. Therefore, the scattered wave from one body is
| |
| − | incident upon all the others and, given that they are not too far apart,
| |
| − | this notably changes the total incident wave upon them.
| |
| − | Therefore, the diffraction calculation must be conducted
| |
| − | for all bodies simultaneously. Since each body must be discretised this
| |
| − | can lead to a very large number of unknowns. However, the scattered
| |
| − | wavefield can be represented in an eigenfunction basis with a
| |
| − | comparatively small number of unknowns. If we can express the problem in this
| |
| − | basis, using what is known as an {\em Interaction Theory},
| |
| − | the number of unknowns can be much reduced, especially if there is a large
| |
| − | number of bodies.
| |
| − |
| |
| − | The first interaction theory that was not based on an approximation was
| |
| − | the interaction theory
| |
| − | developed by [[kagemoto86]]. Kagemoto and Yue found an exact
| |
| − | algebraic method to solve the linear wave problem for vertically
| |
| − | non-overlapping bodies in water of finite depth. The only restriction
| |
| − | of their theory was that the smallest escribed circle for each body must not
| |
| − | overlap any other body. The interaction of the bodies was
| |
| − | accounted for by taking the scattered wave of each body to be the
| |
| − | incident wave upon all other bodies (in addition to the ambient
| |
| − | incident wave). Furthermore, since the cylindrical eigenfunction expansions
| |
| − | are local, these were mapped from one body to another using
| |
| − | Graf's addition theorem for Bessel functions.
| |
| − | Doing this for all bodies,
| |
| − | \citeauthor{kagemoto86} were able to solve for the
| |
| − | coefficients of the scattered wavefields of all bodies simultaneously.
| |
| − | The only difficulty with this method was that the
| |
| − | solutions of the single diffraction problems had to be available in
| |
| − | the cylindrical eigenfunction expansion of an outgoing
| |
| − | wave. \citeauthor{kagemoto86} therefore only solved for axisymmetric
| |
| − | bodies because the single diffraction solution for axisymmetric
| |
| − | bodies was
| |
| − | available in the required representation.
| |
| − |
| |
| − | The extension of the Kagemoto and Yue scattering theory to bodies of
| |
| − | arbitrary geometry was performed by [[goo90]] who found a general
| |
| − | way to solve the single diffraction problem in the required
| |
| − | cylindrical eigenfunction representation. They used a
| |
| − | representation of the finite depth free surface Green's function in
| |
| − | the eigenfunction expansion of cylindrical outgoing waves
| |
| − | centred at an arbitrary point of the water surface (above the
| |
| − | body's mean centre position in this case). This Green's function was
| |
| − | presented by
| |
| − | [[black75]] and further investigated by [[fenton78]]
| |
| − | who corrected some statements about the Green's function which Black had
| |
| − | made. This Green's function is
| |
| − | based on the cylindrical
| |
| − | eigenfunction expansion of the finite depth free surface Green's
| |
| − | function given by [[john2]]. The results of
| |
| − | \citeauthor{goo90} were recently used by [[chakrabarti00]] to
| |
| − | solve for arrays of cylinders which can be divided into modules.
| |
| − |
| |
| − | The development of the Kagemoto and Yue interaction theory was
| |
| − | motivated by problems
| |
| − | in off-shore engineering. However, the theory can also be applied
| |
| − | to the geophysical problem of wave scattering by ice floes. At the
| |
| − | interface of the open and frozen ocean an interfacial region known
| |
| − | as the Marginal Ice Zone (MIZ) forms. The MIZ largely controls
| |
| − | the interaction of the open and frozen ocean, especially the interaction
| |
| − | through wave processors. This is because
| |
| − | the MIZ consists of vast fields
| |
| − | of ice floes whose size is comparable to the dominant wavelength, which
| |
| − | means that it strongly scatters incoming waves. A method of
| |
| − | solving for the wave response of a single ice floe of arbitrary
| |
| − | geometry in water of infinite depth was presented by
| |
| − | [[JGR02]]. The ice floe was modelled as a floating, flexible
| |
| − | thin plate and its motion was expanded in the free plate modes of
| |
| − | vibration. Converting the problem for the water into an integral
| |
| − | equation and substituting the free modes, a system of equations for
| |
| − | the coefficients in the modal expansion was obtained. However,
| |
| − | to understand wave propagation and scattering in the MIZ we need to
| |
| − | understand the way in which large numbers of interacting
| |
| − | ice floes scatter waves. For this reason, we require an interaction
| |
| − | theory. While the Kagemoto and Yue interaction theory could be used,
| |
| − | their theory requires that the water depth is finite.
| |
| − | While the water depth in the Marginal Ice Zone varies,
| |
| − | it is generally located far from shore above the deep ocean. This means
| |
| − | that the finite depth must be chosen large in order to be able to
| |
| − | apply their theory. Furthermore, when ocean waves propagate beneath
| |
| − | an ice floe the wavelength is increased so that it becomes more
| |
| − | difficult to make the
| |
| − | water depth sufficiently deep that it may be approximated as infinite. For this
| |
| − | reason, in this paper we develop the equivalent interaction theory to
| |
| − | Kagemoto and Yue's in infinite depth. Also, because of
| |
| − | the complicated geometry of an ice floe, this interaction theory is
| |
| − | for bodies of arbitrary geometry.
| |
| − |
| |
| − | In the first part of this paper Kagemoto and Yue's interaction theory
| |
| − | is extended to water of infinite depth. We represent the incident and
| |
| − | scattered potentials in the cylindrical eigenfunction expansions
| |
| − | and we use an analogous infinite depth Green's function
| |
| − | to the one used by \citeauthor{goo90} \cite[given by][]{malte03}. We
| |
| − | show how the infinite
| |
| − | depth diffraction transfer matrices can be obtained with the use of
| |
| − | this Green's function and we illustrate how the rotation of a body
| |
| − | about its mean centre position in the plane can be accounted for without
| |
| − | recalculating the diffraction transfer matrix.
| |
| − |
| |
| − | In the second part of the paper, using \citeauthor{JGR02}'s single
| |
| − | floe result,
| |
| − | the full diffraction calculation for the motion and scattering from many
| |
| − | interacting ice floes is calculated and presented. For two square
| |
| − | interacting ice floes the convergence of the method obtained from the
| |
| − | developed interaction theory is compared to the result of the full
| |
| − | diffraction calculation. The solutions of more than two interacting
| |
| − | ice floes and of other shapes in different arrangements are presented as well.
| |
| − | We also compare the convergence of the finite depth and infinite
| |
| − | depth methods in
| |
| − | deep water.
| |
| − |
| |
| − | \section{The extension of Kagemoto and Yue's interaction
| |
| − | theory to bodies of arbitrary shape in water of infinite depth}
| |
| − |
| |
| − | [[kagemoto86]] developed an interaction theory for
| |
| − | vertically non-overlapping axisymmetric structures in water of finite
| |
| − | depth. While their theory was valid for bodies of
| |
| − | arbitrary geometry, they did not develop all the necessary
| |
| − | details to apply the theory to arbitrary bodies.
| |
| − | The only requirements to apply this scattering theory is
| |
| − | that the bodies are vertically non-overlapping and
| |
| − | that the smallest cylinder which completely contains each body does not
| |
| − | intersect with any other body.
| |
| − | In this section we will extend their theory to bodies of
| |
| − | arbitrary geometry in water of infinite depth. The extension of
| |
| − | \citeauthor{kagemoto86}'s finite depth interaction theory to bodies of
| |
| − | arbitrary geometry was accomplished by [[goo90]].
| |
| − |
| |
| − |
| |
| − | The interaction theory begins by representing the scattered potential
| |
| − | of each body in the cylindrical eigenfunction expansion. Furthermore,
| |
| − | the incoming potential is also represented in the cylindrical
| |
| − | eigenfunction expansion. The operator which maps the incoming and
| |
| − | outgoing representation is called the diffraction transfer matrix and
| |
| − | is different for each body.
| |
| − | Since these representations are local to each body, a mapping of
| |
| − | the eigenfunction representations between different bodies
| |
| − | is required. This operator is called the coordinate transformation
| |
| − | matrix.
| |
| − |
| |
| − | The cylindrical eigenfunction expansions will be introduced before we
| |
| − | derive a system of
| |
| − | equations for the coefficients of the scattered wavefields. Analogously to
| |
| − | [[kagemoto86]], we represent the scattered wavefield of
| |
| − | each body as an incoming wave upon all other bodies. The addition of
| |
| − | the ambient incident wave yields the complete incident potential and
| |
| − | with the use of diffraction transfer matrices which relate the
| |
| − | coefficients of the incident potential to those of the scattered
| |
| − | wavefield a system of equations for the unknown coefficients of the
| |
| − | scattered wavefields of all bodies is derived.
| |
| − |
| |
| − |
| |
| − | ===Eigenfunction expansion of the potential===
| |
| − | The equations of motion for the water are derived from the linearised
| |
| − | inviscid theory. Under the assumption of irrotational motion the
| |
| − | velocity vector field of the water can be written as the gradient
| |
| − | field of a scalar velocity potential <math>\Phi</math>. Assuming that the motion
| |
| − | is time-harmonic with the radian frequency <math>\omega</math> the
| |
| − | velocity potential can be expressed as the real part of a complex
| |
| − | quantity,
| |
| − | <center><math> (time)
| |
| − | \Phi(\mathbf{y},t) = \Re \{\phi (\mathbf{y}) \e^{-\i \omega t} \}.
| |
| − | </math></center>
| |
| − | To simplify notation, <math>\mathbf{y} = (x,y,z)</math> will always denote a point
| |
| − | in the water, which is assumed infinitely deep, while <math>\mathbf{x}</math> will
| |
| − | always denote a point of the undisturbed water surface assumed at <math>z=0</math>.
| |
| − |
| |
| − | The problem consists of <math>N</math> vertically non-overlapping bodies, denoted
| |
| − | by <math>\Delta_j</math>, which are sufficiently far apart that there is no
| |
| − | intersection of the smallest cylinder which contains each body with
| |
| − | any other body. Each body is subject to an incident wavefield which is
| |
| − | incoming, responds to this wavefield and produces a scattered wave field which
| |
| − | is outgoing. Both the incident and scattered potential corresponding
| |
| − | to these wavefields can be represented in the cylindrical
| |
| − | eigenfunction expansion valid outside of the escribed cylinder of the
| |
| − | body. Let <math>(r_j,\theta_j,z)</math> be the local cylindrical coordinates of
| |
| − | the <math>j</math>th body, <math>\Delta_j</math>, <math>j \in \{1, \ldots , N\}</math>, and </math>\alpha =
| |
| − | \omega^2/g<math> where </math>g<math> is the acceleration due to gravity. Figure
| |
| − | (fig:floe_tri) shows these coordinate systems for two bodies.
| |
| − |
| |
| − | \begin{figure}
| |
| − | \begin{center}
| |
| − | \includegraphics[height=5.5cm]{floe_tri}
| |
| − | \caption{Plan view of the relation between two bodies.} (fig:floe_tri)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | The scattered potential of body <math>\Delta_j</math> can be expanded in
| |
| − | cylindrical eigenfunctions,
| |
| − | <center><math> (basisrep_out)\begin{split}
| |
| − | \phi_j^\mathrm{S} (r_j,\theta_j,z) &= \e^{\alpha z} \sum_{\nu = -
| |
| − | \infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (\alpha r_j) \e^{\i \nu
| |
| − | \theta_j}\\
| |
| − | &\quad + \int\limits_0^{\infty} \big( \cos \eta z + \frac{\alpha}{\eta}
| |
| − | \sin \eta z \big) \sum_{\nu = -
| |
| − | \infty}^{\infty} A_{\nu}^j (\eta) K_\nu (\eta r_j) \e^{\i \nu
| |
| − | \theta_j} \d\eta,
| |
| − | \end{split}</math></center>
| |
| − | where the coefficients <math>A_{0 \nu}^j</math> for the propagating modes are
| |
| − | discrete and the coefficients <math>A_{\nu}^j (\cdot)</math> for the decaying
| |
| − | modes are functions. <math>H_\nu^{(1)}</math> and <math>K_\nu</math> are the Hankel function
| |
| − | of the first kind and the modified Bessel function of the second kind
| |
| − | respectively, both of order <math>\nu</math> as defined in [[abr_ste]].
| |
| − | The incident potential upon body <math>\Delta_j</math> can be expanded in
| |
| − | cylindrical eigenfunctions,
| |
| − | <center><math> (basisrep_in)\begin{split}
| |
| − | \phi_j^\mathrm{I} (r_j,\theta_j,z) &= \e^{\alpha z} \sum_{\mu = -
| |
| − | \infty}^{\infty} D_{0 \mu}^j J_\mu (\alpha r_j) \e^{\i \mu
| |
| − | \theta_j}\\
| |
| − | & \quad + \int\limits_0^{\infty} \big( \cos \eta z + \frac{\alpha}{\eta}
| |
| − | \sin \eta z \big) \sum_{\mu = -
| |
| − | \infty}^{\infty} D_{\mu}^j (\eta) I_\mu (\eta r_j) \e^{\i \mu
| |
| − | \theta_j} \d\eta,
| |
| − | \end{split}</math></center>
| |
| − | where the coefficients <math>D_{0 \mu}^j</math> for the propagating modes are
| |
| − | discrete and the coefficients <math>D_{\mu}^j (\cdot)</math> for the decaying
| |
| − | modes are functions. <math>J_\mu</math> and <math>I_\mu</math> are the Bessel function and
| |
| − | the modified Bessel function respectively, both of the first kind and
| |
| − | order <math>\mu</math>. To simplify the notation, from now on <math>\psi(z,\eta)</math> will
| |
| − | denote the vertical eigenfunctions corresponding to the decaying modes,
| |
| − | <center><math>
| |
| − | \psi(z,\eta) = \cos \eta z + \alpha / \eta \, \sin \eta z.
| |
| − | </math></center>
| |
| − |
| |
| − | ===The interaction in water of infinite depth===
| |
| − | Following the ideas of [[kagemoto86]], a system of equations for the unknown
| |
| − | coefficients and coefficient functions of the scattered wavefields
| |
| − | will be developed. This system of equations is based on transforming the
| |
| − | scattered potential of <math>\Delta_j</math> into an incident potential upon
| |
| − | <math>\Delta_l</math> (<math>j \neq l</math>). Doing this for all bodies simultaneously,
| |
| − | and relating the incident and scattered potential for each body, a system
| |
| − | of equations for the unknown coefficients will be developed.
| |
| − |
| |
| − | The scattered potential <math>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be
| |
| − | represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
| |
| − | upon <math>\Delta_l</math>, <math>j \neq l</math>. From figure
| |
| − | (fig:floe_tri) we can see that this can be accomplished by using
| |
| − | Graf's addition theorem for Bessel functions given in
| |
| − | \citet[eq. 9.1.79]{abr_ste},
| |
| − | \begin{subequations} (transf)
| |
| − | \begin{align} (transf_h)
| |
| − | H_\nu^{(1)}(\alpha r_j) \e^{\i \nu (\theta_j - \vartheta_{jl})} &=
| |
| − | \sum_{\mu = - \infty}^{\infty} H^{(1)}_{\nu + \mu} (\alpha R_{jl}) \,
| |
| − | J_\mu (\alpha r_l) \e^{\i \mu (\pi - \theta_l + \vartheta_{jl})},
| |
| − | \quad j \neq l,\\
| |
| − | (transf_k)
| |
| − | K_\nu(\eta r_j) \e^{\i \nu (\theta_j - \vartheta_{jl})} &= \sum_{\mu = -
| |
| − | \infty}^{\infty} K_{\nu + \mu} (\eta R_{jl}) \, I_\mu (\eta r_l) \e^{\i \mu
| |
| − | (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | which is valid provided that <math>r_l < R_{jl}</math>. This limitation
| |
| − | only requires that the escribed cylinder of each body <math>\Delta_l</math> does
| |
| − | not enclose any other origin <math>O_j</math> (<math>j \neq l</math>). However, the
| |
| − | expansion of the scattered and incident potential in cylindrical
| |
| − | eigenfunctions is only valid outside the escribed cylinder of each
| |
| − | body. Therefore the condition that the
| |
| − | escribed cylinder of each body <math>\Delta_l</math> does not enclose any other
| |
| − | origin <math>O_j</math> (<math>j \neq l</math>) is superseded by the more rigorous
| |
| − | restriction that the escribed cylinder of each body may not contain any
| |
| − | other body. Making use of the equations \eqref{transf}
| |
| − | the scattered potential of <math>\Delta_j</math> can be expressed in terms of the
| |
| − | incident potential upon <math>\Delta_l</math>,
| |
| − | \begin{align*}
| |
| − | \phi_j^{\mathrm{S}} (r_l,\theta_l,z) &=
| |
| − | \e^{\alpha z} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
| |
| − | \sum_{\mu = -\infty}^{\infty} H_{\nu - \mu}^{(1)} (\alpha R_{jl})
| |
| − | J_\mu (\alpha r_l) \e^{\i \mu \theta_l} \e^{\i (\nu-\mu)
| |
| − | \vartheta_{jl}}\\
| |
| − | & \quad + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\nu = -
| |
| − | \infty}^{\infty} A_{\nu}^j (\eta) \sum_{\mu = -\infty}^{\infty}
| |
| − | (-1)^\mu K_{\nu-\mu} (\eta R_{jl}) I_\mu (\eta r_l) \e^{\i \mu
| |
| − | \theta_l} \e^{\i (\nu-\mu) \vartheta_{jl}} \d\eta\\
| |
| − | &= \e^{\alpha z} \sum_{\mu = -\infty}^{\infty} \Big[ \sum_{\nu = -
| |
| − | \infty}^{\infty} A_{0\nu}^j H_{\nu - \mu}^{(1)} (\alpha R_{jl}) \e^{\i
| |
| − | (\nu-\mu) \vartheta_{jl}} \Big] J_\mu (\alpha r_l) \e^{\i \mu \theta_l}\\
| |
| − | & + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\mu = -\infty}^{\infty}
| |
| − | \Big[ \sum_{\nu = - \infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu-\mu}
| |
| − | (\eta R_{jl}) \e^{\i (\nu-\mu)
| |
| − | \vartheta_{jl}} \Big] I_\mu (\eta r_l) \e^{\i \mu \theta_l} \d\eta.
| |
| − | \end{align*}
| |
| − | The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
| |
| − | expanded in the eigenfunctions corresponding to the incident wavefield upon
| |
| − | <math>\Delta_l</math>. A detailed illustration of how to accomplish this will be given
| |
| − | later. Let <math>D_{l0\mu}^{\mathrm{In}}</math> denote the coefficients of this
| |
| − | ambient incident wavefield corresponding to the propagating modes and
| |
| − | <math>D_{l\mu}^{\mathrm{In}} (\cdot)</math> denote the coefficients functions
| |
| − | corresponding to the decaying modes (which are identically zero) of
| |
| − | the incoming eigenfunction expansion for <math>\Delta_l</math>. The total
| |
| − | incident wavefield upon body <math>\Delta_j</math> can now be expressed as
| |
| − | \begin{align*}
| |
| − | &\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
| |
| − | \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \, \phi_j^{\mathrm{S}}
| |
| − | (r_l,\theta_l,z)\\
| |
| − | &= \e^{\alpha z} \sum_{\mu = -\infty}^{\infty} \Big[
| |
| − | D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
| |
| − | \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
| |
| − | H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \e^{\i
| |
| − | (\nu - \mu) \vartheta_{jl}} \Big] J_\mu (\alpha r_l) \e^{\i \mu \theta_l}\\
| |
| − | & + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\mu =
| |
| − | -\infty}^{\infty} \Big[ D_{l\mu}^{\mathrm{In}}(\eta) +
| |
| − | \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu =
| |
| − | -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta
| |
| − | R_{jl}) \e^{\i (\nu - \mu) \vartheta_{jl}} \Big] I_\mu (\eta r_l)
| |
| − | \e^{\i \mu \theta_l} \d\eta.
| |
| − | \end{align*}
| |
| − | The coefficients of the total incident potential upon <math>\Delta_l</math> are
| |
| − | therefore given by
| |
| − | \begin{subequations} (inc_coeff)
| |
| − | \begin{align}
| |
| − | D_{0\mu}^l &= D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
| |
| − | \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
| |
| − | H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \e^{\i
| |
| − | (\nu - \mu) \vartheta_{jl}},\\
| |
| − | D_{\mu}^l(\eta) &= D_{l\mu}^{\mathrm{In}}(\eta) +
| |
| − | \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu =
| |
| − | -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta
| |
| − | R_{jl}) \e^{\i (\nu - \mu) \vartheta_{jl}}.
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − |
| |
| − | In general, it is possible to relate the total incident and scattered
| |
| − | partial waves for any body through the diffraction characteristics of
| |
| − | that body in isolation. There exist diffraction transfer operators
| |
| − | <math>B_l</math> that relate the coefficients of the incident and scattered
| |
| − | partial waves, such that
| |
| − | <center><math> (eq_B)
| |
| − | A_l = B_l (D_l), \quad l=1, \ldots, N,
| |
| − | </math></center>
| |
| − | where <math>A_l</math> are the scattered modes due to the incident modes <math>D_l</math>.
| |
| − | In the case of a countable number of modes, (i.e. when
| |
| − | the depth is finite), <math>B_l</math> is an infinite dimensional matrix. When
| |
| − | the modes are functions of a continuous variable (i.e. infinite
| |
| − | depth), <math>B_l</math> is the kernel of an integral operator.
| |
| − | For the propagating and the decaying modes respectively, the scattered
| |
| − | potential can be related by diffraction transfer operators acting in the
| |
| − | following ways,
| |
| − | \begin{subequations} (diff_op)
| |
| − | \begin{align}
| |
| − | A_{0\nu}^l &= \sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{pp} D_{0\mu}^l
| |
| − | + \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty}
| |
| − | B_{l\nu\mu}^\mathrm{pd} (\xi) D_{\mu}^l (\xi) \d\xi,\\
| |
| − | A_\nu^l (\eta) &= \sum_{\mu = -\infty}^{\infty}
| |
| − | B_{l\nu\mu}^\mathrm{dp} (\eta) D_{0\mu}^l + \int\limits_{0}^{\infty}
| |
| − | \sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{dd} (\eta;\xi)
| |
| − | D_{\mu}^l (\xi) \d\xi.
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | The superscripts <math>\mathrm{p}</math> and <math>\mathrm{d}</math> are used to distinguish
| |
| − | between propagating and decaying modes, the first superscript denotes the kind
| |
| − | of scattered mode, the second one the kind of incident mode.
| |
| − | If the diffraction transfer operators are known (their calculation
| |
| − | will be discussed later), the substitution of
| |
| − | equations \eqref{inc_coeff} into equations \eqref{diff_op} give the
| |
| − | required equations to determine the coefficients and coefficient
| |
| − | functions of the scattered wavefields of all bodies,
| |
| − | \begin{subequations} (eq_op)
| |
| − | <center><math>\begin{matrix}
| |
| − | &\begin{aligned}
| |
| − | &A_{0n}^l = \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pp}
| |
| − | \Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
| |
| − | \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
| |
| − | H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \e^{\i
| |
| − | (\nu - \mu) \vartheta_{jl}} \Big]\\
| |
| − | & \ + \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty}
| |
| − | B_{ln\mu}^\mathrm{pd} (\xi) \Big[D_{l\mu}^{\mathrm{In}}(\eta) +
| |
| − | \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu =
| |
| − | -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta
| |
| − | R_{jl}) \e^{\i (\nu - \mu) \vartheta_{jl}} \Big] \d\xi,
| |
| − | \end{aligned}\\
| |
| − | &\begin{aligned}
| |
| − | &A_n^l (\eta) = \sum_{\mu = -\infty}^{\infty}
| |
| − | B_{ln\mu}^\mathrm{dp} (\eta) \Big[
| |
| − | D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
| |
| − | \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
| |
| − | H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \e^{\i
| |
| − | (\nu - \mu) \vartheta_{jl}}\Big]\\
| |
| − | & \ + \int\limits_{0}^{\infty}
| |
| − | \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{dd} (\eta;\xi)
| |
| − | \Big[ D_{l\mu}^{\mathrm{In}}(\eta) +
| |
| − | \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu =
| |
| − | -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta
| |
| − | R_{jl}) \e^{\i (\nu - \mu) \vartheta_{jl}}\Big] \d\xi,
| |
| − | \end{aligned}
| |
| − | \end{matrix}</math></center>
| |
| − | \end{subequations}
| |
| − | <math>n \in \mathds{Z},\, l = 1, \ldots, N</math>. It has to be noted that all
| |
| − | equations are coupled so that it is necessary to solve for all
| |
| − | scattered coefficients and coefficient functions simultaneously.
| |
| − |
| |
| − | For numerical calculations, the infinite sums have to be truncated and
| |
| − | the integrals must be discretised. Implying a suitable truncation, the
| |
| − | four different diffraction transfer operators can be represented by
| |
| − | matrices which can be assembled in a big matrix <math>\mathbf{B}_l</math>,
| |
| − | <center><math>
| |
| − | \mathbf{B}_l = \left[
| |
| − | \begin{matrix}{cc} \mathbf{B}_l^{\mathrm{pp}} & \mathbf{B}_l^{\mathrm{pd}}\\
| |
| − | \mathbf{B}_l^{\mathrm{dp}} & \mathbf{B}_l^{\mathrm{dd}}
| |
| − | \end{matrix} \right],
| |
| − | </math></center>
| |
| − | the infinite depth diffraction transfer matrix.
| |
| − | Truncating the coefficients accordingly, defining <math>{\bf a}^l</math> to be the
| |
| − | vector of the coefficients of the scattered potential of body
| |
| − | <math>\Delta_l</math>, <math>\mathbf{d}_l^{\mathrm{In}}</math> to be the vector of
| |
| − | coefficients of the ambient wavefield, and making use of a coordinate
| |
| − | transformation matrix <math>{\bf T}_{jl}</math> given by
| |
| − | \begin{subequations} (T_elem_deep)
| |
| − | \begin{align}
| |
| − | ({\bf T}_{jl})_{pq} &= H_{p-q}^{(1)}(\alpha R_{jl}) \, \e^{\i (p-q)
| |
| − | \vartheta_{jl}}\\
| |
| − | \intertext{for the propagating modes, and}
| |
| − | ({\bf T}_{jl})_{pq} &= (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \e^{\i
| |
| − | (p-q) \vartheta_{jl}}
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | for the decaying modes, a linear system of equations
| |
| − | for the unknown coefficients follows from equations \eqref{eq_op},
| |
| − | <center><math> (eq_B_inf)
| |
| − | {\bf a}_l = {\bf \hat{B}}_l \Big( {\bf d}_l^{\mathrm{In}} +
| |
| − | \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \trans {\bf T}_{jl} \,
| |
| − | {\bf a}_j \Big), \quad l=1, \ldots, N,
| |
| − | </math></center>
| |
| − | where the left superscript <math>\mathrm{t}</math> indicates transposition.
| |
| − | The matrix <math>{\bf \hat{B}}_l</math> denotes the infinite depth diffraction
| |
| − | transfer matrix <math>{\bf B}_l</math> in which the elements associated with
| |
| − | decaying scattered modes have been multiplied with the appropriate
| |
| − | integration weights depending on the discretisation of the continuous variable.
| |
| − |
| |
| − |
| |
| − |
| |
| − | \subsection{Calculation of the diffraction transfer matrix for bodies
| |
| − | of arbitrary geometry}
| |
| − |
| |
| − | Before we can apply the interaction theory we require the diffraction
| |
| − | transfer matrices <math>\mathbf{B}_j</math> which relate the incident and the
| |
| − | scattered potential for a body <math>\Delta_j</math> in isolation.
| |
| − | The elements of the diffraction transfer matrix, <math>({\bf B}_j)_{pq}</math>,
| |
| − | are the coefficients of the
| |
| − | <math>p</math>th partial wave of the scattered potential due to a single
| |
| − | unit-amplitude incident wave of mode <math>q</math> upon <math>\Delta_j</math>.
| |
| − |
| |
| − | While \citeauthor{kagemoto86}'s interaction theory was valid for
| |
| − | bodies of arbitrary shape, they did not explain how to actually obtain the
| |
| − | diffraction transfer matrices for bodies which did not have an axisymmetric
| |
| − | geometry. This step was performed by [[goo90]] who came up with an
| |
| − | explicit method to calculate the diffraction transfer matrices for bodies of
| |
| − | arbitrary geometry in the case of finite depth. Utilising a Green's
| |
| − | function they used the standard
| |
| − | method of transforming the single diffraction boundary-value problem
| |
| − | to an integral equation for the source strength distribution function
| |
| − | over the immersed surface of the body.
| |
| − | However, the representation of the scattered potential which
| |
| − | is obtained using this method is not automatically given in the
| |
| − | cylindrical eigenfunction
| |
| − | expansion. To obtain such cylindrical eigenfunction expansions of the
| |
| − | potential [[goo90]] used the representation of the free surface
| |
| − | finite depth Green's function given by [[black75]] and
| |
| − | [[fenton78]]. \citeauthor{black75} and
| |
| − | \citeauthor{fenton78}'s representation of the Green's function was based
| |
| − | on applying Graf's addition theorem to the eigenfunction
| |
| − | representation of the free surface finite depth Green's function given
| |
| − | by [[john2]]. Their representation allowed the scattered potential to be
| |
| − | represented in the eigenfunction expansion with the cylindrical
| |
| − | coordinate system fixed at the point of the water surface above the
| |
| − | mean centre position of the body.
| |
| − |
| |
| − | It should be noted that, instead of using the source strength distribution
| |
| − | function, it is also possible to consider an integral equation for the
| |
| − | total potential and calculate the elements of the diffraction transfer
| |
| − | matrix from the solution of this integral equation.
| |
| − | An outline of this method for water of finite
| |
| − | depth is given by [[kashiwagi00]]. We will present
| |
| − | here a derivation of the diffraction transfer matrices for the case
| |
| − | infinite depth based on a solution
| |
| − | for the source strength distribution function. However,
| |
| − | an equivalent derivation would be possible based on the solution
| |
| − | for the total velocity potential.
| |
| − |
| |
| − | To calculate the diffraction transfer matrix in infinite depth, we
| |
| − | require the representation of the infinite depth free surface Green's
| |
| − | function in cylindrical eigenfunctions,
| |
| − | <center><math>\begin{split} (green_inf)
| |
| − | G(r,\theta,z;s,\varphi,c) &= \frac{\i \alpha}{2} \, \e^{\alpha (z+c)}
| |
| − | \sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(\alpha r) J_\nu(\alpha s) \e^{\i \nu
| |
| − | (\theta - \varphi)}\\ &\quad + \frac{1}{\pi^2} \int\limits_0^{\infty}
| |
| − | \psi(z,\eta) \frac{\eta^2}{\eta^2+\alpha^2} \psi(c,\eta)
| |
| − | \sum_{\nu=-\infty}^{\infty} K_\nu(\eta r) I_\nu(\eta s) \e^{\i \nu
| |
| − | (\theta - \varphi)} \d\eta,
| |
| − | \end{split}</math></center>
| |
| − | <math>r > s</math>, given by [[malte03]].
| |
| − |
| |
| − | We assume that we have represented the scattered potential in terms of
| |
| − | the source strength distribution <math>\varsigma^j</math> so that the scattered
| |
| − | potential can be written as
| |
| − | <center><math> (int_eq_1)
| |
| − | \phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G
| |
| − | (\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta})
| |
| − | \d\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D,
| |
| − | </math></center>
| |
| − | where <math>D</math> is the volume occupied by the water and <math>\Gamma_j</math> is the
| |
| − | immersed surface of body <math>\Delta_j</math>. The source strength distribution
| |
| − | function <math>\varsigma^j</math> can be found by solving an
| |
| − | integral equation. The integral equation is described in
| |
| − | [[Weh_Lait]] and numerical methods for its solution are outlined in
| |
| − | [[Sarp_Isa]].
| |
| − | Substituting the eigenfunction expansion of the Green's function
| |
| − | \eqref{green_inf} into \eqref{int_eq_1}, the scattered potential can
| |
| − | be written as
| |
| − | \begin{align*}
| |
| − | &\phi_j^\mathrm{S}(r_j,\theta_j,z) = \e^{\alpha z} \sum_{\nu = -
| |
| − | \infty}^{\infty} \bigg[ \frac{\i \alpha}{2}
| |
| − | \int\limits_{\Gamma_j} \e^{\alpha c} J_\nu(\alpha s) \e^{-\i \nu
| |
| − | \varphi} \varsigma^j(\mathbf{\zeta})
| |
| − | \d\sigma_\mathbf{\zeta} \bigg] H_\nu^{(1)} (\alpha r_j) \e^{\i \nu \theta_j}\\
| |
| − | & \, + \int\limits_{0}^{\infty} \psi(z,\eta) \sum_{\nu = -
| |
| − | \infty}^{\infty} \bigg[ \frac{1}{\pi^2} \frac{\eta^2
| |
| − | }{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_\nu(\eta s)
| |
| − | \e^{-\i \nu \varphi} \varsigma^j({\bf{\zeta}})
| |
| − | \d\sigma_{\bf{\zeta}} \bigg] K_\nu (\eta r_j) \e^{\i \nu \theta_j} \d\eta,
| |
| − | \end{align*}
| |
| − | where
| |
| − | <math>\mathbf{\zeta}=(s,\varphi,c)</math> and <math>r>s</math>.
| |
| − | This restriction implies that the eigenfunction expansion is only valid
| |
| − | outside the escribed cylinder of the body.
| |
| − |
| |
| − | The columns of the diffraction transfer matrix are the coefficients of
| |
| − | the eigenfunction expansion of the scattered wavefield due to the
| |
| − | different incident modes of unit-amplitude. The elements of the
| |
| − | diffraction transfer matrix of a body of arbitrary shape are therefore given by
| |
| − | \begin{subequations} (B_elem)
| |
| − | \begin{align}
| |
| − | ({\bf B}_j)_{pq} &= \frac{\i \alpha}{2} \int\limits_{\Gamma_j}
| |
| − | \e^{\alpha c} J_p(\alpha s) \e^{-\i p \varphi} \varsigma_q^j(\mathbf{\zeta})
| |
| − | \d\sigma_\mathbf{\zeta}\\
| |
| − | \intertext{and}
| |
| − | ({\bf B}_j)_{pq} &= \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 +
| |
| − | \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \e^{-\i p
| |
| − | \varphi} \varsigma_q^j(\mathbf{\zeta}) \d\sigma_\mathbf{\zeta}
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | for the propagating and the decaying modes respectively, where
| |
| − | <math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution
| |
| − | due to an incident potential of mode <math>q</math> of the form
| |
| − | \begin{subequations} (test_modes_inf)
| |
| − | \begin{align}
| |
| − | \phi_q^{\mathrm{I}}(s,\varphi,c) &= \e^{\alpha c} H_q^{(1)} (\alpha
| |
| − | s) \e^{\i q \varphi}\\
| |
| − | \intertext{for the propagating modes, and}
| |
| − | \phi_q^{\mathrm{I}}(s,\varphi,c) &= \psi(c,\eta) K_q (\eta s) \e^{\i q \varphi}
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | for the decaying modes.
| |
| − |
| |
| − | ===The diffraction transfer matrix of rotated bodies===
| |
| − |
| |
| − | For a non-axisymmetric body, a rotation about the mean
| |
| − | centre position in the <math>(x,y)</math>-plane will result in a
| |
| − | different diffraction transfer matrix. We will show how the
| |
| − | diffraction transfer matrix of a body rotated by an angle <math>\beta</math> can
| |
| − | be easily calculated from the diffraction transfer matrix of the
| |
| − | non-rotated body. The rotation of the body influences the form of the
| |
| − | elements of the diffraction transfer matrices in two ways. Firstly, the
| |
| − | angular dependence in the integral over the immersed surface of the
| |
| − | body is altered and, secondly, the source strength distribution
| |
| − | function is different if the body is rotated. However, the source
| |
| − | strength distribution function of the rotated body can be obtained by
| |
| − | calculating the response of the non-rotated body due to rotated
| |
| − | incident potentials. It will be shown that the additional angular
| |
| − | dependence can be easily factored out of the elements of the
| |
| − | diffraction transfer matrix.
| |
| − |
| |
| − | The additional angular dependence caused by the rotation of the
| |
| − | incident potential can be factored out of the normal derivative of the
| |
| − | incident potential such that
| |
| − | <center><math>
| |
| − | \frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} =
| |
| − | \frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n}
| |
| − | \e^{\i q \beta},
| |
| − | </math></center>
| |
| − | where <math>\phi_{q\beta}^{\mathrm{I}}</math> is the rotated incident potential.
| |
| − | Since the integral equation for the determination of the source
| |
| − | strength distribution function is linear, the source strength
| |
| − | distribution function due to the rotated incident potential is thus just
| |
| − | given by
| |
| − | <center><math>
| |
| − | \varsigma_{q\beta}^j = \varsigma_q^j \, \e^{\i q \beta}.
| |
| − | </math></center>
| |
| − | This is also the source strength distribution function of the rotated
| |
| − | body due to the standard incident modes.
| |
| − |
| |
| − | The elements of the diffraction transfer matrix <math>\mathbf{B}_j</math> are
| |
| − | given by equations \eqref{B_elem}. Keeping in mind that the body is
| |
| − | rotated by the angle <math>\beta</math>, the elements of the diffraction transfer
| |
| − | matrix of the rotated body are given by
| |
| − | \begin{subequations} (B_elem_rot)
| |
| − | \begin{align}
| |
| − | (\mathbf{B}_j^\beta)_{pq} &= \frac{\i \alpha}{2} \int\limits_{\Gamma_j}
| |
| − | \e^{\alpha c} J_p(\alpha s) \e^{-\i p (\varphi+\beta)}
| |
| − | \varsigma_{q\beta}^j(\mathbf{\zeta}) \d\sigma_\mathbf{\zeta},\\
| |
| − | \intertext{and}
| |
| − | (\mathbf{B}_j^\beta)_{pq} &= \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 +
| |
| − | \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \e^{-\i p
| |
| − | (\varphi+\beta)} \varsigma_{q\beta}^j(\mathbf{\zeta}) \d\sigma_\mathbf{\zeta},
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | for the propagating and decaying modes respectively.
| |
| − |
| |
| − | Thus the additional angular dependence caused by the rotation of
| |
| − | the body can be factored out of the elements of the diffraction
| |
| − | transfer matrix. The elements of the diffraction transfer matrix
| |
| − | corresponding to the body rotated by the angle <math>\beta</math>,
| |
| − | <math>\mathbf{B}_j^\beta</math>, are given by
| |
| − | <center><math> (B_rot)
| |
| − | (\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \e^{\i (q-p) \beta}.
| |
| − | </math></center>
| |
| − | As before, <math>(\mathbf{B})_{pq}</math> is understood to be the element of
| |
| − | <math>\mathbf{B}</math> which corresponds to the coefficient of the <math>p</math>th scattered
| |
| − | mode due to a unit-amplitude incident wave of mode <math>q</math>. Equation
| |
| − | \eqref{B_rot} applies to propagating and decaying modes likewise.
| |
| − |
| |
| − |
| |
| − | \subsection{Representation of the ambient wavefield in the eigenfunction
| |
| − | representation}
| |
| − | In Cartesian coordinates centred at the origin, the ambient wavefield is
| |
| − | given by
| |
| − | <center><math>
| |
| − | \phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \, \e^{\i \alpha (x
| |
| − | \cos \chi + y \sin \chi)+ \alpha z},
| |
| − | </math></center>
| |
| − | where <math>A</math> is the amplitude (in displacement) and <math>\chi</math> is the
| |
| − | angle between the <math>x</math>-axis and the direction in which the wavefield travels.
| |
| − | The interaction theory requires that the ambient wavefield, which is
| |
| − | incident upon
| |
| − | all bodies, is represented in the eigenfunction expansion of an
| |
| − | incoming wave in the local coordinates of the body. The ambient wave
| |
| − | can be represented in an eigenfunction expansion centred at the origin
| |
| − | as
| |
| − | <center><math>
| |
| − | \phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \e^{\alpha z}
| |
| − | \sum_{\mu = -\infty}^{\infty} \e^{\i \mu (\pi/2 - \theta + \chi)}
| |
| − | J_\mu(\alpha r)
| |
| − | </math></center>
| |
| − | \cite[p. 169]{linton01}.
| |
| − | Since the local coordinates of the bodies are centred at their mean
| |
| − | centre positions <math>O_l = (O_x^l,O_y^l)</math>, a phase factor has to be defined
| |
| − | which accounts for the position from the origin. Including this phase
| |
| − | factor the ambient wavefield at the <math>l</math>th body is given
| |
| − | by
| |
| − | <center><math>
| |
| − | \phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\i \alpha (O_x^l
| |
| − | \cos \chi + O_x^l \sin \chi)} \, \e^{\alpha z}
| |
| − | \sum_{\mu = -\infty}^{\infty} \e^{\i \mu (\pi/2 - \chi)}
| |
| − | J_\mu(\alpha r_l) \e^{\i \mu \theta_l}.
| |
| − | </math></center>
| |
| − |
| |
| − | ===Solving the resulting system of equations===
| |
| − | After the coefficient vector of the ambient incident wavefield, the
| |
| − | diffraction transfer matrices and the coordinate
| |
| − | transformation matrices have been calculated, the system of
| |
| − | equations \eqref{eq_B_inf},
| |
| − | has to be solved. This system can be represented by the following
| |
| − | matrix equation,
| |
| − | <center><math>
| |
| − | \left[ \begin{matrix}{c}
| |
| − | {\bf a}_1\\ {\bf a}_2\\ \\ \vdots \\ \\ {\bf a}_N
| |
| − | \end{matrix} \right]
| |
| − | = \left[ \begin{matrix}{c}
| |
| − | \hat{{\bf B}}_1 {\bf d}_1^\mathrm{In}\\ \hat{{\bf B}}_2 {\bf
| |
| − | d}_2^\mathrm{In}\\ \\ \vdots \\ \\ \hat{{\bf B}}_N {\bf d}_N^\mathrm{In}
| |
| − | \end{matrix} \right]+
| |
| − | \left[ \begin{matrix}{ccccc}
| |
| − | \mathbf{0} & \hat{{\bf B}}_1 \trans {\bf T}_{21} & \hat{{\bf B}}_1
| |
| − | \trans {\bf T}_{31} & \dots & \hat{{\bf B}}_1 \trans {\bf T}_{N1}\\
| |
| − | \hat{{\bf B}}_2 \trans {\bf T}_{12} & \mathbf{0} & \hat{{\bf B}}_2
| |
| − | \trans {\bf T}_{32} & \dots & \hat{{\bf B}}_2 \trans {\bf T}_{N2}\\
| |
| − | & & \mathbf{0} & &\\
| |
| − | \vdots & & & \ddots & \vdots\\
| |
| − | & & & & \\
| |
| − | \hat{{\bf B}}_N \trans {\bf T}_{1N} & & \dots &
| |
| − | & \mathbf{0}
| |
| − | \end{matrix} \right]
| |
| − | \left[ \begin{matrix}{c}
| |
| − | {\bf a}_1\\ {\bf a}_2\\ \\ \vdots \\ \\ {\bf a}_N
| |
| − | \end{matrix} \right],
| |
| − | </math></center>
| |
| − | where <math>\mathbf{0}</math> denotes the zero-matrix which is of the same
| |
| − | dimension as <math>\hat{{\bf B}}_j</math>, say <math>n</math>. This matrix equation can be
| |
| − | easily transformed into a classical <math>(N \, n)</math>-dimensional linear system of
| |
| − | equations.
| |
| − |
| |
| − |
| |
| − | ==Finite Depth Interaction Theory==
| |
| − |
| |
| − | We will compare the performance of the infinite depth interaction theory
| |
| − | with the equivalent theory for finite
| |
| − | depth. As we have stated previously, the finite depth theory was
| |
| − | developed by [[kagemoto86]] and extended to bodies of arbitrary
| |
| − | geometry by [[goo90]]. We will briefly present this theory in
| |
| − | our notation and the comparisons will be made in a later section.
| |
| − |
| |
| − | In water of constant finite depth <math>d</math>, the scattered potential of a body
| |
| − | <math>\Delta_j</math> can be expanded in cylindrical eigenfunctions,
| |
| − | <center><math> (basisrep_out_d)\begin{split}
| |
| − | \phi_j^\mathrm{S} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd}
| |
| − | \sum_{\nu = - \infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (k r_j) \e^{\i \nu
| |
| − | \theta_j}\\
| |
| − | &\quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\nu = -
| |
| − | \infty}^{\infty} A_{m \nu}^j K_\nu (k_m r_j) \e^{\i \nu
| |
| − | \theta_j},
| |
| − | \end{split}</math></center>
| |
| − | with discrete coefficients <math>A_{m \nu}^j</math>. The positive wavenumber <math>k</math>
| |
| − | is related to <math>\alpha</math> by the dispersion relation
| |
| − | <center><math> (eq_k)
| |
| − | \alpha = k \tanh k d,
| |
| − | </math></center>
| |
| − | and the values of <math>k_m</math>, <math>m>0</math>, are given as positive real roots of
| |
| − | the dispersion relation
| |
| − | <center><math> (eq_k_m)
| |
| − | \alpha + k_m \tan k_m d = 0.
| |
| − | </math></center>
| |
| − | The incident potential upon body <math>\Delta_j</math> can be also be expanded in
| |
| − | cylindrical eigenfunctions,
| |
| − | <center><math> (basisrep_in_d)\begin{split}
| |
| − | \phi_j^\mathrm{I} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd}
| |
| − | \sum_{\mu = - \infty}^{\infty} D_{0 \mu}^j J_\mu (k r_j) \e^{\i \mu
| |
| − | \theta_j}\\
| |
| − | & \quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\mu = -
| |
| − | \infty}^{\infty} D_{m\mu}^j I_\mu (k_m r_j) \e^{\i \mu
| |
| − | \theta_j},
| |
| − | \end{split}</math></center>
| |
| − | with discrete coefficients <math>D_{m\mu}^j</math>. A system of equations for the
| |
| − | coefficients of the scattered wavefields for the bodies are derived
| |
| − | in an analogous way to the infinite depth case. The derivation is
| |
| − | simpler because all the coefficients are discrete and the
| |
| − | diffraction transfer operator can be represented by an
| |
| − | infinite dimensional matrix.
| |
| − | Truncating the infinite dimensional matrix as well as the
| |
| − | coefficient vectors appropriately, the resulting system of
| |
| − | equations is given by
| |
| − | <center><math>
| |
| − | {\bf a}_l = {\bf B}_l \Big( {\bf d}_l^\mathrm{In} +
| |
| − | \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \trans {\bf T}_{jl} \,
| |
| − | {\bf a}_j \Big), \quad l=1, \ldots, N,
| |
| − | </math></center>
| |
| − | where <math>{\bf a}_l</math> is the coefficient vector of the scattered
| |
| − | wave, <math>{\bf d}_l^\mathrm{In}</math> is the coefficient vector of the
| |
| − | ambient incident wave, <math>{\bf B}_l</math> is the diffraction transfer
| |
| − | matrix of <math>\Delta_l</math> and <math>{\bf T}_{jl}</math> is the coordinate transformation
| |
| − | matrix analogous to \eqref{T_elem_deep}.
| |
| − |
| |
| − | The calculation of the diffraction transfer matrices is
| |
| − | also similar to the infinite depth case. The finite depth
| |
| − | Green's function
| |
| − | <center><math> (green_d)\begin{split}
| |
| − | &G(r,\theta,z;s,\varphi,c)\\ &= \frac{\i}{2} \,
| |
| − | \frac{\alpha^2-k^2}{d(\alpha^2-k^2)-\alpha}\, \cosh k(z+d) \cosh k(c+d)
| |
| − | \sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(k r) J_\nu(k s) \e^{\i \nu
| |
| − | (\theta - \varphi)}\\
| |
| − | & \quad + \frac{1}{\pi} \sum_{m=1}^{\infty}
| |
| − | \frac{k_m^2+\alpha^2}{d(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+d) \cos
| |
| − | k_m(c+d) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \e^{\i \nu
| |
| − | (\theta - \varphi)},
| |
| − | \end{split}</math></center>
| |
| − | given by [[black75]] and [[fenton78]], needs to be used instead
| |
| − | of the infinite depth Green's function \eqref{green_inf}.
| |
| − | The elements of <math>{\bf B}_j</math> are therefore given by
| |
| − | \begin{subequations} (B_elem_d)
| |
| − | \begin{align}
| |
| − | ({\bf B}_j)_{pq} &= \frac{\i}{2} \,
| |
| − | \frac{(\alpha^2-k^2)\cosh^2 kd}{d(\alpha^2-k^2)-\alpha} \int\limits_{\Gamma_j}
| |
| − | \cosh k(c+d) J_p(\alpha s) \e^{-\i p \varphi} \varsigma_q^j(\mathbf{\zeta})
| |
| − | \d\sigma_\mathbf{\zeta}\\
| |
| − | \intertext{and}
| |
| − | ({\bf B}_j)_{pq} &= \frac{1}{\pi}
| |
| − | \frac{(k_m^2+\alpha^2)\cos^2 k_md}{d(k_m^2+\alpha^2)-\alpha}
| |
| − | \int\limits_{\Gamma_j} \cos k_m(c+d) I_p(\eta s) \e^{-\i p
| |
| − | \varphi} \varsigma_q^j(\mathbf{\zeta}) \d\sigma_\mathbf{\zeta}
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | for the propagating and the decaying modes respectively, where
| |
| − | <math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution
| |
| − | due to an incident potential of mode <math>q</math> of the form
| |
| − | \begin{subequations} (test_modes_d)
| |
| − | \begin{align}
| |
| − | \phi_q^{\mathrm{I}}(s,\varphi,c) &= \frac{\cosh k_m(c+d)}{\cosh kd}
| |
| − | H_q^{(1)} (k s) \e^{\i q \varphi}\\
| |
| − | \intertext{for the propagating modes, and}
| |
| − | \phi_q^{\mathrm{I}}(s,\varphi,c) &= \frac{\cos k_m(c+d)}{\cos kd} K_q
| |
| − | (k_m s) \e^{\i q \varphi}
| |
| − | \end{align}
| |
| − | \end{subequations}
| |
| − | for the decaying modes.
| |
| − |
| |
| − |
| |
| − |
| |
| − |
| |
| − | ==Wave forcing of an ice floe of arbitrary geometry==
| |
| − |
| |
| − | The interaction theory which has been developed so far has been
| |
| − | for arbitrary bodies. No assumption has been made about the body
| |
| − | geometry or its equations of motion. However, we will now use this
| |
| − | interaction theory to make calculations for the specific case of ice
| |
| − | floes. Ice floes form in vast fields consisting of hundreds if not
| |
| − | thousands of individual floes and furthermore most ice floe fields
| |
| − | occur in the deep ocean. For this reason they are ideally suited to
| |
| − | the application of the scattering theory we have just developed.
| |
| − | Furthermore, the presence of the ice lengthens the wavelength
| |
| − | making it more difficult to determine how deep the water must
| |
| − | be to be approximately infinite.
| |
| − |
| |
| − | ===Mathematical model for ice floes===
| |
| − | We will briefly describe the mathematical model which is used to
| |
| − | describe ice floes. A more detailed account can be found in
| |
| − | [[Squire_review]]. We assume that the ice floe is sufficiently thin
| |
| − | that we may apply the shallow draft approximation, which essentially
| |
| − | applies the boundary conditions underneath the floe at the water
| |
| − | surface. The ice floe is modelled as a thin plate rather than a rigid
| |
| − | body since the floe flexure is significant owing to the ice floe
| |
| − | geometry. This model has been applied to a single ice floe by
| |
| − | [[JGR02]]. Assuming the ice floe is in contact with the water
| |
| − | surface at all times, its displacement
| |
| − | <math>W</math> is that of the water surface and <math>W</math> is required to satisfy the linear
| |
| − | plate equation in the area occupied by the ice floe <math>\Delta</math>. In analogy to
| |
| − | \eqref{time}, <math>w</math> denotes the time-independent surface displacement
| |
| − | (with the same radian frequency as the water velocity potential due to
| |
| − | linearity) and the plate equation becomes
| |
| − | <center><math> (plate_non)
| |
| − | D \, \nabla^4 w - \omega^2 \, \rho_\Delta \, h \, w = \i \, \omega \, \rho
| |
| − | \, \phi - \rho \, g \, w, \quad {\bf{x}} \in \Delta,
| |
| − | </math></center>
| |
| − | with the density of the water <math>\rho</math>, the modulus of rigidity of the
| |
| − | ice floe <math>D</math>, its density <math>\rho_\Delta</math> and its
| |
| − | thickness <math>h</math>. The right-hand-side of \eqref{plate_non} arises from the
| |
| − | linearised Bernoulli equation. It needs to be recalled that
| |
| − | <math>\mathbf{x}</math> always denotes a point of the undisturbed water surface.
| |
| − | Free edge boundary conditions apply, namely
| |
| − | <center><math>
| |
| − | \frac{\partial^2 w}{\partial n^2} + \nu \frac{\partial^2 w}{\partial
| |
| − | s^2} = 0 \quad =and= \quad \frac{\partial^3 w}{\partial n^3} + (2 - \nu)
| |
| − | \frac{\partial^3 w}{\partial n \partial s^2} = 0, \quad
| |
| − | \mathbf{x} \in \partial \Delta,
| |
| − | </math></center>
| |
| − | where <math>n</math> and <math>s</math> denote the normal and tangential directions on
| |
| − | <math>\partial \Delta</math> (where they exist) respectively and <math>\nu</math> is
| |
| − | Poisson's ratio.
| |
| − |
| |
| − | Non-dimensional variables (denoted with an overbar) are introduced,
| |
| − | <center><math>
| |
| − | (\bar{x},\bar{y},\bar{z}) = \frac{1}{a} (x,y,z), \quad \bar{w} =
| |
| − | \frac{w}{a}, \quad \bar{\alpha} = a\, \alpha, \quad \bar{\omega} = \omega
| |
| − | \sqrt{\frac{a}{g}} \quad =and= \quad \bar{\phi} = \frac{\phi}{a
| |
| − | \sqrt{a g}},
| |
| − | </math></center>
| |
| − | where <math>a</math> is a length parameter associated with the floe.
| |
| − | In non-dimensional variables, the equation for the ice floe
| |
| − | \eqref{plate_non} reduces to
| |
| − | <center><math> (plate_final)
| |
| − | \beta \nabla^4 \bar{w} - \bar{\alpha} \gamma \bar{w} = \i
| |
| − | \sqrt{\bar{\alpha}} \bar{\phi} - \bar{w}, \quad
| |
| − | \bar{\mathbf{x}} \in \bar{\Delta}
| |
| − | </math></center>
| |
| − | with
| |
| − | <center><math>
| |
| − | \beta = \frac{D}{g \rho a^4} \quad =and= \quad \gamma =
| |
| − | \frac{\rho_\Delta h}{ \rho a}.
| |
| − | </math></center>
| |
| − | The constants <math>\beta</math> and <math>\gamma</math> represent the stiffness and the
| |
| − | mass of the plate respectively. For convenience, the overbars will be
| |
| − | dropped and non-dimensional variables will be assumed in the sequel.
| |
| − |
| |
| − | The standard boundary-value problem applies to the water.
| |
| − | The water velocity potential must satisfy the boundary value problem
| |
| − | \begin{subequations} (water)
| |
| − | \begin{align}
| |
| − | \nabla^2 \phi &= 0, \; & & \mathbf{y} \in D,\\
| |
| − | (water_freesurf)
| |
| − | \frac{\partial \phi}{\partial z} &= \alpha \phi, \; & &
| |
| − | {\bf{x}} \not\in \Delta,\\
| |
| − | (water_depth)
| |
| − | \sup_{\mathbf{y} \in D} \abs{\phi} &< \infty.
| |
| − | \intertext{The linearised kinematic boundary condition is applied under
| |
| − | the ice floe,}
| |
| − | (water_body)
| |
| − | \frac{\partial \phi}{\partial z} &= - \i \sqrt{\alpha} w, \; && {\bf{x}}
| |
| − | \in \Delta,
| |
| − | \end{align}
| |
| − | and the Sommerfeld radiation condition
| |
| − | <center><math>
| |
| − | \lim_{\tilde{r} \rightarrow \infty} \sqrt{\tilde{r}} \, \Big(
| |
| − | \frac{\partial}{\partial \tilde{r}} - \i k
| |
| − | \Big) (\phi - \phi^{\mathrm{In}}) = 0,
| |
| − | </math></center>
| |
| − | \end{subequations}
| |
| − | where <math>\tilde{r}^2=x^2+y^2</math> and <math>k</math> is the wavenumber is imposed.
| |
| − |
| |
| − | Since the numerical convergence will be compared to the finite depth
| |
| − | theory later, a formulation for the finite depth problem will be
| |
| − | required. However, the differences to the infinite depth
| |
| − | formulation are few. For water of constant finite depth <math>d</math>, the volume
| |
| − | occupied by the water changes, the vertical dimension being reduced to
| |
| − | <math>(-d,0)</math>, (still denoted by <math>D</math>),
| |
| − | and the depth condition \eqref{water_depth} is replaced by the bed
| |
| − | condition,
| |
| − | <center><math>
| |
| − | \frac{\partial \phi}{\partial z} = 0, \quad \mathbf{y} \in D,\: z=-d.
| |
| − | </math></center>
| |
| − | In water of finite depth, the positive real wavenumber <math>k</math> is related
| |
| − | to the radian frequency by the dispersion relations \eqref{eq_k}.
| |
| − |
| |
| − |
| |
| − | ===The wavelength under the ice floe=== (sec:kappa)
| |
| − | For the case of a floating thin plate of shallow draft, which we have
| |
| − | used here to model ice floes, waves can propagate under the plate.
| |
| − | These
| |
| − | waves can be understood by considering an infinite sheet of ice
| |
| − | and they satisfy a complex dispersion relation given by
| |
| − | [[FoxandSquire]]. In non-dimensional form it states
| |
| − | <center><math>
| |
| − | \kappa^* \tan \kappa^* d = - \frac{\alpha}{\beta \kappa^{*4} - \gamma
| |
| − | \alpha +1},
| |
| − | </math></center>
| |
| − | where <math>\kappa^*</math> is the wavenumber under the plate. The purely imaginary
| |
| − | roots of this dispersion relation correspond to the propagating modes
| |
| − | and their absolute value is given as the positive root of
| |
| − | <center><math>
| |
| − | \kappa \tanh \kappa d = \frac{\alpha}{\beta \kappa^4 - \gamma
| |
| − | \alpha + 1}.
| |
| − | </math></center>
| |
| − | For realistic values of the parameters, the effect of the plate is to
| |
| − | make <math>\kappa</math> smaller than <math>k</math> (the open water wavenumber), which
| |
| − | increases the wavelength. The effect of the increased wavelength is to
| |
| − | increase the depth at which the water may be approximated as
| |
| − | infinite.
| |
| − |
| |
| − | ===Transformation into an integral equation===
| |
| − | The problem for the water is converted to an integral equation in the
| |
| − | following way. Let <math>G</math> be the three-dimensional free surface
| |
| − | Green's function for water of infinite depth.
| |
| − | The Green's function allows the representation of the scattered water
| |
| − | velocity potential in the standard way,
| |
| − | <center><math> (int_eq)
| |
| − | \phi^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma}
| |
| − | \left( \phi^\mathrm{S} (\mathbf{\zeta}) \, \frac{\partial G}{\partial
| |
| − | n_\mathbf{\zeta}} (\mathbf{y};\mathbf{\zeta}) - G
| |
| − | (\mathbf{y};\mathbf{\zeta}) \, \frac{\partial
| |
| − | \phi^\mathrm{S}}{\partial n_\mathbf{\zeta}} (\mathbf{\zeta}) \right)
| |
| − | \d\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D.
| |
| − | </math></center>
| |
| − | In the case of a shallow draft, the fact that the Green's function is
| |
| − | symmetric and therefore satisfies the free surface boundary condition
| |
| − | with respect to the second variable as well can be used to
| |
| − | drastically simplify \eqref{int_eq}. Due to the linearity of the problem
| |
| − | the ambient incident potential can just be added to the equation to obtain the
| |
| − | total water velocity potential,
| |
| − | <math>\phi=\phi^{\mathrm{I}}+\phi^{\mathrm{S}}</math>. Limiting the result to
| |
| − | the water surface leaves the integral equation for the water velocity
| |
| − | potential under the ice floe,
| |
| − | <center><math> (int_eq_hs)
| |
| − | \phi(\mathbf{x}) = \phi^{\mathrm{I}}(\mathbf{x}) +
| |
| − | \int\limits_{\Delta} G (\mathbf{x};\mathbf{\xi}) \big( \alpha
| |
| − | \phi(\mathbf{\xi}) + \i \sqrt{\alpha} w(\mathbf{\xi}) \big)
| |
| − | \d\sigma_\mathbf{\xi}, \quad \mathbf{x} \in \Delta.
| |
| − | </math></center>
| |
| − | Since the surface displacement of the ice floe appears in this
| |
| − | integral equation, it is coupled with the plate equation \eqref{plate_final}.
| |
| − | A method of solution is discussed in detail by [[JGR02]] but a short
| |
| − | outline will be given. The surface displacement of the ice floe is
| |
| − | expanded into its modes of vibration by calculating the eigenfunctions
| |
| − | and eigenvalues of the biharmonic operator. The integral equation for
| |
| − | the potential is then solved for every eigenfunction which gives a
| |
| − | corresponding potential to each eigenfunction. The expansion in the
| |
| − | eigenfunctions simplifies the biharmonic equation and, by using the
| |
| − | orthogonality of the eigenfunctions, a system of equations for the
| |
| − | unknown coefficients of the eigenfunction expansion is obtained.
| |
| − |
| |
| − | ===The coupled ice floe - water equations===
| |
| − |
| |
| − | Since the operator <math>\nabla^4</math>, subject to the free edge boundary
| |
| − | conditions, is self-adjoint a thin plate must possess a set of modes <math>w^k</math>
| |
| − | which satisfy the free boundary conditions and the eigenvalue
| |
| − | equation
| |
| − | <center><math>
| |
| − | \nabla^4 w^k = \lambda_k w^k.
| |
| − | </math></center>
| |
| − | The modes which correspond to different eigenvalues <math>\lambda_k</math> are
| |
| − | orthogonal and the eigenvalues are positive and real. While the plate will
| |
| − | always have repeated eigenvalues, orthogonal modes can still be found and
| |
| − | the modes can be normalised. We therefore assume that the modes are
| |
| − | orthonormal, i.e.
| |
| − | <center><math>
| |
| − | \int\limits_\Delta w^j (\mathbf{\xi}) w^k (\mathbf{\xi})
| |
| − | \d\sigma_{\mathbf{\xi}} = \delta _{jk},
| |
| − | </math></center>
| |
| − | where <math>\delta _{jk}</math> is the Kronecker delta. The eigenvalues <math>\lambda_k</math>
| |
| − | have the property that <math>\lambda_k \rightarrow \infty</math> as </math>k \rightarrow
| |
| − | \infty<math> and we order the modes by increasing eigenvalue. These modes can be
| |
| − | used to expand any function over the wetted surface of the ice floe <math>\Delta</math>.
| |
| − |
| |
| − | We expand the displacement of the floe in a finite number of modes <math>M</math>, i.e.
| |
| − | <center><math> (expansion)
| |
| − | w(\mathbf{x}) =\sum_{k=1}^{M} c_k w^k (\mathbf{x}).
| |
| − | </math></center>
| |
| − | >From the linearity of \eqref{int_eq_hs} the potential can be
| |
| − | written in the form
| |
| − | <center><math> (expansionphi)
| |
| − | \phi(\mathbf{x}) =\phi^0(\mathbf{x}) + \sum_{k=1}^{M} c_k \phi^k (\mathbf{x}),
| |
| − | </math></center>
| |
| − | where <math>\phi^0</math> and <math>\phi^k</math> respectively satisfy the integral equations
| |
| − | \begin{subequations} (phi)
| |
| − | <center><math> (phi0)
| |
| − | \phi^0(\mathbf{x}) = \phi^{\mathrm{I}} (\mathbf{x}) +
| |
| − | \int\limits_\Delta \alpha G (\mathbf{x};\mathbf{\xi}) \phi^0
| |
| − | (\mathbf{\xi}) d\sigma_\mathbf{\xi}
| |
| − | </math></center>
| |
| − | and
| |
| − | <center><math> (phii)
| |
| − | \phi^k (\mathbf{x}) = \int\limits_{\Delta} G (\mathbf{x};\mathbf{\xi})
| |
| − | \left( \alpha \phi^k (\mathbf{\xi}) + \i \sqrt{\alpha} w^k
| |
| − | (\mathbf{\xi})\right) \d\sigma_{\mathbf{\xi}}.
| |
| − | </math></center>
| |
| − | \end{subequations}
| |
| − | The potential <math>\phi^0</math> represents the potential due to the incoming wave
| |
| − | assuming that the displacement of the ice floe is zero. The potential
| |
| − | <math>\phi^k</math> represents the potential which is generated by the plate
| |
| − | vibrating with the <math>k</math>th mode in the absence of any input wave forcing.
| |
| − |
| |
| − | We substitute equations \eqref{expansion} and \eqref{expansionphi} into
| |
| − | equation \eqref{plate_final} to obtain
| |
| − | <center><math> (expanded)
| |
| − | \beta \sum_{k=1}^{M} \lambda_k c_k w^k -\alpha \gamma
| |
| − | \sum_{k=1}^{M} c_k w^k = \i \sqrt{\alpha} \big( \phi^0 +
| |
| − | \sum_{k=1}^{M} c_k \phi^k \big) - \sum_{k=1}^{M} c_k w^k.
| |
| − | </math></center>
| |
| − | To solve equation \eqref{expanded} we multiply by <math>w^j</math> and integrate over
| |
| − | the plate (i.e. we take the inner product with respect to <math>w^j</math>) taking
| |
| − | into account the orthogonality of the modes <math>w^j</math> and obtain
| |
| − | <center><math> (final)
| |
| − | \beta \lambda_k c_k + \left( 1-\alpha \gamma \right) c_k =
| |
| − | \int\limits_{\Delta} \i \sqrt{\alpha} \big( \phi^0 (\mathbf{\xi})
| |
| − | + \sum_{j=1}^{N} c_j \phi^j (\mathbf{\xi}) \big) w^k (\mathbf{\xi})
| |
| − | \d\sigma_{\mathbf{\xi}},
| |
| − | </math></center>
| |
| − | which is a matrix equation in <math>c_k</math>.
| |
| − |
| |
| − | Equation \eqref{final} cannot be solved without determining the modes of
| |
| − | vibration of the thin plate <math>w^k</math> (along with the associated
| |
| − | eigenvalues <math>\lambda_k</math>) and solving the integral equations
| |
| − | \eqref{phi}. We use the finite element method to
| |
| − | determine the modes of vibration \cite[]{Zienkiewicz} and the integral
| |
| − | equations \eqref{phi} are solved by a constant panel
| |
| − | method \cite[]{Sarp_Isa}. The same set of nodes is used for the finite
| |
| − | element method and to define the panels for the integral equation.
| |
| − |
| |
| − |
| |
| − | ===Full diffraction calculation for multiple ice floes===
| |
| − |
| |
| − | The interaction theory is a method to allow more rapid solutions to
| |
| − | problems involving multiple bodies. The principle advantage is that the
| |
| − | potential is represented in the cylindrical eigenfunctions and
| |
| − | therefore fewer unknowns are required. However, every problem which
| |
| − | can be solved by the interaction theory can also be solved by applying
| |
| − | the full diffraction theory and solving an integral equation over the
| |
| − | wetted surface of all the bodies. In this section we will briefly show
| |
| − | how this extension can be performed for the ice floe situation. The
| |
| − | full diffraction calculation will be used to check the performance and
| |
| − | convergence of our interaction theory. Also, because the interaction
| |
| − | theory is only valid when the escribed cylinder for each ice floe does
| |
| − | not contain any other floe, the full diffraction calculation is
| |
| − | required for a very dense arrangement of ice floes.
| |
| − |
| |
| − | We can solve the full diffraction problem for multiple ice floes by the
| |
| − | following extension. The displacement of the <math>j</math>th floe is expanded in a
| |
| − | finite number of modes <math>M_j</math> (since the number of modes may not
| |
| − | necessarily be the same), i.e.
| |
| − | <center><math> (expansion_f)
| |
| − | w_j \left( \mathbf{x}\right) =\sum_{k=1}^{M_j} c_{jk} w_j^k (\mathbf{x}).
| |
| − | </math></center>
| |
| − | >From the linearity of \eqref{int_eq_hs} the potential can be
| |
| − | written in the form
| |
| − | <center><math> (expansionphi_f)
| |
| − | \phi(\mathbf{x}) =\phi_0(\mathbf{x}) + \sum_{n=1}^{N} \sum_{k=1}^{M_n}
| |
| − | c_{nk} \phi_n^k(\mathbf{x}),
| |
| − | </math></center>
| |
| − | where <math>\phi _{0}</math> and <math>\phi_j^k</math> respectively satisfy the integral equations
| |
| − | \begin{subequations} (phi_f)
| |
| − | <center><math> (phi0_f)
| |
| − | \phi_j^0 (\mathbf{x}) = \phi^{\mathrm{I}} (\mathbf{x}) + \sum_{n=1}^{N}
| |
| − | \int\limits_{\Delta_n} \alpha G (\mathbf{x};\mathbf{\xi})
| |
| − | \phi_j^0(\mathbf{\xi}) \d\sigma_{\mathbf{\xi}}
| |
| − | </math></center>
| |
| − | and
| |
| − | <center><math> (phii_f)
| |
| − | \phi_j^k(\mathbf{x}) = \sum_{n=1}^{N} \int\limits_{\Delta_n} G
| |
| − | (\mathbf{x};\mathbf{\xi}) \left( \alpha \phi_j^k (\mathbf{\xi}) +
| |
| − | \i\sqrt{\alpha} w_j^k (\mathbf{\xi})\right) \d\sigma_{\mathbf{\xi}}.
| |
| − | </math></center>
| |
| − | \end{subequations}
| |
| − | The potential <math>\phi_j^{0}</math> represents the potential due the incoming wave
| |
| − | assuming that the displacement of the ice floe is zero,
| |
| − | <math>\phi_j^k</math> represents the potential which is generated by the <math>j</math>th plate
| |
| − | vibrating with the <math>k</math>th mode in the absence of any input wave forcing. It
| |
| − | should be noted that <math>\phi_j^k(\mathbf{x})</math> is, in general, non-zero
| |
| − | for <math>\mathbf{x}\in \Delta_{n}</math> (since the vibration of the <math>j</math>th
| |
| − | plate will result in potential under the <math>n</math>th plate).
| |
| − |
| |
| − | We substitute equations \eqref{expansion_f} and \eqref{expansionphi_f} into
| |
| − | equation \eqref{plate_final} to obtain
| |
| − | <center><math> (expanded_f)
| |
| − | \beta_j \sum_{k=1}^{M_j} \lambda_{jk} c_{jk} w_j^k - \alpha \gamma_j
| |
| − | \sum_{k=1}^{M_j} c_{jk} w_j^k = \i \sqrt{\alpha} \,
| |
| − | \big( \phi_j^0 + \sum_{n=1}^{N} \sum_{k=1}^{M_n} c_{nk} \phi_n^k \big)
| |
| − | - \sum_{k=1}^{M_n} c_{jk} w_j^k.
| |
| − | </math></center>
| |
| − | To solve equation \eqref{expanded_f} we multiply by <math>w_j^{l}</math> and integrate
| |
| − | over the plate (as before)
| |
| − | taking into account the orthogonality of the modes <math>w_j^l</math> and obtain
| |
| − | <center><math> (final_f)
| |
| − | \beta_j \lambda_{jk} c_{jk} + \big(1- \alpha \gamma_j \big)
| |
| − | c_{jk} = \int\limits_{\Delta_j} \i \sqrt{\alpha} \, \big( \phi_0
| |
| − | (\mathbf{\xi}) + \sum_{n=1}^{N} \sum_{l=1}^{M_n} c_{nl} \phi_n^l
| |
| − | (\mathbf{\xi}) \big) \, w_j^l (\mathbf{\xi}) \d\sigma_{\mathbf{\xi}},
| |
| − | </math></center>
| |
| − | which holds for all <math>j= 1, \ldots, N</math> and therefore gives a matrix
| |
| − | equation for <math>c_{jk}</math>.
| |
| − |
| |
| − | ==Numerical Results==
| |
| − |
| |
| − | In this section we will present some calculations using the interaction
| |
| − | theory in finite and infinite depth and the full
| |
| − | diffraction method in finite and infinite depth.
| |
| − | These will be based on calculations for ice floes. We begin with some
| |
| − | convergence tests which aim to compare the various methods. It needs
| |
| − | to be noted that this comparison is only of numerical nature since the
| |
| − | interactions methods as well as the full diffraction calculations
| |
| − | are exact in an analytical sense. However, numerical calculations
| |
| − | require truncations which affect the different methods in different
| |
| − | ways. Especially the dependence on these truncations will be investigated.
| |
| − |
| |
| − | ===Convergence Test===
| |
| − | We will present some convergence tests that aim to compare the
| |
| − | performance of the interaction theory with the full diffraction
| |
| − | calculations and to compare the
| |
| − | performance of the finite and infinite depth interaction methods in deep water.
| |
| − | The comparisons will be conducted for the case of two square ice floes
| |
| − | in three different arrangements.
| |
| − | In the full diffraction calculation the ice floes
| |
| − | are discretised in <math>24 \times 24 = 576</math> elements. For the full diffraction
| |
| − | calculation the resulting linear system of equations to be solved is
| |
| − | therefore 1152. As will be seen, once the diffraction
| |
| − | transfer matrix has been calculated (and saved), the dimension of the
| |
| − | linear system of equations to be solved in the interaction method is
| |
| − | considerably smaller. It is given by twice the dimension of the
| |
| − | diffraction transfer matrix. The most challenging situation for the
| |
| − | interaction theory is when the bodies are close together. For this
| |
| − | reason we choose the distance such that the escribed circles
| |
| − | of the two ice floes just overlap. It must be recalled that the
| |
| − | interaction theory is valid as long as the escribed cylinder of a body
| |
| − | does not intersect with any other body.
| |
| − |
| |
| − | Both ice floes have non-dimensionalised
| |
| − | stiffness <math>\beta = 0.02</math>, mass <math>\gamma = 0.02</math> and Poisson's ratio
| |
| − | is chosen as <math>\nu=0.3333</math>. The wavelength of
| |
| − | the ambient incident wave is <math>\lambda = 2</math>. Each ice floe has
| |
| − | side length 2. The ambient
| |
| − | wavefield is of unit amplitude and propagates in the <math>x</math>-direction.
| |
| − | Three different arrangements are chosen to compare the results of the
| |
| − | finite depth interaction method in deep water and the infinite depth
| |
| − | interaction method with the corresponding full diffraction
| |
| − | calculations. In the first arrangement the second ice floe is located
| |
| − | behind the first, in the second arrangement it is located
| |
| − | beside, and the third arrangement it is both
| |
| − | beside and behind. The exact positions of the ice floes
| |
| − | are given in table (tab:pos).
| |
| − |
| |
| − | \begin{table}
| |
| − | \begin{center}
| |
| − | \begin{tabular}{@{}ccc@{}}
| |
| − | arrangement & <math>O_1</math> & <math>O_2</math>\<center><math>3pt]
| |
| − | 1 & <math>(-1.4,0)</math> & <math>(1.4,0)</math>\\
| |
| − | 2 & <math>(0,-1.4)</math> & <math>(0,1.4)</math>\\
| |
| − | 3 & <math>(-1.4,-0.6)</math> & <math>(1.4,0.6)</math>
| |
| − | \end{tabular}
| |
| − | \caption{Positions of the ice floes in the different arrangements.} (tab:pos)
| |
| − | \end{center}
| |
| − | \end{table}
| |
| − |
| |
| − | Figure (fig:tsf) shows the
| |
| − | solutions corresponding to the three arrangements in the case of water
| |
| − | of infinite depth. To illustrate the effect on the water in the
| |
| − | vicinity of the ice floes, the water displacement is also shown.
| |
| − | It is interesting
| |
| − | to note that the ice floe in front is barely influenced by the
| |
| − | floe behind while the motion of the floe behind is quite
| |
| − | different from its motion in the absence of the floe in front.
| |
| − |
| |
| − | \begin{figure}
| |
| − | \begin{center}
| |
| − |
| |
| − | \includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi2}\<center><math>0.4cm]
| |
| − | \includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi0}\<center><math>0.4cm]
| |
| − | \includegraphics[width=8.2cm]{int_n11_x12_y28_v5_b02_chi2}
| |
| − |
| |
| − | \end{center}
| |
| − | \caption{Surface displacement of the ice floes
| |
| − | and the water in their vicinity,\newline arrangements 1, 2 and 3.} (fig:tsf)
| |
| − | \end{figure}
| |
| − |
| |
| − | To compare the results, a measure of
| |
| − | the error from the full diffraction calculation is used. We calculate
| |
| − | the full diffraction solution with a sufficient number of points
| |
| − | so that we may use it to approximate the exact solution.
| |
| − | <center><math>
| |
| − | E_2 = \left( \, \int\limits_{\Delta}
| |
| − | \big| w_{i}(\mathbf{x})-w_{f}(\mathbf{x}) \big|^2 \d\mathbf{x} \,
| |
| − | \right)^{1/2},
| |
| − | </math></center>
| |
| − | where <math>w_{i}</math> and <math>w_{f}</math> are the solutions of the
| |
| − | interaction method and the corresponding full diffraction calculation
| |
| − | respectively. It would also be possible to compare other errors, the
| |
| − | maximum difference of the solutions for example, but the results are
| |
| − | very similar.
| |
| − |
| |
| − | It is worth noting that the finite depth interaction method
| |
| − | only converges up to a certain depth if used with the
| |
| − | eigenfunction expansion of the finite depth Green's function \eqref{green_d}.
| |
| − | This is because of the factor
| |
| − | <math>\alpha^2-k^2</math> in the term of propagating modes of the Green's
| |
| − | function. The Green's function can
| |
| − | be rewritten by making use of the dispersion relation \eqref{eq_k}
| |
| − | \cite[as suggested by][p. 26, for example]{linton01}
| |
| − | and the depth restriction of the finite depth interaction method for
| |
| − | bodies of arbitrary geometry can be circumvented.
| |
| − |
| |
| − | The truncation parameters for the interaction methods will
| |
| − | now be considered for both finite and infinite depth.
| |
| − | The number of propagating modes and angular decaying
| |
| − | components are free parameters in both methods. In
| |
| − | finite depth, the number of decaying roots of the dispersion relation
| |
| − | needs to be chosen while in infinite depth the discretisation of
| |
| − | a continuous variable must be selected.
| |
| − | In the infinite depth case we are free to choose the number of
| |
| − | points as well as the points themselves. In water of finite depth, the depth
| |
| − | can also be considered a free parameter as long as it is chosen large
| |
| − | enough to account for deep water.
| |
| − |
| |
| − | Truncating the infinite sums in the eigenfunction expansion of the
| |
| − | outgoing water velocity potential for infinite depth with
| |
| − | truncation parameters <math>T_H</math> and <math>T_K</math> and discretising the integration
| |
| − | by defining a set of nodes, <math>0\leq\eta_1 < \ldots < \eta_m < \ldots <
| |
| − | \eta_{_{T_R}}<math>, with weights </math>h_m</math>, the potential for infinite depth
| |
| − | can be approximated by
| |
| − | <center><math>\begin{split}
| |
| − | \phi (r,\theta,z) &= \e^{\alpha z} \sum_{\nu = -
| |
| − | T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (\alpha r) \e^{\i \nu
| |
| − | \theta}\\
| |
| − | &\quad + \sum_{m=1}^{T_R} h_m \, \psi (z,\eta_m) \sum_{\nu = -
| |
| − | T_K}^{T_K} A_{\nu} (\eta_m) K_\nu (\eta_m r) \e^{\i \nu \theta}.
| |
| − | \end{split}</math></center>
| |
| − | In the following, the integration weights are chosen to be <math>h_m =
| |
| − | 1/2\,(\eta_{m+1}-\eta_{m-1})<math>, </math>m=2, \ldots, T_R-1<math> and </math>h_1 =
| |
| − | \eta_2-\eta_1<math> as well as </math>h_{_{T_R}} =
| |
| − | \eta_{_{T_R}}-\eta_{_{T_R-1}}</math>, which corresponds to the mid-point
| |
| − | quadrature rule.
| |
| − | Different quadrature rules such as Gaussian quadrature
| |
| − | could be considered. Although in general this would lead to better
| |
| − | results, the mid-point rule allows a clever
| |
| − | choice of the discretisation points so that the convergence with
| |
| − | Gaussian quadrature is no better.
| |
| − | In finite depth, the analogous truncation leads to
| |
| − | <center><math>\begin{split}
| |
| − | \phi (r,\theta,z) &= \frac{\cosh k (z+d)}{\cosh kd} \sum_{\nu = -
| |
| − | T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (k r) \e^{\i \nu \theta}\\
| |
| − | & \quad + \sum_{m = 1}^{T_R} \frac{\cos k_m(z+d)}{\cos k_m d}
| |
| − | \sum_{\nu = - T_K}^{T_K} A_{m\nu} K_\nu (k_m r) \e^{\i \nu \theta}.
| |
| − | \end{split}</math></center>
| |
| − | In both cases, the dimension of the diffraction transfer matrix,
| |
| − | <math>\mathbf{B}</math>, is given by <math>2 \, T_H+1+T_R \, (2 \, T_K+1)</math>.
| |
| − |
| |
| − | Since the choice of the number of propagating
| |
| − | modes and angular decaying components affects the finite and
| |
| − | infinite depth methods in similar ways, the dependence on these
| |
| − | parameters will not be further presented. Thorough convergence tests
| |
| − | have shown that in the settings investigated here, it is sufficient to
| |
| − | choose <math>T_H</math> to be 11 and <math>T_K</math> to be 5. Further increasing these
| |
| − | parameter values does not result in smaller errors (as compared
| |
| − | to the full diffraction calculation with 576 elements per floe).
| |
| − | We will now compare the convergence of the infinite depth and
| |
| − | the finite depth methods if <math>T_H</math> and <math>T_K</math> are
| |
| − | fixed (with the previously mentioned values) and <math>T_R</math> is varied. To be able to
| |
| − | compare the results, the discretisation of the continuous variable
| |
| − | will always be the same for fixed <math>T_R</math> and these are
| |
| − | shown in table (tab:discr).
| |
| − | It should be noted that if only one node is used the integration
| |
| − | weight is chosen to be 1.
| |
| − |
| |
| − | \begin{table}
| |
| − | \begin{center}
| |
| − | \begin{tabular}{@{}cl@{}}
| |
| − | <math>T_R</math> & discretisation of <math>\eta</math>\<center><math>3pt]
| |
| − | 1 & \{ 2.1 \}\\
| |
| − | 2 & \{ 1.2, 2.7 \}\\
| |
| − | 3 & \{ 0.8, 1.8, 3.0 \}\\
| |
| − | 4 & \{ 0.4, 1.4, 2.2, 3.2 \}\\
| |
| − | 5 & \{ 0.2, 1.0, 1.8, 3.0, 4.6 \}
| |
| − | \end{tabular}
| |
| − | \caption{The different discretisations used in the convergence tests.} (tab:discr)
| |
| − | \end{center}
| |
| − | \end{table}
| |
| | | | |
| − | Figures (fig:behind), (fig:beside) and (fig:shifted) show the convergence for arrangement 1, arrangement 2 and arrangement 3, respectively, for
| |
| − | the infinite depth method and the finite depth method with depth 2
| |
| − | (plot (a)) and depth 4 (plot (b)).
| |
| − | Since the ice floes are located beside each other
| |
| − | in arrangement 2 the average errors are the same for both floes.
| |
| − | As can be seen from figures (fig:behind), (fig:beside) and
| |
| − | (fig:shifted) the convergence of the infinite depth method
| |
| − | is similar to that of the finite depth method. Used with depth 2, the
| |
| − | convergence of the finite depth method is generally better than that
| |
| − | of the infinite depth method while used with depth 4, the infinite depth
| |
| − | method achieves the better results. Tests with other depths show that
| |
| − | the performance of the finite depth method decreases with increasing
| |
| − | water depth as expected. In general, since the wavelength is 2, a depth
| |
| − | of <math>d=2</math> should approximate infinite depth and hence there is no
| |
| − | advantage to using the infinite depth theory. However, as mentioned
| |
| − | previously, for certain situations such as ice floes it is not necessarily
| |
| − | true that <math>d=2</math> will approximate infinite depth.
| |
| − |
| |
| − | \begin{figure}
| |
| − | \begin{center}
| |
| − | \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}}
| |
| − | \includegraphics[height=0.38\columnwidth]{behind_d2}&&
| |
| − | \includegraphics[height=0.38\columnwidth]{behind_d4}
| |
| − | \end{tabular}
| |
| − | \caption{Development of the errors as <math>T_R</math> is increased in
| |
| − | arrangement 1.} (fig:behind)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | \begin{center}
| |
| − | \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}}
| |
| − | \includegraphics[height=0.38\columnwidth]{beside_d2}&&
| |
| − | \includegraphics[height=0.38\columnwidth]{beside_d4}
| |
| − | \end{tabular}
| |
| − | \caption{Development of the errors as <math>T_R</math> is increased in
| |
| − | arrangement 2.} (fig:beside)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | \begin{center}
| |
| − | \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}}
| |
| − | \includegraphics[height=0.38\columnwidth]{shifted_d2}&&
| |
| − | \includegraphics[height=0.38\columnwidth]{shifted_d4}
| |
| − | \end{tabular}
| |
| − | \caption{Development of the errors as <math>T_R</math> is increased in
| |
| − | arrangement 3.} (fig:shifted)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | ===Multiple ice floe results===
| |
| − | We will now present results for multiple ice floes of different
| |
| − | geometries and in different arrangements on water of infinite depth.
| |
| − | We choose the floe arrangements arbitrarily, since there are
| |
| − | no known special ice floe arrangements, such as those that give
| |
| − | rise to resonances in the infinite limit.
| |
| − | In all plots, the wavelength <math>\lambda</math> has been chosen to
| |
| − | be <math>2</math>, the stiffness <math>\beta</math> and the mass <math>\gamma</math> of the ice
| |
| − | floes to be 0.02 and Poisson's ratio <math>\nu</math> is <math>0.3333</math>. The ambient
| |
| − | wavefield of amplitude 1 propagates in
| |
| − | the positive direction of the <math>x</math>-axis, thus it travels from left to
| |
| − | right in the plots.
| |
| − |
| |
| − | Figure (fig:int_arb) shows the
| |
| − | displacements of multiple interacting ice floes of different shapes and
| |
| − | in different arrangements. Since square elements have been used to
| |
| − | represent the floes, non-rectangular geometries are approximated.
| |
| − | All ice floes have an area of 4 and the escribing circles do not
| |
| − | intersect with any of the other ice floes.
| |
| − | The plots show the displacement of the ice floes at time <math>t=0</math>.
| |
| − |
| |
| − | \begin{figure}
| |
| − | \begin{center}
| |
| − | \begin{tabular}{p{0.46\columnwidth}p{0.03\columnwidth}p{0.46\columnwidth}}
| |
| − | \includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_in} &&
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| − | \includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_out}\<center><math>0.2cm]
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| − | \includegraphics[width=0.45\columnwidth]{mult_n15_rh_rh_rh} &&
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| − | \includegraphics[width=0.45\columnwidth]{mult_n15_sq_sq_sq_sq_sq}\\
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| − | \end{tabular}
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| − | \end{center}
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| − | \caption{Surface displacement of interacting ice floes of different geometries.} (fig:int_arb)
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| − | \end{figure}
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| − |
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| − | ==Summary==
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| − | The finite depth interaction theory developed by
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| − | [[kagemoto86]] has been extended to water of infinite
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| − | depth. Furthermore, using the eigenfunction
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| − | expansion of the infinite depth free surface Green's function we have
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| − | been able to calculate the diffraction transfer matrices for bodies of
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| − | arbitrary geometry. We also showed how the diffraction transfer
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| − | matrices can be calculated efficiently for different orientations of
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| − | the body.
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| − |
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| − | The convergence of the infinite depth interaction method is similar to
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| − | that of the finite depth method. Generally, it can be said that the
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| − | greater the water depth in the finite depth method the poorer its
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| − | performance. Since bodies in the water can change the water depth
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| − | which is required to allow the water to be approximated as infinitely
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| − | deep (ice floes for example) it is recommendable to use the infinite
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| − | depth method if the water depth may be considered
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| − | infinite. Furthermore, the infinite depth method requires the infinite
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| − | depth single diffraction solutions which are easier to
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| − | compute than the finite depth solutions.
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| − | It is also possible that the
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| − | convergence of the infinite depth method may be further improved
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| − | by a novel to optimisation of the discretisation of the continuous variable.
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| − |
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| − |
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| − |
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| − |
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| − |
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| − | \appendix
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| − |
| |
| − | ==Notation==
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| − | \begin{supertabular}{cl}
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| − | <math>\Delta</math> & a floating or submerged body, <math>\Delta \subset
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| − | \mathds{R}^3</math>\\
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| − | <math>D</math> & area occupied by the water\\
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| − | <math>\Gamma</math> & the immersed surface of <math>\Delta</math>\\
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| − | <math>\mathds{R}_-^3</math> & the lower three-dimensional half-space\\
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| − | <math>(x,y,z)</math> & Cartesian coordinates (<math>z</math> pointing vertically upwards)\\
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| − | <math>(r,\theta,z)</math> & cylindrical coordinates\\
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| − | <math>\mathbf{y}</math> & a point of the water, <math>\mathbf{y}=(x,y,z)</math>\\
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| − | <math>\mathbf{x}</math> & a point of the undisturbed water surface,
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| − | <math>\mathbf{x}=(x,y) \; (=(x,y,0))</math>\\
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| − | <math>O_j</math> & mean centre position of <math>\Delta_j</math>\\
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| − | <math>\mathbf{\zeta}=(s,\varphi,c)</math> & cylindrical coordinates of a source point\\
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| − | <math>(R_{jl},\vartheta_{jl})</math> & polar coordinates of mean centre position of
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| − | <math>\Delta_l</math> in coordinates of <math>\Delta_j</math>\\
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| − | &\\
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| − | <math>\Phi</math> & the water velocity potential, </math>\Phi: D \times S
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| − | \rightarrow \mathds{R}<math>\\
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| − | <math>W</math> & the surface displacement off the undisturbed position, </math>W:
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| − | \Gamma \times S \rightarrow \mathds{R}<math>\\
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| − | <math>\phi</math> & the time-independent water velocity potential,\\ & </math>\Phi(x,y,z,t)
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| − | = \Re \{\phi (x,y,z) \e^{-\i \omega t} \}<math>\\
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| − | <math>w</math> & the time-independent surface displacement,\\ & </math>W(x,y,t) = \Re \{w (x,y)
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| − | \e^{-\i \omega t} \}<math>\\
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| − | <math>\phi^{\mathrm{In}}</math> & the time-independent ambient wavefield,
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| − | <math>\phi^{\mathrm{In}}: D \rightarrow \mathds{C}</math>\\
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| − | <math>G</math> & a Green's function\\
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| − | <math>\varsigma</math> & the source strength distribution function, </math>\varsigma: \Gamma
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| − | \rightarrow \mathds{C}<math>\\
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| − | &\\
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| − | <math>J_\nu</math> & Bessel function of the first kind of integer order <math>\nu</math>\\
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| − | <math>Y_\nu</math> & Bessel function of the second kind of integer order <math>\nu</math>\\
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| − | <math>I_\nu</math> & modified Bessel function of the first kind of integer order
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| − | <math>\nu</math>\\
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| − | <math>K_\nu</math> & modified Bessel function of the second kind of integer order
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| − | <math>\nu</math>\\
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| − | <math>H_\nu^{(1)}</math> & Hankel function of the first kind of integer order <math>\nu</math>\\
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| − | <math>\mathbf{H}_0</math> & Struve function of order zero\\
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| − | <math>\Psi</math> & </math>\Psi(z,\eta) = \cos(\eta z) + \alpha / \eta \,
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| − | \sin(\eta z), \ (z,\eta) \in \mathds{R}_{\leq 0} \times \mathds{R}_{>0}<math>\\
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| − | &\\
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| − | <math>\i</math> & the imaginary unit, <math>\i = \sqrt{-1}</math>\\
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| − | <math>g</math> & the acceleration due to gravity\\
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| − | <math>d</math> & the water depth\\
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| − | <math>\omega</math> & the radian frequency\\
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| − | <math>T</math> & the wave period, <math>T = 2\pi/\omega</math>\\
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| − | <math>\lambda</math> & the wavelength\\
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| − | <math>\alpha</math> & <math>\alpha = \omega^2/g</math>\\
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| − | <math>k_m</math> & positive real roots of the dispersion relation, equation
| |
| − | \eqref{eq_k_m}\\
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| − | <math>k</math> & imaginary part of the positive imaginary root of the dispersion
| |
| − | relation,\\ & <math>k = \i \, k_0</math>\\
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| − | <math>\kappa</math> & wavenumber under a flexible floating thin plate\\
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| − | <math>\chi</math> & angle of the <math>x</math>-axis with the direction of propagation of the
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| − | ambient\\ & wavefield\\
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| − | &\\
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| − | <math>A_{0\nu}^j</math> & coefficients of outgoing water velocity potential\\ &
| |
| − | corresponding to propagating modes\\
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| − | <math>A_{\nu}^j(\cdot)</math> & coefficient functions of outgoing water velocity
| |
| − | potential\\ & corresponding to decaying modes\\
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| − | <math>D_{0\nu}^j</math> & coefficients of incident water velocity potential\\ &
| |
| − | corresponding to propagating modes\\
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| − | <math>D_{\nu}^j(\cdot)</math> & coefficients of incident water velocity
| |
| − | potential\\ & corresponding to decaying modes\\
| |
| − | &\\
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| − | <math>\mathbf{a}_j</math> & truncated vector of amplitudes of the scattered
| |
| − | wavefield of <math>\Delta_j</math>\\
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| − | <math>\mathbf{d}_j</math> & truncated vector of amplitudes of the incident
| |
| − | wavefield upon <math>\Delta_j</math>\\
| |
| − | <math>\mathbf{d}_j^{\mathrm{In}}</math> & truncated vector of amplitudes of the
| |
| − | ambient incident wavefield upon <math>\Delta_j</math>\\
| |
| − |
| |
| − |
| |
| − |
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| − |
| |
| − | <math>\mathbf{T}_{jl}</math> & coordinate transformation matrix from <math>\Delta_j</math>
| |
| − | to <math>\Delta_l</math>, <math>j \neq l</math>\\
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| − | <math>\mathbf{B}_j</math> & diffraction transfer matrix\\
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| − | <math>\hat{\mathbf{B}}_j</math> & diffraction transfer matrix with integration
| |
| − | weights\\
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| − | &\\
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| − | <math>\rho</math> & the density of the water\\
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| − | <math>\rho_\Delta</math> & the density of the ice floe\\
| |
| − | <math>h</math> & the thickness of the ice floe\\
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| − | <math>D</math> & the modulus of rigidity of the ice floe\\
| |
| − | <math>\beta</math> & the non-dimensionalised stiffness of the ice floe\\
| |
| − | <math>\gamma</math> & the non-dimensionalised mass of the ice floe\\
| |
| − | <math>\nu</math> & Poisson's ratio for the ice floe\\
| |
| − | \end{supertabular}
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| − |
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| − | \bibliography{mike,others}
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| − | \bibliographystyle{jfm}
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| | [[Category:Linear Water-Wave Theory]] | | [[Category:Linear Water-Wave Theory]] |
