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| | We show here a solution for a semi-infinite [[:Category:Floating Elastic Plate|Floating Elastic Plate]] on [[Finite Depth]]. | | We show here a solution for a semi-infinite [[:Category:Floating Elastic Plate|Floating Elastic Plate]] on [[Finite Depth]]. |
| − | THe problem was solved by A solution | + | THe problem was solved by |
| − | for [[Shallow Depth]] was given in [[Zilman_Miloh_2000a|Zilman and Miloh 2000]] and we will also show this.
| |
| | | | |
| − | =Governing Equations=
| + | [[Category:Eigenfunction Matching Method]] |
| − | | |
| − | We begin with the [[Frequency Domain Problem]] for a [[Floating Elastic Plate]]
| |
| − | in the non-dimensional form of [[Tayler_1986a|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]])
| |
| − | We will use a cylindrical coordinate system, <math>(r,\theta,z)</math>,
| |
| − | assumed to have its origin at the centre of the circular
| |
| − | plate which has radius <math>a</math>. The water is assumed to have
| |
| − | constant finite depth <math>H</math> and the <math>z</math>-direction points vertically
| |
| − | upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-H</math>. The
| |
| − | boundary value problem can therefore be expressed as
| |
| − | <center>
| |
| − | <math>
| |
| − | \Delta\phi=0, \,\, -H<z<0,
| |
| − | </math>
| |
| − | </center>
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi_{z}=0, \,\, z=-H,
| |
| − | </math>
| |
| − | </center>
| |
| − | <center><math>
| |
| − | \phi_{z}=\alpha\phi, \,\, z=0,\,r>a,
| |
| − | </math></center>
| |
| − | <center>
| |
| − | <math>
| |
| − | (\beta\Delta^{2}+1-\alpha\gamma)\phi_{z}=\alpha\phi, \,\, z=0,\,r<a
| |
| − | </math>
| |
| − | </center>
| |
| − | where the constants <math>\beta</math> and <math>\gamma</math> are given by
| |
| − | <center>
| |
| − | <math>
| |
| − | \beta=\frac{D}{\rho\,L^{4}g}, \gamma=\frac{\rho_{i}h}{\rho\,L}
| |
| − | </math>
| |
| − | </center>
| |
| − | and <math>\rho_{i}</math> is the density of the plate. We
| |
| − | must also apply the edge conditions for the plate and the [[Sommerfeld Radiation Condition]]
| |
| − | as <math>r\rightarrow\infty</math>. The subscript <math>z</math>
| |
| − | denotes the derivative in <math>z</math>-direction.
| |
| − | | |
| − | =Solution Method=
| |
| − | | |
| − | == Separation of variables==
| |
| − | | |
| − | We now separate variables, noting that since the problem has
| |
| − | circular symmetry we can write the potential as
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi(r,\theta,z)=\zeta(z)\sum_{n=-\infty}^{\infty}\rho_{n}(r)e^{i n \theta}
| |
| − | </math>
| |
| − | </center>
| |
| − | Applying Laplace's equation we obtain
| |
| − | <center>
| |
| − | <math>
| |
| − | \zeta_{zz}+\mu^{2}\zeta=0
| |
| − | </math>
| |
| − | </center>
| |
| − | so that:
| |
| − | <center>
| |
| − | <math>
| |
| − | \zeta=\cos\mu(z+H)
| |
| − | </math>
| |
| − | </center>
| |
| − | where the separation constant <math>\mu^{2}</math> must
| |
| − | satisfy the [[Dispersion Relation for a Free Surface]]
| |
| − | <center>
| |
| − | <math>
| |
| − | k\tan\left( kH\right) =-\alpha,\quad r>a\,\,\,(1)
| |
| − | </math>
| |
| − | </center>
| |
| − | and the [[Dispersion Relation for a Floating Elastic Plate]]
| |
| − | <center>
| |
| − | <math>
| |
| − | \kappa\tan(\kappa H)=\frac{-\alpha}{\beta\kappa^{4}+1-\alpha\gamma},\quad
| |
| − | r<a \,\,\,(2)
| |
| − | </math>
| |
| − | </center>
| |
| − | Note that we have set <math>\mu=k</math> under the free
| |
| − | surface and <math>\mu=\kappa</math> under the plate. We denote the
| |
| − | positive imaginary solution of (1) by <math>k_{0}</math> and
| |
| − | the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
| |
| − | (2) will be denoted by
| |
| − | <math>\kappa_{m}</math>, <math>m\geq-2</math>. The fully complex
| |
| − | solutions with positive imaginary part are <math>\kappa_{-2}</math> and
| |
| − | <math>\kappa_{-1}</math> (where <math>\kappa_{-1}=\overline{\kappa_{-2}}</math>),
| |
| − | the negative imaginary solution is <math>\kappa_{0}</math> and the positive real
| |
| − | solutions are <math>\kappa_{m}</math>, <math>m\geq1</math>. We define
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi_{m}\left( z\right) =\frac{\cos k_{m}(z+H)}{\cos k_{m}H},\quad m\geq0
| |
| − | </math>
| |
| − | </center>
| |
| − | as the vertical eigenfunction of the potential in the open
| |
| − | water region and
| |
| − | <center>
| |
| − | <math>
| |
| − | \psi_{m}\left( z\right) =\frac{\cos\kappa_{m}(z+H)}{\cos\kappa_{m}H},\quad
| |
| − | m\geq-2
| |
| − | </math>
| |
| − | </center>
| |
| − | as the vertical eigenfunction of the potential in the plate
| |
| − | covered region. For later reference, we note that:
| |
| − | <center>
| |
| − | <math>
| |
| − | \int\nolimits_{-H}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
| |
| − | </math>
| |
| − | </center>
| |
| − | where
| |
| − | <center>
| |
| − | <math>
| |
| − | A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}H\sin k_{m}H+k_{m}H}{k_{m}\cos
| |
| − | ^{2}k_{m}H}\right)
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | \int\nolimits_{-H}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
| |
| − | </math>
| |
| − | </center>
| |
| − | where
| |
| − | <center><math>
| |
| − | B_{mn}=\frac{k_{n}\sin k_{n}H\cos\kappa_{m}H-\kappa_{m}\cos k_{n}H\sin
| |
| − | \kappa_{m}H}{\left( \cos k_{n}H\cos\kappa_{m}H\right) \left( k_{n}
| |
| − | ^{2}-\kappa_{m}^{2}\right) }
| |
| − | </math></center>
| |
| − | | |
| − | We now solve for the function <math>\rho_{n}(r)</math>.
| |
| − | Using Laplace's equation in polar coordinates we obtain
| |
| − | <center>
| |
| − | <math>
| |
| − | \frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r}
| |
| − | \frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left(
| |
| − | \frac{n^{2}}{r^{2}}+\mu^{2}\right) \rho_{n}=0
| |
| − | </math>
| |
| − | </center>
| |
| − | where <math>\mu</math> is <math>k_{m}</math> or
| |
| − | <math>\kappa_{m},</math> depending on whether <math>r</math> is
| |
| − | greater or less than <math>a</math>. We can convert this equation to the
| |
| − | standard form by substituting <math>y=\mu r</math> to obtain
| |
| − | <center>
| |
| − | <math>
| |
| − | y^{2}\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}y^{2}}+y\frac{\mathrm{d}\rho_{n}
| |
| − | }{\rm{d}y}-(n^{2}+y^{2})\rho_{n}=0
| |
| − | </math>
| |
| − | </center>
| |
| − | The solution of this equation is a linear combination of the
| |
| − | modified Bessel functions of order <math>n</math>, <math>I_{n}(y)</math> and
| |
| − | <math>K_{n}(y)</math> [[Abramowitz and Stegun 1964]]. Since the solution must be bounded
| |
| − | we know that under the plate the solution will be a linear combination of
| |
| − | <math>I_{n}(y)</math> while outside the plate the solution will be a
| |
| − | linear combination of <math>K_{n}(y)</math>. Therefore the potential can
| |
| − | be expanded as
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}a_{mn}K_{n}
| |
| − | (k_{m}r)e^{i n\theta}\phi_{m}(z), \;\;r>a
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=-2}^{\infty}b_{mn}
| |
| − | I_{n}(\kappa_{m}r)e^{i n\theta}\psi_{m}(z), \;\;r<a
| |
| − | </math>
| |
| − | </center>
| |
| − | where <math>a_{mn}</math> and <math>b_{mn}</math>
| |
| − | are the coefficients of the potential in the open water and
| |
| − | the plate covered region respectively.
| |
| − | | |
| − | ==Incident potential==
| |
| − | | |
| − | The incident potential is a wave of amplitude <math>A</math>
| |
| − | in displacement travelling in the positive <math>x</math>-direction.
| |
| − | The incident potential can therefore be written as
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi^{\mathrm{I}} =\frac{A}{i\sqrt{\alpha}}e^{k_{0}x}\phi_{0}\left(
| |
| − | z\right)
| |
| − | =\sum\limits_{n=-\infty}^{\infty}e_{n}I_{n}(k_{0}r)\phi_{0}\left(z\right)
| |
| − | e^{i n \theta}
| |
| − | </math>
| |
| − | </center>
| |
| − | where <math>e_{n}=A/\left(i\sqrt{\alpha}\right)</math>
| |
| − | (we retain the dependence on <math>n</math> for situations
| |
| − | where the incident potential might take another form).
| |
| − | | |
| − | ==Boundary conditions==
| |
| − | | |
| − | The boundary conditions for the plate also have to be
| |
| − | considered. The vertical force and bending moment must vanish, which can be
| |
| − | written as
| |
| − | <center>
| |
| − | <math>
| |
| − | \left[\bar{\Delta}-\frac{1-\nu}{r}\left(\frac{\partial}{\partial r}
| |
| − | +\frac{1}{r}\frac{\partial^{2}}{\partial\theta^{2}}\right)\right]
| |
| − | w=0\,\,\,(3)
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | \left[ \frac{\partial}{\partial r}\bar{\Delta}-\frac{1-\nu}{r^{2}}\left(
| |
| − | \frac{\partial}{\partial r}+\frac{1}{r}\right) \frac{\partial^{2}}
| |
| − | {\partial\theta^{2}}\right] w=0 \,\,\,(4)
| |
| − | </math>
| |
| − | </center>
| |
| − | where <math>w</math> is the time-independent surface
| |
| − | displacement, <math>\nu</math> is Poisson's ratio, and <math>\bar{\Delta}</math> is the
| |
| − | polar coordinate Laplacian
| |
| − | <center>
| |
| − | <math>
| |
| − | \bar{\Delta}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial
| |
| − | }{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}
| |
| − | </math>
| |
| − | </center>
| |
| − | == Displacement of the plate ==
| |
| − | | |
| − | The surface displacement and the water velocity potential at
| |
| − | the water surface are linked through the kinematic boundary condition
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi_{z}=-i\sqrt{\alpha}w,\,\,\,z=0
| |
| − | </math>
| |
| − | </center>
| |
| − | From equations (\ref{bvp_plate}) the potential and the surface
| |
| − | displacement are therefore related by
| |
| − | <center>
| |
| − | <math>
| |
| − | w=i\sqrt{\alpha}\phi,\quad r>a
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | (\beta\bar{\Delta}^{2}+1-\alpha\gamma)w=i\sqrt{\alpha}\phi,\quad r<a
| |
| − | </math>
| |
| − | </center>
| |
| − | The surface displacement can also be expanded in eigenfunctions
| |
| − | as
| |
| − | <center>
| |
| − | <math>
| |
| − | w(r,\theta)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}i\sqrt{\alpha}
| |
| − | a_{mn}K_{n}(k_{m}r)e^{i n\theta},\;\;r>a
| |
| − | </math>
| |
| − | </center>
| |
| − | and:
| |
| − | <center>
| |
| − | <math>
| |
| − | w(r,\theta)=
| |
| − | \sum_{n=-\infty}^{\infty}\sum_{m=-2}^{\infty}i\sqrt{\alpha}(\beta\kappa
| |
| − | _{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}I_{n}(\kappa_{m}r)e^{i
| |
| − | n\theta},\; r<a
| |
| − | </math>
| |
| − | </center>
| |
| − | using the fact that
| |
| − | <center>
| |
| − | <math>
| |
| − | \bar{\Delta}\left( I_{n}(\kappa_{m}r)e^{i n\theta}\right) =\kappa_{m}
| |
| − | ^{2}I_{n}(\kappa_{m}r)e^{i n\theta}\,\,\,(5)
| |
| − | </math>
| |
| − | </center>
| |
| − | | |
| − | ==An infinite dimensional system of equations==
| |
| − | | |
| − | The boundary conditions (3) and
| |
| − | (4) can be expressed in terms of the potential
| |
| − | using (5). Since the angular modes are uncoupled the
| |
| − | conditions apply to each mode, giving
| |
| − | <center>
| |
| − | <math>
| |
| − | \sum_{m=-2}^{\infty}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
| |
| − | \left(\kappa_{m}^{2}I_{n}(\kappa_{m}a)-\frac{1-\nu}{a}\left(\kappa
| |
| − | _{m}I_{n}^{\prime}(\kappa_{m}a)-\frac{n^{2}}{a}I_{n}(\kappa_{m}a)\right)
| |
| − | \right) =0\,\,\,(6)
| |
| − | </math>
| |
| − | </center>
| |
| − | <center>
| |
| − | <math>
| |
| − | \sum_{m=-2}^{\infty}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
| |
| − | \left( \kappa_{m}^{3}I_{n}^{\prime}(\kappa_{m}a)+n^{2}\frac{1-\nu}{a^{2}
| |
| − | }\left( \kappa_{m}I_{n}^{\prime}(\kappa_{m}a)+\frac{1}{a}I_{n}(\kappa
| |
| − | _{m}a)\right) \right)
| |
| − | =0\,\,\,(7)
| |
| − | </math>
| |
| − | </center>
| |
| − | The potential and its derivative must be continuous across the
| |
| − | transition from open water to the plate covered region. Therefore, the
| |
| − | potentials and their derivatives at <math>r=a</math> have to be equal.
| |
| − | Again we know that this must be true for each angle and we obtain
| |
| − | <center>
| |
| − | <math>
| |
| − | e_{n}I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
| |
| − | a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right)
| |
| − | =\sum_{m=-2}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\psi_{m}(z)
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | e_{n}k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum
| |
| − | _{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)
| |
| − | =\sum_{m=-2}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\psi
| |
| − | _{m}(z)
| |
| − | </math>
| |
| − | </center>
| |
| − | for each <math>n</math>.
| |
| − | We solve these equations by multiplying both equations by
| |
| − | <math>\phi_{l}(z)</math> and integrating from <math>-H</math> to <math>0</math> to obtain:
| |
| − | <center>
| |
| − | <math>
| |
| − | e_{n}I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}
| |
| − | =\sum_{m=-2}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml} \,\,\,(8)
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | e_{n}k_{0}I_{n}^{\prime}(k_{0}a)A_{0}\delta_{0l}+a_{ln}k_{l}K_{n}^{\prime
| |
| − | }(k_{l}a)A_{l}
| |
| − | =\sum_{m=-2}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}
| |
| − | a)B_{ml} \,\,\,(9)
| |
| − | </math>
| |
| − | </center>
| |
| − | Equation (8) can be solved for the open water
| |
| − | coefficients <math>a_{mn}</math>
| |
| − | <center>
| |
| − | <math>
| |
| − | a_{ln}=-e_{n}\frac{I_{n}(k_{0}a)}{K_{n}(k_{0}a)}\delta_{0l}+\sum
| |
| − | _{m=-2}^{\infty}b_{mn}\frac{I_{n}(\kappa_{m}a)B_{ml}}{K_{n}(k_{l}a)A_{l}}
| |
| − | </math>
| |
| − | </center>
| |
| − | which can then be substituted into equation
| |
| − | (9) to give us
| |
| − | <center>
| |
| − | <math>
| |
| − | \left( k_{0}I_{n}^{\prime}(k_{0}a)-k_{0}\frac{K_{n}^{\prime}(k_{0}
| |
| − | a)}{K_{n}(k_{0}a)}I_{n}(k_{0}a)\right) e_{n}A_{0}\delta_{0l}
| |
| − | =\sum_{m=-2}^{\infty}\left( \kappa_{m}I_{n}^{\prime}(\kappa_{m}
| |
| − | a)-k_{l}\frac{K_{n}^{\prime}(k_{l}a)}{K_{n}(k_{l}a)}I_{n}(\kappa
| |
| − | _{m}a)\right) B_{ml}b_{mn}\,\,\,(10)
| |
| − | </math>
| |
| − | </center>
| |
| − | for each <math>n</math>.
| |
| − | Together with equations (6) and (7)
| |
| − | equation (10) gives the required equations to solve for the
| |
| − | coefficients of the water velocity potential in the plate covered region.
| |
| − | | |
| − | =Numerical Solution=
| |
| − | | |
| − | To solve the system of equations (10) together
| |
| − | with the boundary conditions (6 and 7) we set the upper limit of <math>l</math> to
| |
| − | be <math>M</math>. We also set the angular expansion to be from
| |
| − | <math>n=-N</math> to <math>N</math>. This gives us
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi(r,\theta,z)=\sum_{n=-N}^{N}\sum_{m=0}^{M}a_{mn}K_{n}(k_{m}r)e^{i
| |
| − | n\theta }\phi_{m}(z), \;\;r>a
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | \phi(r,\theta,z)=\sum_{n=-N}^{N}\sum_{m=-2}^{M}b_{mn}I_{n}(\kappa
| |
| − | _{m}r)e^{i n\theta}\psi_{m}(z), \;\;r<a
| |
| − | </math>
| |
| − | </center>
| |
| − | Since <math>l</math> is an integer with <math>0\leq l\leq
| |
| − | M</math> this leads to a system of <math>M+1</math> equations.
| |
| − | The number of unknowns is <math>M+3</math> and the two extra equations
| |
| − | are obtained from the boundary conditions for the free plate (6)
| |
| − | and (7). The equations to be solved for each <math>n</math> are
| |
| − | <center>
| |
| − | <math>
| |
| − | \left( k_{0}I_{n}^{\prime}(k_{0}a)-k_{0}\frac{K_{n}^{\prime}(k_{0}
| |
| − | a)}{K_{n}(k_{0}a)}I_{n}(k_{0}a)\right) e_{n}A_{0}\delta_{0l}
| |
| − | =\sum_{m=-2}^{M}\left( \kappa_{m}I_{n}^{\prime}(\kappa_{m}a)-k_{l}
| |
| − | \frac{K_{n}^{\prime}(k_{l}a)}{K_{n}(k_{l}a)}I_{n}(\kappa_{m}a)\right)
| |
| − | B_{ml}b_{mn}
| |
| − | </math>
| |
| − | </center>
| |
| − | <center>
| |
| − | <math>
| |
| − | \sum_{m=-2}^{M}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
| |
| − | \left( \kappa_{m}^{2}I_{n}(\kappa_{m}a)-\frac{1-\nu}{a}\left( \kappa
| |
| − | _{m}I_{n}^{\prime}(\kappa_{m}a)-\frac{n^{2}}{a}I_{n}(\kappa_{m}a)\right)
| |
| − | \right) =0
| |
| − | </math>
| |
| − | </center>
| |
| − | and
| |
| − | <center>
| |
| − | <math>
| |
| − | \sum_{m=-2}^{M}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
| |
| − | \left( \kappa_{m}^{3}I_{n}^{\prime}(\kappa_{m}a)+n^{2}\frac{1-\nu}{a^{2}
| |
| − | }\left( \kappa_{m}I_{n}^{\prime}(\kappa_{m}a)+\frac{1}{a}I_{n}(\kappa
| |
| − | _{m}a)\right) \right) =0
| |
| − | </math>
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| − | </center>
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| − | It should be noted that the solutions for positive and negative
| |
| − | <math>n</math> are identical so that they do not both need to be
| |
| − | calculated. There are some minor simplifications which are a consequence of
| |
| − | this which are discussed in more detail in [[Zilman_Miloh 2000a|Zilman and Miloh 2000]].
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| − | | |
| − | =The [[Shallow Depth]] Theory of [[Zilman_Miloh_2000a|Zilman and Miloh 2000]]=
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| − | | |
| − | The shallow water theory of [[Zilman_Miloh_2000a|Zilman and Miloh 2000]] can be recovered by
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| − | simply setting the depth shallow enough that the shallow water theory is valid
| |
| − | and setting <math>M=0</math>. If the shallow water theory is valid then
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| − | the first three roots of the dispersion equation for the ice will be exactly
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| − | the same roots found in the shallow water theory by solving the polynomial
| |
| − | equation. The system of equations has four unknowns (three under the plate and
| |
| − | one in the open water) exactly as for the theory of [[Zilman_Miloh_2000a|Zilman and Miloh 2000]].
| |
| − | | |
| − | =Numerical Results=
| |
| − | | |
| − | [[Image:ComparisionH25.jpg|thumb|right|300px|Figure 1]]
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| − | | |
| − | We present solutions for a plate of radius <math>a=100</math>. The wavelength is
| |
| − | <math>\lambda=50</math> (recall that <math>\alpha=2\pi/\lambda\tanh\left( 2\pi
| |
| − | H/\lambda\right)</math>), <math>\beta=10^{5}</math> and
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| − | <math>\gamma=0</math>.
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| − | We compare with the method
| |
| − | presented in [[Meylan_2002a|Meylan 2002]] for an arbitrary shaped plate modified to compute
| |
| − | the solution for finite depth. The circle is represented in this scheme by
| |
| − | square panels which are arranged to, as nearly as possible, form a circular
| |
| − | shape.
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| − | | |
| − | Figure 1 shows the
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| − | real part (a and c) and imaginary part (b and d) of the displacement for depth
| |
| − | <math>H=25</math>. The number
| |
| − | of points in the angular expansion is <math>N=16</math>. The number of
| |
| − | roots of the dispersion equation is <math>M=8</math>. Plots (a) and
| |
| − | (b) are calculated using the circular plate method described here. Plots (c)
| |
| − | and (d) are calculated using an arbitrary shaped plate method, with the
| |
| − | panels shown being the actual panels used in the calculation. We see the
| |
| − | expected agreement between the two methods.
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| − | | |
| − | | |
| − | The table below shows the values of the coefficients
| |
| − | <math>b_{mn}</math> for the case for previous case (<math>\lambda=50</math>,
| |
| − | <math>a=100</math>, <math>\beta=10^5</math>, <math>\gamma=0</math>, and <math>H=25</math>). The very rapid
| |
| − | decay of the higher evanescent modes is apparent. This shows how efficient this method of
| |
| − | solution is since only a small number of modes are required.
| |
| − | | |
| − | <blockquote style="background: white; padding: 0em;">
| |
| − | <table border="1">
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| − | <tr>
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| − | <td> <math>b_{mn}</math> </td><td> <math>n=0</math> </td>
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| − | <td> <math>n=1</math> </td><td> <math>n=2</math> </td><td> <math>n=3</math> </td>
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| − | </tr>
| |
| − | <tr>
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| − | <td><math>m=-2</math></td> <td><math>1.32 \!\times\!10^{-1}-9.71 \!\times\!10^{-1}i</math> </td>
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| − | <td> <math>6.85 \!\times\!10^{-1} -6.37 \!\times\!10^{-1}i</math></td>
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| − | <td> <math>2.95 \!\times\!10^{-1}-1.12 \!\times\!10^{0}i</math> </td><td> <math>6.09 \!\times\!10^{-1}
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| − | -4.95 \!\times\!10^{-1}i</math> </td></tr>
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| − | | |
| − | <tr><td><math>m=-1</math> </td><td> <math>-6.38 \!\times\!10^{-5}+1.47 \!\times\!10^{-3}i</math> </td><td>
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| − | <math>-3.92 \!\times\!10^{-3} + 3.99 \!\times\!10^{-3}i</math></td>
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| − | <td> <math>1.41 \!\times\!10^{-3}+2.82 \!\times\!10^{-3}i</math> </td><td>
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| − | <math>-4.28 \!\times\!10^{-3} +3.89 \!\times\!10^{-3}i</math></td></tr>
| |
| − | | |
| − | <tr><td><math>m=0</math> </td><td>
| |
| − | <math>-3.29 \!\times\!10^{-4}+1.43 \!\times\!10^{-3}i</math> </td><td> <math>4.26 \!\times\!10^{-3}
| |
| − | -3.62 \!\times\!10^{-3}i</math></td><td>
| |
| − | <math>-2.62 \!\times\!10^{-3}+1.76 \!\times\!10^{-3}i</math> </td><td> <math>4.68 \!\times\!10^{-3}
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| − | -3.39 \!\times\!10^{-3}i</math></td></tr>
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| − | | |
| − | <tr><td><math>m=1</math> </td><td> <math>4.31 \!\times\!10^{-7}-3.18 \!\times\!10^{-6}i</math> </td><td>
| |
| − | <math>-6.64 \!\times\!10^{-6} -7.14 \!\times\!10^{-6}i</math></td>
| |
| − | <td> <math>2.07 \!\times\!10^{-7}-7.89 \!\times\!10^{-7}i</math> </td><td>
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| − | <math>-6.30 \!\times\!10^{-6} -7.74 \!\times\!10^{-6}i</math></td></tr>
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| − | | |
| − | <tr><td><math>m=2</math> </td><td>
| |
| − | <math>6.79 \!\times\!10^{-13}-5.01 \!\times\!10^{-12}i</math> </td><td>
| |
| − | <math>-5.78 \!\times\!10^{-12}-6.21 \!\times\!10^{-12}i</math></td>
| |
| − | <td> <math>8.87 \!\times\!10^{-13}-3.38 \!\times\!10^{-12}i</math> </td><td>
| |
| − | <math>-5.54 \!\times\!10^{-12}-6.81 \!\times\!10^{-12}i</math></td></tr>
| |
| − | | |
| − | <tr><td><math>m=3</math> </td><td> <math>1.35 \!\times\!10^{-18}-9.95 \!\times\!10^{-18}i</math> </td><td>
| |
| − | <math>-9.69 \!\times\!10^{-18}-1.04 \!\times\!10^{-17}i</math></td>
| |
| − | <td> <math>1.94 \!\times\!10^{-18}-7.39 \!\times\!10^{-18}i</math> </td><td>
| |
| − | <math>-9.37 \!\times\!10^{-18}-1.15 \!\times\!10^{-17}i</math></td>
| |
| − | </tr>
| |
| − | </table>
| |
| − | </blockquote>
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| − | | |
| − | | |
| − | [[Category:Problems with Cylindrical Symmetry]] | |
| | [[Category:Linear Hydroelasticity]] | | [[Category:Linear Hydroelasticity]] |
