\chapter{Mathematical Formulation of the Problem}(1)

In constructing a model for wave propagation in a sea-ice sheet, we
are mindful of the considerable literature that suggests that
the behaviour of the sheet may be represented as a thin elastic
plate at the strains and strain rates involved, as originally
proposed by Greenhill (1887, 1916), and discussed in \C1.
While some damping of the
waves undoubtedly occurs---indeed viscoelastic plates have
occasionally been employed---the additional complication of
including viscosity is unwarranted here. We are helped
considerably by the observation that the wavelength of
ice-coupled, flexural-gravity oscillations is always much
greater than the ice thickness or the submergence even when the
wave period is quite short. Accordingly, we are generally at ease that
sea ice can be described by a Euler-Bernoulli thin plate
floating with zero draft on the water surface at <math>z=0</math>. Refer to the beginnings of Chapters~6 and 8 for further discussions on the implications of this assumption.\\

Figure~2 shows the physical situation to be modelled,
where the axes have been displaced to the right to avoid
clutter. A monochromatic sinusoidal wave travelling beneath a uniform
ice sheet of thickness <math>h_0</math>, approaches a region with variable
properties at an angle <math>\theta</math> to normal incidence. There it is
partially reflected and partially transmitted into the region beneath a
second uniform ice sheet, of thickness <math>h_2</math>.
The purpose of this thesis is to determine how the relative amounts
of reflection and transmission that occur vary with parameters such as
<math>\theta</math>, <math>h_0</math>, <math>h_2</math> and wave period, and also with the properties of the variable region.

\begin{figure}[H]\begin{center}
\includegraphics{figs/ramp/schem-ramp.eps}
\caption[Schematic showing the general physical situation to be modelled in this thesis.]{The general physical situation to be modelled in this thesis: the scattering that occurs as an obliquely incident plane wave travelling beneath one uniform ice sheet arrives at a region of ice with a variable thickness, and is partially reflected and partially transmitted into a second uniform ice sheet. The ice on the left has a constant thickness of <math>h_0</math>, the ice on the right has a constant thickness of <math>h_2</math>, and the thickness of the central feature is represented by the function <math>h_1(x)</math>. The sea water has a finite, constant depth of <math>H</math>. The coordinate axes are oriented as shown, but should be displaced to the left so that the <math>x=0</math> line actually corresponds to the left hand limit of the ramp. <math>|I|</math>, which will generally be taken to be 1, is the amplitude of the velocity potential corresponding to the incident wave, while <math>|R|</math> and <math>|T|</math>, the moduli of reflection and transmission coefficients, are the amplitudes of the velocity potentials corresponding to the reflected and transmitted waves (respectively).}
(2)\end{center}
\end{figure}

=Equations and boundary conditions=(3)
The water beneath the ice is taken to have constant density and
to be of depth <math>H</math>. Assuming also that the fluid flow is irrotational,
it may be represented by a velocity potential <math>\Phi</math> so that
the fluid velocity <math>{\mathbf v}=\nabla\Phi</math>. The potential <math>\Phi</math> then
satisfies a continuity equation (Laplace's equation), Bernoulli's equation (linearized), requirements that horizontal flow be continuous, a kinematic condition (also linearized), and a no-flow condition at the sea-floor, as follows:
\begin{mathletters}(4)
<center><math>\begin{matrix}
\nabla_{xyz}^2\Phi(x,y,z,t)=0,(5)\\
\rho_w\Phi_t(x,y,0,t)+P(x,y,t)-\rho_w g\eta(x,y,t)=0,(6)\\
\Phi_x(x^+,y,z,t)-\Phi_x(x^-,y,z,t)
=\Phi(x^+,y,z,t)-\Phi(x^-,y,z,t)=0,(7)\\
\Phi_x(x,y^+,z,t)-\Phi_x(x,y^-,z,t)
=\Phi(x,y^+,z,t)-\Phi(x,y^-,z,t)=0,(8)\\
\Phi_z(x,y,0,t)-\eta_t(x,y,t)=0,(9)\\
\Phi_z(x,y,H,t)=0(10).
\end{matrix}</math></center>\end{mathletters}
In the above <math>\rho_w</math> is the density of water, <math>P</math> is the water pressure at the
surface, <math>g</math> is the acceleration due to gravity, <math>H</math> is the water depth, and
<math>\eta</math> is the vertical displacement of the free surface. In (7), <math>\Phi(x^\pm,y,z,t)</math> is shorthand for <math>\lim_{\xi\ar x^\pm}\Phi(\xi,y,z,t)</math>, and similar notation is used for <math>\Phi_x(x^\pm,y,z,t)</math>; similarly for <math>y^\pm</math> in (8). These conditions generally only need to be applied at dicontinuities in the ice or in its properties.\\

The pressure is related to the displacement by the Euler-Bernoulli thin plate equation (derived for a spatially varying flexural rigidity in Appendix A) by
<center><math>(11)
m(x)\eta_{tt}(x,y,t)+\Lop_{\rm plate}(x,\pa_x,\pa_y)\eta(x,y,t)+P(x,y,t)-P_\textrm{a}=0,
</math></center>
where <math>P_\textrm{a}</math> is the atmospheric pressure, <math>m(x)=[\rho h](x)</math>
is the mass per unit surface area of the ice, and the plate operator <math>\Lop_{\rm plate}</math> is given by
<center><math>\begin{matrix}
\Lop_{\rm plate}(x,\pa_x,\pa_y)&=\nabla_{xy}^2\big(D(x)\nabla_{xy}^2\big)
-(1-\nu)D''(x)\pa_y^2.
\end{matrix}</math></center>
<math>D(x)=[E\,h^3/12(1-\nu^2)](x)</math> is the flexural rigidity of the
ice; <math>E</math>, <math>h</math>, <math>\nu</math> and <math>\rho</math> are respectively the effective
(low strain rate) elastic modulus, the thickness, Poisson's
ratio, and the density of the ice. Any of these quantities may
be varied but, as the most common source of variation in sea-ice
is its thickness, the others will be treated as constants. The values used
for the different physical constants are given in Table~12
below.\\

\begin{table}[H]
\begin{center}
\begin{tabular}{|c|l|l|}\hline
Symbol       &Description           &Value Used\\\hline
<math>E</math>      &Young's modulus of ice        &<math>6\e9\,{\rm Pa}</math>\\
<math>\nu</math>        &Poisson's ratio of ice        &0.3\\
<math>\rho</math>       &Density of ice            &<math>922.5\,{\rm kg\,m^{-3}}</math>\\
<math>\rho_w</math>     &Density of water          &<math>1025\,{\rm kg\,m^{-3}}</math>\\
<math>g</math>      &Acceleration due to gravity   &<math>9.81\,{\rm m\,s^{-2}}</math>\\\hline
\end{tabular}\end{center}
\caption{Values for physical constants used in this thesis.}
(12)
\end{table}

The thickness will be given in a piecewise manner by
<center><math>\begin{matrix}
h(x)=\begin{cases}
h_0\quad&=for <math>x<0</math>= \\
h_1(x)\quad&=for <math>0<x<a</math>= \\
h_2\quad&=for <math>a<x</math>= ,
\end{cases}
\end{matrix}</math></center>
where <math>h_0</math> and <math>h_2</math> are constant, and the rigidity <math>D(x)</math> and the mass per unit area <math>m(x)</math> are given analogously. Inside
<math>(0,a)</math> the only restriction on possible thickness profiles is
that the function <math>D_1(x)</math> must have an integrable second
derivative, be piecewise continuous itself, and must have a
piecewise continuous first derivative.\\

Without the imposition of additional conditions, the system of equations (4) and (11) does not have a unique solution. To reduce the dimension of the problem to two instead of four we assume that our potential and displacement are harmonic in time and the <math>y</math> coordinate. We assume we can describe the y dependence of the problem in this way because it is invariant under translations in that direction. This is made more explicit in equation (27) below (cf. p\pageref{assum}).\\

We must also apply some radiation conditions as <math>x\ar\pm\infty</math> and some conditions at any edges where <math>D(x)</math> and/or <math>D'(x)</math>  are discontinuous. These edge conditions are described below, but we will until \Sr{sec-greens-thm} to make the radiation conditions explicit.

==Edge Conditions==(13)
Let <math>X_1</math> be the set of points at which there is a free edge, <math>X_0</math> be the set of points at which there is no free edge but either a discontinuity in <math>D(x)</math> or <math>D'(x)</math>, or both, and let <math>X_c=X_0\cup X_1</math>. The members of <math>X_c</math> will loosely be referred to as critical points. In most situations we will consider <math>X_c=\{0,a\}</math>, although in Chapter~\ref{chap-ridge} we will also present results for when <math>X_c</math>, has up to three other members inside <math>(0,a)</math>. In addition, its size doubles in \C9 when the scattering by two features is considered. Both sets of edge conditions used are energy conserving, i.e. no work is done at the edges, so that no energy is lost (or gained) there.\\

From Appendix~\ref{app-eqns}, at points <math>x_c\in X_0</math> the displacement <math>\eta</math> should satisfy the following conditions, which will be referred to as the frozen edge conditions:
\begin{mathletters}(14)
<center><math>\begin{matrix}
\eta(x_c^+,y,t)&=\eta(x_c^-,y,t),(15)\\
\eta_x(x_c^+,y,t)&=\eta_x(x_c^-,y,t),(16)\\
\Mop_x(x_c^+,\pa_x,\pa_y)\,\eta(x_c^+,y,t)&=
\Mop_x(x_c^-,\pa_x,\pa_y)\,\eta(x_c^-,y,t),(17)\\
\Sop_{xy}(x_c^+,\pa_x,\pa_y)\,\eta(x_c^+,y,t)&=
\Sop_{xy}(x_c^-,\pa_x,\pa_y)\,\eta(x_c^-,y,t),(18)
\end{matrix}</math></center>
\end{mathletters}
where the operators <math>\Mop_x</math> and <math>\Sop_{xy}</math> are
respectively the bending moment and
vertical edge force operators, defined
\begin{mathletters}(19)
<center><math>\begin{matrix}
\Mop_x(x,\pa_x,\pa_y)&=-D(x)\Lop_x^-(\pa_x,\pa_y),(20)\\
\Sop_{xy}(x,\pa_x,\pa_y)&=D(x)\Lop_x^+(\pa_x,\pa_y)
\pa_x+{D'(x)}\Lop_x^-(\pa_x,\pa_y),(21)\\
\Lop_x^\pm(\pa_x,\pa_y)&=\nabla^2_{xy}\pm(1-\nu)\pa^2_y.(22)
\end{matrix}</math></center>
\end{mathletters}
The subscripts in the above operators are used only for consistency with Appendix~\ref{app-eqns} (although they do have some small significance there).\\

Referring to Figure~\ref{fig-moments}, the moments <math>\Mop_x(x_c^\pm,\pa_x,\pa_y)\,\eta(x_c^\pm,y,t)</math> respectively produce clockwise and anticlockwise rotations at the crack edges. Thus (17) implies that these moments cancel each other out and consequently are unable to do any work. Similarly, the vertical edge forces <math>\Sop_{xy}(x_c^\pm,\pa_x,\pa_y)\,\eta(x_c^\pm,y,t)</math> respectively produce downwards and upwards displacements in the edges, so (18) implies that no translational work is done there either. Conditions (15) and (16) require that the displacement and its slope are continuous across the extremities.\\

At free edges, that is at all points <math>x_c\in X_1</math>, the bending moments and vertical edge forces should actually vanish independently, as opposed to just cancelling. Consequently
\begin{mathletters}(23)
<center><math>\begin{matrix}
\Mop_x(x_c^+,\pa_x,\pa_y)\,\eta(x_c^+,y,t)&=
\Mop_x(x_c^-,\pa_x,\pa_y)\,\eta(x_c^-,y,t)=0,(24)\\
\Sop_{xy}(x_c^+,\pa_x,\pa_y)\,\eta(x_c^+,y,t)&=
\Sop_{xy}(x_c^-,\pa_x,\pa_y)\,\eta(x_c^-,y,t)=0.(25)
\end{matrix}</math></center>
\end{mathletters}
Both sets of equations are derived in Appendix~\ref{app-eqns}.

=Nondimensionalization=(26)
For the <math>j^{\rm th}</math> region of ice we define the
characteristic length <math>L_j=\sqrt[4]{D_j/\rho_w g}</math> and the
characteristic time <math>\tau_j=\sqrt{L_j/g}</math>, after \can{Fox00}.
These quantities will be important in the parameterization of the
results obtained.\\

Now, let <math>h_M={\rm max}\{h_j|\,j=0,1,2\,\&\,h'_j(x)=0\}</math>. Also denote by <math>D_M</math>, <math>m_M</math>, <math>L_M</math> and <math>\tau_M</math> the flexural rigidity, mass per unit surface area, characteristic length and characteristic time of ice with thickness <math>h_M</math>. We shall scale times with respect to <math>\tau_M.</math>\\

Then let us take the incident wave to be monochromatic, with radial
frequency <math>\omega</math> and wave period <math>\tau</math>. The natural length that we shall nondimensionalize lengths with respect to is defined as <math>L=L_M\bar\omega^{-2/5}</math>, where
<math>\bar\omega=\omega\tau_M</math>. <math>L</math> has the property that <math>L^5=D_M/\rho_w\om^2.</math> The reason for choosing this natural length is that it allows the dispersion relation for a uniform sheet of ice to be characterized by a single parameter (other than the water depth) that combines the effects of wave period and ice thickness. This is discussed further in Sections~44 and \ref{sec-ck-res-pram}.\\

We now define the following nondimensionalized quantities
<center><math>(\bar x,\bar y,\bar z)=(x,y,z)/L,\quad\bar t=t/\tau_M,\quad\bar\Phi=
\tau_M\Phi/{L^2},\quad\bar\eta=\eta/L,</math></center>
<center><math>\overline{D}_j=D_j/D_M,\quad\overline m_j=m_j/m_M,\quad\bar a=a/L,\quad\overline H=H/L,\quad\overline X_c=X_c/L</math></center>
(the notation <math>X_c/L</math> indicates that each element of the set <math>X_c</math> should be scaled by <math>L</math>), and
<center><math>\lambda=\bar\omega^{-8/5},\quad\sigma(h_M)=m_M/\rho_wL_M,\quad\mu(h_M,\lambda)=\sigma\bar\omega^{2/5}
=\sigma/\lambda^{1/4},</math></center>
We have nondimensionalized with respect to <math>h_M</math> because the natural length and other quantities such as the wavelength increase with ice thickness. Consequently, a distance that is quite large with respect to thinner ice, may not be as significant in comparison with the characteristic lengths of thicker ice. For example, in \Sr{sec-ck-res-H}, we establish how large the nondimensional water depth <math>\overline{H}</math> must be for the infinite depth approximation to be valid, so by choosing to nondimensionalize with respect to the maximum average thickness we are ensuring that the water is deep with respect to each different region of ice.\\

Since the geometry of the problem is invariant in the
<math>y</math>-direction the component of the wave propagating in that
direction is unaffected by the feature, so we may concentrate on
velocity potentials of the form
<center><math>(27)
\bar\Phi(x,y,z,t)=\re{\left[\phi(\bar x,\bar z)
\rme^{\ri(l\bar y-\bar\omega\bar t\,)}\right]},
</math></center>
where <math>\re[.]</math> denotes the real part and we have also
taken into account the timewise periodicity  of the forcing from the
incident wave. <math>l</math> is the wavenumber of the incident wave in the <math>y</math> direction. Note that assuming (27) forces <math>\Phi</math> to automatically satisfy (8).\\

Dropping the overbars for clarity, the function
<math>\phi(x,z)</math> must satisfy the following system of equations
\begin{mathletters}(28)
<center><math>\begin{matrix}
\big(\nabla^2_{xz}-l^2\big)\phi(x,z)&=0, (29)\\
\Lop(x,\pa_x)\phi_z(x,0)+\phi(x,0)&=0,(30)\\
\phi_x(x^+,z)-\phi_x(x^-,z)=\phi(x^+,z)-\phi(x^-,z)&=0,(31)\\
\phi_z(x,H)&=0.(32)
\end{matrix}</math></center>
\end{mathletters}
The differential operator <math>\Lop</math> is given by
<center><math>\begin{matrix}(33)
\Lop(x,\pa_x)&=(\pa_x^2-l^2)\big(D(x)(\pa_x^2-l^2)\big)
+(1-\nu)l^2D''(x)+\lambda-m(x)\mu-\ri\eps,
\end{matrix}</math></center>
where <math>\eps=0</math>. Although it seems pointless to define <math>\eps</math> in this way, it was found to be easier to find a <math>\phi</math> which satisfies (30) for an arbitrarily small positive value of <math>\eps</math> first, and then to take the limit as <math>\eps\ar0</math>. (In particular, the application of Green's theorem in \Sr{sec-greens-thm} is simplified greatly by doing this.) This can be done analytically, and we verify the legitimacy of this procedure in \Sr{sec-greens-thm}.\\

When <math>x\not\in(0,a)</math> (i.e. <math>j=0,2</math>), and the ice properties are constant, <math>\Lop</math> simplifies to <math>\Lop_j(\pa_x)=D_j(\pa_x^2-l^2)^2+\lambda-m_j\mu-\ri\eps</math>
and is referred to as <math>\Lop_1(x,\pa_x)</math> when <math>x\in(0,a)</math>. It can be broken up into differential operators that are independent of <math>x</math> by writing
<center><math>
\Lop_1(x,\pa_x)=\Lop_0(\pa_x)+\sum_{j=1}^4d_j(x)\Lop_{1j}(\pa_x),
</math></center>
where <math>\Lop_{11}(\pa_x)=(\pa_x^2-l^2)^2</math>, <math>\Lop_{12}(\pa_x)=(\pa_x^2-l^2)\pa_x</math>, <math>\Lop_{13}(\pa_x)=\bs{\cal L}^-(\pa_x)</math> and <math>\Lop_{14}(\pa_x)=-\mu</math>. Their multipliers are <math>d_1(x)=D_0(d_0(x)-1)</math>, <math>d_2(x)=2D_1'(x)</math>, <math>d_3(x)=D_1''(x)</math> and
<math>d_4(x)=m_1(x)-m_0</math>, where <math>d_0(x)=D_1(x)/D_0</math>. Also, the operator <math>\Lop^-</math> in <math>\Lop_{13}</math> can be obtained from
<center><math>\begin{matrix}(34)
\Lop^\pm(\pa_x)&=\Lop_x^\pm(\pa_x,\ri l)=(\pa_x^2-l^2)\mp(1-\nu)l^2
\end{matrix}</math></center>
(cf. equation 22), while equation (30) results from differentiating (6) with respect to time, and then using (9) and (11) to eliminate <math>\eta</math> and <math>P</math>.\\

The edge conditions to be satisfied can also be put in terms of <math>\phi_z(x,0)</math>. The frozen edge conditions become
\begin{mathletters}(35)
<center><math>\begin{matrix}
\phi_z(x_c^+,0)&=\phi_z(x_c^-,0),(36)\\
\phi_{zx}(x_c^+,0)&=\phi_{zx}(x_c^-,0),(37)\\
\Mop(x_c^+,\pa_x)\,\phi_z(x_c^+,0)&=
\Mop(x_c^-,\pa_x)\,\phi_z(x_c^-,0),(38)\\
\Sop(x_c^+,\pa_x)\,\phi_z(x_c^+,0)&=
\Sop(x_c^-,\pa_x)\,\phi_z(x_c^-,0),(39)
\end{matrix}</math></center>
\end{mathletters}
while the free edge conditions become
\begin{mathletters}(40)
<center><math>\begin{matrix}
\Mop(x_c^+,\pa_x)\,\phi_z(x_c^+,0)&=
\Mop(x_c^-,\pa_x)\,\phi_z(x_c^-,0)=0,(41)\\
\Sop(x_c^+,\pa_x)\,\phi_z(x_c^+,0)&=
\Sop(x_c^-,\pa_x)\,\phi_z(x_c^-,0)=0.(42)
\end{matrix}</math></center>
\end{mathletters}
The non-dimensional operators <math>\Mop</math> and <math>\Sop</math> are given by
\begin{mathletters}(43)
<center><math>\begin{matrix}
\Mop(x,\pa_x)&=\frac1{D_M}\Mop_x(x,\pa_x,\ri l)=-D(x)\Lop^-(\pa_x),\\
\Sop(x,\pa_x)&=\frac1{D_M}\Sop_{xy}(x,\pa_x,\ri l)
=D(x)\Lop^+(\pa_x)\pa_x+D'(x)\Lop^-(\pa_x),
\end{matrix}</math></center>
\end{mathletters}
referring to (19) and (34).

==Important Parameters==(44)
Of the parameters in the differential operators <math>\Lop_j</math>, it will turn out that
the ratios <math>m_j=h_j/h_M</math> (which also determine the nondimensional rigidities <math> D_j=m_j^3</math>) will be the most important in summarizing the scattering properties of the different features modelled. Note that when talking about <math>m_1(x)</math> and <math>D_1(x)</math>, we will discuss them in terms of properties such as their average value, their own and their derivatives' continuity, and the width <math>a</math> of the region that they vary over. This will be discussed further in Chapter~\ref{chap-ridge}.\\

The properties of the wave itself and of the background ice sheet will turn out to be best described by the quantities <math>l</math>, and <math>\lambda</math>. Using the former as a parameter is equivalent to using the angle of incidence <math>\theta</math>, while the latter incorporates the effects of both the wave period and the background thickness.\\

The remaining parameter <math>\mu</math> is less important because it will be shown to be almost entirely a function of <math>\lambda</math>. From its definition it depends on the actual value of the background thickness <math>h_M</math> as well (through the function <math>\sig(h_M)</math>); however <math>\sig</math> is so weakly dependent on its argument that it may essentially be taken as being constant. (<math>\sigma\varpropto h_M^{1/4}</math>, and we will generally only use values for <math>h_M</math> between 0.5\,m and 10\,m; this corresponds to the very small range of <math>\sig</math> values of between about 0.05 and 0.11, compared to the much larger range for <math>\lam</math> of between about <math>10^{-4}</math> and 4.5.) Accordingly, by choosing a representative value for <math>h_M</math>, which we shall denote by <math>h_M'</math>, <math>\sig</math> may be fixed by setting it to <math>\sig'=\sig(h_M')</math>, allowing us to summarize approximately the combined effects of <math>h_0</math> and wave period with the single parameter <math>\lambda</math>. We shall generally choose <math>h_M'</math> so that <math>h_0</math>, which we shall refer to as the background ice thickness, is equal to 1\,m, although certain situations will be discussed where a small or even zero value for <math>h_0</math> is required; in that case a more appropriate thickness shall be forced to be 1\,m. (For example, when discussing the scattering by a finite strip of ice in \Sr{sec-floe}, in which <math>h_0=h_2=0</math>, we shall set <math>h_1=1</math>\,m.)\\

Using <math>\lambda</math> in this way is equivalent to using the nondimensional period <math>\tau</math> or the nondimensional frequency <math>\omega</math>. The main consequence of being able to do this is that scattering results for a maximum thickness <math>h_M</math> (with characteristic time <math>\tau_M</math>) may be approximated extremely easily if the results for our reference thickness <math>h'_M</math> (with characteristic time <math>\tau'_M</math>) are already known. For example, say we have a plot of <math>|R|</math> against wave period for <math>h_M'</math>. Since the nondimensional period is only the dimensional period divided by the characteristic time of the ice, results for the second value may be estimated by simply scaling the period axis by <math>\tau_M/\tau'_M=(h_M/h'_M)^{3/8}</math>.\\

\begin{figure}[h]
\begin{center}
\psfrag{vi and l}{<math>\varpi</math> and <math>\lam</math>}\psfrag{L, m}{<math>L</math>, m}\psfrag{H}{<math>H</math>}
\psfrag{(a)}{({\it a})}\psfrag{(b)}{({\it b})}\psfrag{(c)}{({\it c})}
\psfrag{Wave Period, s}{Wave Period, s}
\includegraphics[width=140mm,height=100mm]{figs/Chaps124/pram_values.eps}
\caption[The behaviour of some important nondimensional parameters with wave period.]{The behaviour of some important nondimensional parameters with wave period when the maximum ice thickness <math>h_M</math> is 1\,m: ({\it a}) <math>\varpi=\lambda-\mu</math> (solid) and <math>\lam</math> itself (dashed), ({\it b}) the natural length <math>L</math>, ({\it c}) the nondimensional water depth <math>H</math> when the actual depth is 50\,m.}
(45)
\end{center}
\end{figure}\vspace{-5mm}
However, in situations where the three thicknesses <math>h_0</math>, <math>h_1</math>, and <math>h_2</math> are identical, such as in the single-crack problem discussed in Chapter~\ref{chap-crack}, then <math>\lam</math> and <math>\mu</math> can be combined into the single parameter <math>\varpi=\lam-\mu</math>, and there is no need to approximate <math>\mu</math>. This is also useful in characterizing the Green's function derived in Chapter~\ref{chap-Green}, and the roots of the dispersion relation, which are discussed in Appendix~\ref{app-roots}. The solid line in Figure~45{\it a} plots <math>\varpi</math> as a function of wave period for a background ice thickness of 1\,m, showing that for periods between 0.1\,s and 20\,s, <math>\varpi</math> takes values from between about <math>-0.5</math> and 4.5. Such a small range makes this nondimensionalization scheme quite friendly numerically---for example, the routines for finding the roots of the dispersion relation are very reliable over an extremely large range of periods because of this (including very small and very large periods, although admittedly extremely small periods are beyond the applicability of the thin plate model).\\

Section~\ref{sec-ck-res-pram} illustrates that the single-crack scattering results can indeed be described completely by <math>\varpi</math>, and discusses some of the consequences of that. However, it also shows that the simpler parameterization of <math>\lam</math> is perfectly adequate and only becomes less accurate at very small periods when the thin plate model is actually invalid. In addition, by plotting <math>\lam</math> itself on the same graph as <math>\varpi</math>, the dashed line in Figure~45{\it a} shows that it is only at the smaller, ``less physical" periods that <math>\mu\varpropto\lam^{-1/4}</math> becomes more significant anyway, and many researchers ignore the entire term altogether \caf{Tkacheva01inf}{e.g.}.\\

The last parameter that will prove to be important in our results is the nondimensional water depth <math>H</math>. Section~\ref{sec-ck-res-H} discusses the effect of <math>H</math> on the scattering results for a single crack, although mainly with the intention of establishing an infinite depth criterion, as the principal goal of this thesis is to model central Arctic and Antarctic waters away from the coast, where the depth is usually at least 300\,m. Appendix~\ref{app-roots} also shows that the roots of the dispersion relation exhibit some quite interesting behaviour as <math>H</math> changes.\\

Since the nondimensionalization used here will be relatively unfamiliar to most readers, Figure~45{\it b} also plots the natural length <math>L</math> as period varies when <math>h_M=1</math>\,m while Figure~45{\it c} shows how the nondimensional water depth <math>H</math> corresponding to a dimensional depth of 50\,m changes with period. These figures are mostly intended for reference when reading Chapter~\ref{chap-crack}.