Roots of the Dispersion Relation

The distribution of the roots of both the finite depth and the infinite depth dispersion relations have been published previously, by Fox and Squire 1990 and Squire and Dixon 2000 respectively. However, there are interesting exceptions to those rules, particularly for shorter periods. Although the periods at which these deviations from the usual behaviour occur are often small enough to make ice-coupled water waves physically impossible, for interest's sake they are presented below. First, however, the properties of the shallow water roots are discussed, as they help to introduce the small-period behaviour of the finite depth roots. There is also a connection between the shallow water Green's function and the Green's function for an isolated plate (i.e., one that isn't floating), which is explained in Section 5.1.2.

Shallow Water Roots

Recall that the finite depth dispersion relation for the left-hand sheet of ice is

[math]\displaystyle{ f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0, }[/math]

where [math]\displaystyle{ \Lambda_0=\kappa^4+\varpi }[/math]. Since [math]\displaystyle{ \tanh\kappa\approx\kappa H }[/math] for small [math]\displaystyle{ H }[/math], for shallow water we can obtain the approximate dispersion relation [math]\displaystyle{ p_s(\kappa^2)=(\kappa^4+\varpi)\kappa^2H-1=0 }[/math], which is a cubic polynomial in [math]\displaystyle{ \kappa^2 }[/math].

Setting [math]\displaystyle{ \varpi=0 }[/math] means [math]\displaystyle{ p_s(w)=Hw^3-1=0 }[/math] when [math]\displaystyle{ H^{1/3}w }[/math] is a cube root of unity. The roots usually show this kind of distribution---there is always one real root, [math]\displaystyle{ \gamma_0^2 }[/math], and two roots [math]\displaystyle{ \gamma_{-1}^2 }[/math] and [math]\displaystyle{ \gamma_{-2}^2 }[/math] in the left hand half of the complex plane. The latter two roots are usually a complex conjugate pair (if [math]\displaystyle{ \mathfrak{Im}[\gamma_{-1}^2]\gt 0 }[/math]). However, by solving for when

Figure B.1: Small period behaviour of the roots of the shallow water dispersion relation for ice-coupled waves, [math]\displaystyle{ f_0(\kappa)=-p_s(\kappa^2)/\kappa^2H }[/math], where [math]\displaystyle{ p_s(w)=(w^2+\varpi)wH-1 }[/math]. The roots [math]\displaystyle{ \kappa=\pm\gamma_n }[/math] ([math]\displaystyle{ n=-2,-1,0 }[/math]) of [math]\displaystyle{ f_0 }[/math] are the square roots of the roots of the cubic [math]\displaystyle{ p_s }[/math]. The left hand column shows [math]\displaystyle{ p_s }[/math] plotted against [math]\displaystyle{ w }[/math] for a series of values of [math]\displaystyle{ \varpi }[/math]. Note that when [math]\displaystyle{ \varpi=\varpi_s=-3/(2H)^{2/3} }[/math], [math]\displaystyle{ p_s(w) }[/math] has a double root on the negative real axis which separates to form two simple roots for [math]\displaystyle{ \varpi\lt \varpi_s }[/math]. The right hand column shows the locations of the roots of [math]\displaystyle{ f_0 }[/math] in the copmlex plane for the same values of [math]\displaystyle{ \varpi }[/math]; arrows on two of these plots indicate the direction that the complex roots in the upper half plane move in as [math]\displaystyle{ \varpi }[/math] becomes more negative. When [math]\displaystyle{ \varpi }[/math] reaches [math]\displaystyle{ \varpi_s }[/math], the complex roots in the upper half plane meet on the positive imaginary axis and, like the roots of [math]\displaystyle{ p_s }[/math], separate into two single roots as [math]\displaystyle{ \varpi }[/math] becomes more negative (or equivalently, as period decreases and/or [math]\displaystyle{ h_0 }[/math] increases). The value of the nondimensional water depth used is [math]\displaystyle{ H=1 }[/math].

[math]\displaystyle{ p_s'(w)=p_s(w)=0 }[/math], it can be shown that as [math]\displaystyle{ \varpi }[/math] decreases towards [math]\displaystyle{ \varpi_s=-3\times(2H)^{-2/3} }[/math], they both move closer and closer towards the negative real axis. When [math]\displaystyle{ \varpi }[/math] reaches [math]\displaystyle{ \varpi_s }[/math], the complex conjugate pair form a double root at [math]\displaystyle{ w=w_s=-(2H)^{-1/3} }[/math], which then separates to form two simple roots that remain on the negative real axis for all [math]\displaystyle{ \varpi\lt \varpi_s }[/math]. In that case we will assume [math]\displaystyle{ \gamma_{-1}^2\gt \gamma_{-2}^2 }[/math].

Figure B.1 shows the shape of [math]\displaystyle{ p_s(w) }[/math] for real [math]\displaystyle{ w }[/math] as the above transition takes place. As [math]\displaystyle{ \varpi }[/math] becomes negative, [math]\displaystyle{ p_s }[/math] develops a local maximum which moves upwards as [math]\displaystyle{ \varpi\to\varpi_s }[/math], hitting the negative axis when it reaches it. It continues to move upwards as [math]\displaystyle{ \varpi }[/math] decreases further, which produces two simple negative roots.

The behaviour of the roots of the actual dispersion relation is also shown in Figure B.1. Defining the [math]\displaystyle{ \gamma_n }[/math] themselves as having positive real parts (so that [math]\displaystyle{ \gamma_{-2}=\gamma_{-1}^* }[/math] is in the fourth quadrant for [math]\displaystyle{ \varpi\gt \varpi_s }[/math]), the two non-real roots [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ -\gamma_{-2} }[/math] approach the positive imaginary axis to form a pure imaginary double root as [math]\displaystyle{ \varpi\to\varpi_s }[/math]; two simple imaginary roots are then produced as [math]\displaystyle{ \varpi }[/math] becomes more negative ([math]\displaystyle{ \gamma_1 }[/math] being the root with smaller modulus). In addition, since [math]\displaystyle{ f_0 }[/math] is even, an equivalent process takes place in the lower half plane.

Finite Depth Roots

The finite depth dispersion relation for the left-hand sheet of ice is

[math]\displaystyle{ f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0, }[/math]

where [math]\displaystyle{ \Lambda_0=\kappa^4+\varpi }[/math]. Since [math]\displaystyle{ f_0 }[/math] is even, the negative of any zero is also a zero, and so we will only investigate the location of those on the positive real axis or with imaginary part greater than zero.

For [math]\displaystyle{ \kappa\gt 0 }[/math], as [math]\displaystyle{ \kappa }[/math] increases from 0, [math]\displaystyle{ \coth(\kappa H)/\kappa }[/math] decreases from [math]\displaystyle{ +\infty }[/math], while [math]\displaystyle{ \Lambda_0 }[/math] increases from [math]\displaystyle{ \varpi }[/math], so there will always be exactly one root [math]\displaystyle{ \gamma_0 }[/math] on the positive real axis. Additional roots may be found along the imaginary axis by studying the properties of

[math]\displaystyle{ p_H(w)=H^4w\sin w\times f_0(\mbox{i} w/H)=(w^4+\varpi H^4)w\sin w+H^5\cos w, }[/math]

where [math]\displaystyle{ w=-\mbox{i}\kappa H }[/math]. Since [math]\displaystyle{ p(n\pi)=(-1)^nH^5 }[/math] ([math]\displaystyle{ n=0,1,2,\ldots }[/math]) and is continuous for all [math]\displaystyle{ w }[/math], it will always have at least one root [math]\displaystyle{ w_n }[/math] in each interval [math]\displaystyle{ I_n=\pi\times(n-1,n) }[/math] ([math]\displaystyle{ n=1,2,\ldots }[/math]). Moreover, the roots [math]\displaystyle{ w_n }[/math] must satisfy the relation

[math]\displaystyle{ \tan w_n=-\frac{H^5}{(w_n^4+\varpi H^4)w_n}, \qquad (B.1) }[/math]

so as [math]\displaystyle{ w_n }[/math] gets large, [math]\displaystyle{ \tan w_n\to0 }[/math], or equivalently [math]\displaystyle{ w_n\to n\pi^- }[/math] (since for large [math]\displaystyle{ w_n }[/math] the right hand side of Equation B.1 will be negative, even if [math]\displaystyle{ \varpi\lt 0 }[/math]). The error [math]\displaystyle{ \epsilon=n\pi-w_n }[/math] can also be estimated for large [math]\displaystyle{ w_n }[/math] by expanding both sides of (B.1) as Taylor series and ignoring the terms of order greater [math]\displaystyle{ \epsilon^2 }[/math]. This gives

[math]\displaystyle{ \begin{matrix} \epsilon\approx\frac{\big(\tilde\gamma_n^4+\varpi\big)|\tilde\gamma_n|} {\big(\tilde\gamma_n^4+\varpi\big)^2|\tilde\gamma_n|^2-\big(5\tilde\gamma_n^4+\varpi\big)/H}, \end{matrix} }[/math]

where [math]\displaystyle{ \tilde\gamma_n=n\pi \mbox{i}/H }[/math] are the zeros of [math]\displaystyle{ \tanh\kappa }[/math]. For small values of [math]\displaystyle{ H }[/math] or large values of [math]\displaystyle{ n }[/math] it is clear that [math]\displaystyle{ \epsilon }[/math] is vanishing like [math]\displaystyle{ |\gamma_n|^{-5} }[/math]. Thus we can say that as [math]\displaystyle{ n\to\infty }[/math], or as the water becomes shallower, the [math]\displaystyle{ w_n }[/math] become closer and closer to [math]\displaystyle{ n\pi }[/math], and consequently that the [math]\displaystyle{ \gamma_n }[/math] also become closer and closer to the [math]\displaystyle{ \tilde\gamma_n }[/math].

At this point we can say something about how the shallow water roots situation might arise from the finite depth one. As [math]\displaystyle{ H\to0 }[/math], the imaginary zeros of [math]\displaystyle{ f_0(\kappa) }[/math] are becoming closer and closer to the [math]\displaystyle{ \tilde\gamma_n }[/math] ([math]\displaystyle{ n\pi \mbox{i}/ H }[/math], [math]\displaystyle{ n=1,2,\ldots }[/math]), which are themselves becoming infinite. If we suppose that in addition to the real root [math]\displaystyle{ \gamma_0 }[/math] and the imaginary ones that we know about, [math]\displaystyle{ f_0(\kappa) }[/math] has two additional zeros [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ \gamma_{-2} }[/math], then the shallow water case could be reproduced in the limit. We might also anticipate that like their shallow-water counterparts, those roots will be complex for most periods, but may become additional imaginary roots for lower periods. It will be shown that this is indeed the case. First, however, we will prove their existence formally, and then enter into a more numerical discussion of their locations. Again, such a discussion is not particularly practical in that most of the phenomena described only occur at periods where the thin plate model might break down. Nevertheless, there are some exceptions, and the aforementioned phenomena are themselves quite fascinating.

Now, the number of zeros that a given function has in a particular region may be determined using the argument principle. Evans and Davies 1968 and Chung and Fox 2002a applied that principle to effectively show that for a large enough choice of [math]\displaystyle{ N }[/math], [math]\displaystyle{ p_H }[/math] has [math]\displaystyle{ 2N+6 }[/math] zeros inside the square with corners [math]\displaystyle{ \pm(N+\pi/4)\times(1+\mbox{i}) }[/math]. The [math]\displaystyle{ N }[/math] real [math]\displaystyle{ w_n }[/math] and their negatives, and [math]\displaystyle{ \pm w_0 }[/math], which are imaginary, account for [math]\displaystyle{ 2N+2 }[/math] of these roots, leaving four still to be located. Since it is easily shown that [math]\displaystyle{ w_0 }[/math] is a simple root ([math]\displaystyle{ f_0'(\gamma_0)\lt 0 }[/math]), we have verified that [math]\displaystyle{ p_H }[/math] must have two complex roots, [math]\displaystyle{ -w_{-2} }[/math] in the first quadrant, and its complex conjugate [math]\displaystyle{ w_{-1} }[/math], or else two additional real roots. In that case we will distinguish between them by choosing [math]\displaystyle{ w_{-1} }[/math] to have a larger modulus than [math]\displaystyle{ w_{-2} }[/math].

Figure B.2: Small period behaviour of the roots of the finite depth dispersion relation for ice-coupled waves, [math]\displaystyle{ f_0(\kappa)=\coth\kappa H/\kappa-\L_0(\kappa) }[/math], where [math]\displaystyle{ \L_0(\kappa)=\kappa^4+\varpi }[/math]. In particular, this figure demonstrates the existence of double roots to the dispersion relation. The left hand plots show the behaviour of the function [math]\displaystyle{ \bar p_H(w)=p_H(w)/(w^5+H^5) }[/math] for real [math]\displaystyle{ w }[/math] for decreasing values of [math]\displaystyle{ \varpi }[/math] (shorter periods and/or larger ice thicknesses). [math]\displaystyle{ \bar p_H }[/math] has the same zeros as [math]\displaystyle{ p_H(w)=H^4w\sin w\times f_0(iw/H) }[/math] but does not grow as quickly for large [math]\displaystyle{ w }[/math]; if [math]\displaystyle{ w_n }[/math] is a root of [math]\displaystyle{ p_H }[/math] (or [math]\displaystyle{ \bar p_H }[/math]), then [math]\displaystyle{ \gamma_n=iw_n/H }[/math] is a root of [math]\displaystyle{ f_0 }[/math]. Note the formation of double zeros near [math]\displaystyle{ \varpi=-1.51 }[/math] and between [math]\displaystyle{ -60 }[/math] and [math]\displaystyle{ -95 }[/math]. The right hand plots show the locations of the [math]\displaystyle{ w_n }[/math] for the same values of [math]\displaystyle{ \varpi }[/math]. Arrows on two of those plots indicate the direction that the complex roots move in as [math]\displaystyle{ \varpi }[/math] decreases. The formation of one double zero can be seen in them also as the two complex roots shown move onto the real axis and split to form a pair of simple roots; another double root is formed as one of the newly-formed single roots reaches one of the original real roots creating another pair of complex roots. The value of the nondimensional water depth used is [math]\displaystyle{ H=1 }[/math].

Their locations can be narrowed down further by repeating the application of the argument principle, again following Chung and Fox 2002a, but this time to the triangle with vertices 0 and [math]\displaystyle{ (N+\pi/4)\times(1\pm \mbox{i}) }[/math] finding that [math]\displaystyle{ p_H }[/math] has [math]\displaystyle{ N+2 }[/math] zeros inside that region. This implies that [math]\displaystyle{ w_{-1} }[/math] and [math]\displaystyle{ -w_{-2} }[/math] have arguments between [math]\displaystyle{ \pm\pi/4 }[/math]. Since the [math]\displaystyle{ \gamma_n }[/math] are related to the [math]\displaystyle{ w_n }[/math] by [math]\displaystyle{ \gamma_n=iw_n/H }[/math], [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ \gamma_{-2} }[/math] must be closer to the imaginary axis than they are to the real one.

Figure B.2 shows the behaviour of the [math]\displaystyle{ w_n }[/math] with decreasing period (or as [math]\displaystyle{ \varpi }[/math] becomes more negative) for [math]\displaystyle{ H=1 }[/math], and also how the shape of the [math]\displaystyle{ p_H }[/math] curves evolve to produce this behaviour. At [math]\displaystyle{ \varpi=-1.1 }[/math], [math]\displaystyle{ w_{-1} }[/math] and [math]\displaystyle{ -w_{-2} }[/math] are still complex, and the first three real roots are extremely close to the zeros of [math]\displaystyle{ \tan w }[/math] at [math]\displaystyle{ \pi }[/math], [math]\displaystyle{ 2\pi }[/math] and [math]\displaystyle{ 3\pi. }[/math] However, [math]\displaystyle{ p_H }[/math] has a minimum only slightly above the real axis at about [math]\displaystyle{ w=0.28\pi, }[/math] which proceeds to become lower and lower as [math]\displaystyle{ \varpi }[/math] decreases past [math]\displaystyle{ -1.3 }[/math] and [math]\displaystyle{ -1.51 }[/math]. As this happens, [math]\displaystyle{ w_{-1} }[/math] and [math]\displaystyle{ -w_{-2} }[/math] also move closer and closer to the real axis.

When [math]\displaystyle{ \varpi }[/math] reaches about [math]\displaystyle{ -1.54 }[/math], [math]\displaystyle{ p_H }[/math] touches the real axis and [math]\displaystyle{ w_{-1} }[/math] and [math]\displaystyle{ -w_{-2} }[/math] meet to form a double root which quickly separates to form two real roots as [math]\displaystyle{ \varpi }[/math] decreases further and the minimum in [math]\displaystyle{ p_H }[/math] drops below the real axis. As [math]\displaystyle{ \varpi }[/math] sinks past [math]\displaystyle{ -8 }[/math] and [math]\displaystyle{ -60 }[/math], this minimum deepens and become more symmetrical in the interval [math]\displaystyle{ I_1 }[/math]. As this happens, [math]\displaystyle{ -w_{-2} }[/math] moves towards zero while [math]\displaystyle{ w_{-1} }[/math] moves towards [math]\displaystyle{ w_1 }[/math] at the other end of the interval. Note that at the same time the maximum which occurs at about [math]\displaystyle{ w=3\pi/4 }[/math] when [math]\displaystyle{ \varpi=-8 }[/math] has also been moving to the right and moving closer to the real axis.

The maximum crosses the real line as [math]\displaystyle{ \varpi }[/math] decreases past [math]\displaystyle{ -85 }[/math], at the same time as [math]\displaystyle{ w_{-1} }[/math] meets [math]\displaystyle{ w_1 }[/math] to form another double root and lift off the real axis to produce the situation shown when [math]\displaystyle{ \varpi=-95. }[/math] This cycle of roots moving onto and off the real axis is repeated in every interval as the period becomes shorter and shorter.

The behaviour of the complex roots described and illustrated above for [math]\displaystyle{ H=1 }[/math] is typical for all values of [math]\displaystyle{ H }[/math] less than [math]\displaystyle{ H_1\approx2.40 }[/math]. As [math]\displaystyle{ H }[/math] increases towards that value, the two double roots in [math]\displaystyle{ I_1 }[/math] move closer and closer together until they meet and form a triple root. For [math]\displaystyle{ H\gt H_1 }[/math] there are no multiple roots in [math]\displaystyle{ I_1 }[/math].

Figure B.3: Demonstration of the existence of triple roots of the finite depth dispersion relation for ice-coupled waves [math]\displaystyle{ f_0(\kappa) }[/math]. [math]\displaystyle{ \bar p_H(w) }[/math] is defined in Figure B.2; its zeros [math]\displaystyle{ w_n }[/math] are related to [math]\displaystyle{ \gamma_n }[/math], the zeros of [math]\displaystyle{ f_0 }[/math], by [math]\displaystyle{ \gamma_n=iw_n/H }[/math]. The left hand plot shows the presence of a point of inflection in the graph of [math]\displaystyle{ p_H }[/math] for real [math]\displaystyle{ w }[/math] and for [math]\displaystyle{ \varpi=0.087 }[/math] at one of the real [math]\displaystyle{ w_n }[/math]. Consequently [math]\displaystyle{ \bar p_H }[/math] has a triple point at that point, and [math]\displaystyle{ f_0 }[/math] itself must have a triple root on the imaginary axis. The right hand plots show the location of the roots of [math]\displaystyle{ \bar p_H }[/math] for values of [math]\displaystyle{ \varpi }[/math] around 0.087, showing the two complex roots moving towards the real line, meeting one of the original real roots to form the triple root and immediately becoming complex again. The direction of their motion as [math]\displaystyle{ \varpi }[/math] decreases is indicated on the upper and lower figures. The value of the nondimensional water depth used is [math]\displaystyle{ H=2.40 }[/math].}

Figure B.3 shows the behaviour of the roots as [math]\displaystyle{ \varpi }[/math] decreases when [math]\displaystyle{ H=H_1 }[/math], confirming the formation of a triple root at that water depth. The left hand plot clearly shows the presence of a point of inflection when [math]\displaystyle{ \varpi\approx0.087, }[/math] while the series of plots on the right show the two complex roots merging with [math]\displaystyle{ w_1 }[/math] on the real axis before immediately becoming complex again. [math]\displaystyle{ w_{-1} }[/math] and [math]\displaystyle{ -w_{-2} }[/math] will become real again when [math]\displaystyle{ \varpi\approx-6.35 }[/math], forming a double root in [math]\displaystyle{ I_2 }[/math] and from there on exhibiting the same pattern as observed for [math]\displaystyle{ H=1 }[/math].

In order to study this phenomenon of multiple roots further, and to describe the situation for larger water depths, let us attempt to quantify how the locations of the double roots, and the values of [math]\displaystyle{ \varpi }[/math] at which they occur, vary as [math]\displaystyle{ H }[/math] changes. Suppose that an imaginary root does have multiplicity two. In that case we would have [math]\displaystyle{ f_0(\gamma_n) }[/math] and [math]\displaystyle{ f'_0(\gamma_n) }[/math] ([math]\displaystyle{ n=-1,-2 }[/math]) both vanishing. If [math]\displaystyle{ w=-\mbox{i}\gamma_n }[/math], [math]\displaystyle{ p_H(w)=p_H'(w)=0 }[/math] and we can eliminate [math]\displaystyle{ \varpi }[/math] to give a relationship between [math]\displaystyle{ w }[/math] and [math]\displaystyle{ H }[/math] alone.

This is achieved by writing

[math]\displaystyle{ \begin{matrix} -\varpi H^4=w^4+\frac{H^5}{w\tan{w}}=\frac{(5w^4-H^5)\tan{w}+w^5}{\tan{w}+w}, &\qquad (B.3) \end{matrix} }[/math]

or

[math]\displaystyle{ \begin{matrix} q_H(w)=2w^4(1-\cos{2w})-H^5(1+\sin{2w}/2w)=0. &\qquad (B.4) \end{matrix} }[/math]

The zeros of this function are plotted as a function of [math]\displaystyle{ H }[/math] in Figure B.4[math]\displaystyle{ \it{a} }[/math]. The most striking feature of the curves is that each interval contains a certain extremal value of [math]\displaystyle{ H }[/math] at which [math]\displaystyle{ dH/dw=0. }[/math] Let [math]\displaystyle{ H_n }[/math] be the extremal value in the [math]\displaystyle{ n^{\rm th} }[/math] interval. If [math]\displaystyle{ H\gt H_n }[/math], then [math]\displaystyle{ p_H }[/math] contains no double roots in either [math]\displaystyle{ I_n }[/math] itself or any of the other intervals with indices smaller than [math]\displaystyle{ n }[/math]. If [math]\displaystyle{ H\lt H_n }[/math], it has two zeros in [math]\displaystyle{ I_n }[/math], one corresponding to when [math]\displaystyle{ w_{-1} }[/math] and [math]\displaystyle{ w_{-2} }[/math] move onto the real axis, and the other to when they move off it. If [math]\displaystyle{ H=H_n }[/math] there is only one point in [math]\displaystyle{ I_n }[/math] at which a double root forms. Differentiating (10) with respect to [math]\displaystyle{ w }[/math], it can be seen that at such points [math]\displaystyle{ q_H'(w)=q_H(w)=0 }[/math], and so the root [math]\displaystyle{ w }[/math] is actually a double root of [math]\displaystyle{ q_H }[/math] and is consequently a triple root of [math]\displaystyle{ p_H }[/math]. This was discussed above with reference to [math]\displaystyle{ H_1 }[/math].

The solid and chained lines in Figure B.4[math]\displaystyle{ \it{a} }[/math] are used to provide a correspondence between the two curves for each interval in Figure B.4[math]\displaystyle{ \it{b} }[/math], which shows the values of [math]\displaystyle{ \varpi }[/math] at which each double root forms for a given [math]\displaystyle{ H }[/math]. The left hand pair of curves corresponds to [math]\displaystyle{ I_1 }[/math], the second pair from the left corresponds to [math]\displaystyle{ I_2 }[/math], and so on. Comparing the two graphs it can be seen that the root in an interval with smaller modulus always corresponds to a more positive value of [math]\displaystyle{ \varpi }[/math] (a larger period) than the root with larger modulus. Thus, the pattern observed in Figure B.2 can be said to be typical---the two

Figure B.4: ([math]\displaystyle{ \it{a} }[/math]) The first and second double root of [math]\displaystyle{ p_H(w) }[/math] (which correspond to simple roots of [math]\displaystyle{ q_H(w) }[/math]) in each interval [math]\displaystyle{ I_n }[/math] are plotted against the nondimensional water depth [math]\displaystyle{ H }[/math] as solid and chained lines respectively. ([math]\displaystyle{ w/\pi=n }[/math] corresponds to the greater endpoint of [math]\displaystyle{ I_n }[/math], and the first root is initially defined as the closest root to zero.) Note that for small [math]\displaystyle{ H }[/math] there are always two double roots inside each interval but as [math]\displaystyle{ H }[/math] increases these roots become closer and closer together before meeting and becoming complex. Points where the solid and chained lines meet in an interval correspond to triple roots of [math]\displaystyle{ p }[/math] (points of inflexion for [math]\displaystyle{ p }[/math]); if the roots of [math]\displaystyle{ q_H(w) }[/math] in a given interval have become complex then that interval contains no double roots. Also note that the first double root in the first interval is well approximated for values of [math]\displaystyle{ H }[/math] less than about 0.5 by the dashed line, which corresponds to the point at which the complex shallow water roots become pure imaginary for that value of [math]\displaystyle{ H }[/math]. ([math]\displaystyle{ \it{b} }[/math]) plots the value of [math]\displaystyle{ \varpi }[/math] at which double roots occur. As in ([math]\displaystyle{ \it{a} }[/math]), solid lines correspond to the first double root in [math]\displaystyle{ I_n }[/math], and chained lines to the second---note that the first root for a given [math]\displaystyle{ H }[/math] always occurs at a greater value of [math]\displaystyle{ \varpi }[/math] than the second. The [math]\displaystyle{ n^{\rm th} }[/math] curve from the left corresponds to [math]\displaystyle{ I_n }[/math], while the apex of each curve corresponds to the triple root in that interval. The triple root in [math]\displaystyle{ I_1 }[/math] occurs when [math]\displaystyle{ H=2.40 }[/math] and [math]\displaystyle{ \varpi=0.08 }[/math] (a wave period of 2.64s when [math]\displaystyle{ h_0=1 }[/math]m). The dashed line plots the value of [math]\displaystyle{ \varpi }[/math] at which the complex shallow water roots become imaginary---again for [math]\displaystyle{ H\lesssim0.5 }[/math], it gives a good approximation to the point at which the first double root in the first interval occurs. For reference, dotted lines described by [math]\displaystyle{ \varpi=0 }[/math], [math]\displaystyle{ -0.35 }[/math], [math]\displaystyle{ -0.88 }[/math], and [math]\displaystyle{ -2.21 }[/math] (respectively from top) are also plotted. For 1-m-thick ice these correspond to periods of 1.9s, 0.1s, 0.01s and 1ms, which implies that we really only need to allow for double roots in the first interval, or equivalently when [math]\displaystyle{ H\lesssim2.40 }[/math].

complex roots move onto the real axis at one point in [math]\displaystyle{ I_n }[/math], and then [math]\displaystyle{ w_{-1} }[/math] moves to the right before meeting [math]\displaystyle{ w_n }[/math] to form two more complex roots.

The dashed line in Figure B.4[math]\displaystyle{ \it{a} }[/math] is included for interest and compares the point at which the complex shallow water roots become imaginary with the first double root in [math]\displaystyle{ I_1 }[/math], and provides quite a good approximation for that root for [math]\displaystyle{ H }[/math] less than about 0.5. Similarly, in Figure B.4[math]\displaystyle{ \it{b} }[/math] the dashed line corresponds to [math]\displaystyle{ \varpi_s }[/math] and also approximates the value of [math]\displaystyle{ \varpi }[/math] when the first double root forms for small water depths quite well.

The dotted lines in Figure B.4[math]\displaystyle{ \it{b} }[/math] are reference lines, which for 1-m-thick ice correspond to periods of 1.9s ([math]\displaystyle{ \varpi=0 }[/math]), 0.1s ([math]\displaystyle{ \varpi=-0.35 }[/math]), 10ms ([math]\displaystyle{ \varpi=-0.88 }[/math]) and 1ms ([math]\displaystyle{ \varpi=-2.21 }[/math]). They are included to give an idea of when we need to be concerned about double and triple roots when solving the dispersion relation for physical wave periods.

The latter three periods are obviously far too small for the thin plate model for the ice to hold, and so for practical purposes there is only a very small range of water depths where we actually have to worry about the complex roots becoming imaginary. In fact, if we restricted ourselves to values of [math]\displaystyle{ \varpi\gtrsim0.087 }[/math], which for 1-m-thick ice corresponds to periods greater than about 2.64s, we could say that we need never consider them.

If we do wish to calculate results for smaller periods, however, we would need to allow for multiple roots. Fortunately, though, we would generally only need to do this for a small range of values of [math]\displaystyle{ H }[/math]. If our minimum period was about 1.9s, so that [math]\displaystyle{ \varpi\geq0 }[/math], multiple roots could only occur if [math]\displaystyle{ 2.34\lesssim H\leq H_1\approx2.40 }[/math]. (This same range would also apply if we neglected the inertia term in the Euler-Bernoulli equation, so that [math]\displaystyle{ \mu=0 }[/math] and [math]\displaystyle{ \varpi=\lam-\mu }[/math] would be positive for all periods.) The lower limit for this interval would have to be decreased to about 1.50 if we required scattering results for periods down to 0.1s. (This is the lower limit in most graphs presented in this thesis, although usually only infinite depth results are presented in which [math]\displaystyle{ H=5 }[/math] is used if the infinite depth results are not calculated directly.)

Infinite Depth Roots

Letting [math]\displaystyle{ H\to\infty }[/math] in the finite depth dispersion relation (for real [math]\displaystyle{ k }[/math] and thus positive real [math]\displaystyle{ \kappa }[/math]) means that the dispersion relation becomes [math]\displaystyle{ f_0(\kappa)=1/\kappa-\L_0 }[/math], which has zeros when the polynomial [math]\displaystyle{ p_\infty(\kappa)=-\kappa f_0(\kappa)=\kappa^5+\varpi\kappa-1 }[/math] does. The roots of this equation were presented for cases equivalent to [math]\displaystyle{ \varpi\gt 0 }[/math] by Squire and Dixon 2000. Figure B.5 reproduces their figure.

Figure B.5 Schematic diagram showing the possible locations of the roots of the infinite depth dispersion relation [math]\displaystyle{ 1/\kappa-\kappa^4-\varpi=0 }[/math] for positive values of the parameter [math]\displaystyle{ \varpi }[/math]. There is one positive real root [math]\displaystyle{ \gamma_0 }[/math] (indicated by a circle), and four complex roots (indicated by crosses). The complex roots consist of two complex conjugate pairs, [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ \gamma_{-2} }[/math] in the right hand half plane ([math]\displaystyle{ \gamma_{-1} }[/math] is in the first quadrant), and [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] in the left hand half plane ([math]\displaystyle{ \gamma_1 }[/math] is in the second quadrant). The shaded regions show the possible range of arguments that the complex roots may take for [math]\displaystyle{ \varpi\gt 0 }[/math]: [math]\displaystyle{ \pi/4\lt |\Arg[\gamma_j|\leq2\pi/5 }[/math] for [math]\displaystyle{ j=-1,-2 }[/math], and [math]\displaystyle{ 3\pi/4\lt |\Arg[\gamma_j|\leq4\pi/5 }[/math] for [math]\displaystyle{ j=1,2 }[/math]. As [math]\displaystyle{ \varpi }[/math] becomes negative [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ \gamma_{-2} }[/math] move towards the imaginary axis, while [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] move towards and eventually onto the negative real axis.

The bounds shown for the arguments of the complex roots may be deduced by considering the limits as [math]\displaystyle{ \varpi\to0 }[/math] and [math]\displaystyle{ \varpi\to\infty }[/math]. In the former case the roots satisfy [math]\displaystyle{ \kappa^5-1=0 }[/math]; in the latter the transformed roots [math]\displaystyle{ \bar\kappa=\kappa/|\varpi|^{1/4} }[/math] satisfy [math]\displaystyle{ \bar\kappa^5+\bar\kappa=\epsilon }[/math], where [math]\displaystyle{ \epsilon=|\varpi|^{-5/4}\to0 }[/math]. Thus, as [math]\displaystyle{ \varpi }[/math] decreases from [math]\displaystyle{ \infty }[/math] to 0, [math]\displaystyle{ \gamma_{-1} }[/math] moves from the line [math]\displaystyle{ \Arg[\kappa]=\pi/4 }[/math] to [math]\displaystyle{ \Arg[\kappa]=2\pi/5 }[/math], while [math]\displaystyle{ \gamma_1 }[/math] travels from the line [math]\displaystyle{ \Arg[\kappa]=3\pi/4 }[/math] to [math]\displaystyle{ \Arg[\kappa]=4\pi/5. }[/math] [math]\displaystyle{ \gamma_0 }[/math] is asymptotically zero for large [math]\displaystyle{ \varpi }[/math], and it increases to 1 as [math]\displaystyle{ \varpi }[/math] decreases to 0. Also note that [math]\displaystyle{ |\gamma_j|\to\infty }[/math] for [math]\displaystyle{ j\neq0 }[/math] as [math]\displaystyle{ \varpi\to\infty }[/math].

In this section, we demonstrate the behaviour of the infinite depth roots as [math]\displaystyle{ \varpi }[/math] becomes more negative. In particular, we will show that the complex roots [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] actually move onto the negative real line when [math]\displaystyle{ \varpi }[/math] becomes less than or equal to [math]\displaystyle{ \varpi_\infty=-5/4^{4/5}\approx-1.65, }[/math] a proposition which follows easily by noting that [math]\displaystyle{ p_\infty }[/math] and its derivative [math]\displaystyle{ p_\infty'(\kappa)=5\kappa^4+\varpi }[/math] are simultaneously zero when [math]\displaystyle{ \kappa=\kappa_\infty=-1/4^{1/5} }[/math] and [math]\displaystyle{ \varpi=\varpi_\infty }[/math].

Figure B.6 plots the positions of the infinite depth roots for three different values of [math]\displaystyle{ \varpi }[/math]: ([math]\displaystyle{ \it{a} }[/math]) [math]\displaystyle{ \varpi_\infty/2 }[/math], ([math]\displaystyle{ \it{b} }[/math]) [math]\displaystyle{ \varpi_\infty }[/math] and (\itc) [math]\displaystyle{ 2\varpi_\infty }[/math]. It shows that a double root does indeed exist when [math]\displaystyle{ \varpi=\varpi_\infty }[/math], and that it forms in the same way that the double roots of the shallow water and finite depth dispersion relations form on the imaginary axis. In this case [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] merge on the real axis, and separate again to form two simple zeros as [math]\displaystyle{ \varpi }[/math] decreases. The more negative root, which we shall define as [math]\displaystyle{ \gamma_2 }[/math], becomes increasingly more negative as [math]\displaystyle{ \varpi }[/math] decreases further, while the other root [math]\displaystyle{ \gamma_1 }[/math] increases towards 0. It also shows [math]\displaystyle{ \gamma_0 }[/math] getting larger, as do [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ \gamma_{-2} }[/math] as they move towards the imaginary axis.

The behaviour of all the roots as [math]\displaystyle{ \varpi\to-\infty }[/math] may be shown formally by considering the same transformation that we used when considering their behaviour when [math]\displaystyle{ \varpi\to\infty }[/math]. For large, negative [math]\displaystyle{ \varpi }[/math] the roots satisfy [math]\displaystyle{ \bar\kappa^5-\bar\kappa=\epsilon }[/math]. Thus, in agreement with the results observed in Figure B.6, four roots become infinitely large---two, [math]\displaystyle{ \gamma_0 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] travelling on the real axis towards [math]\displaystyle{ \pm\infty }[/math], while another two, [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ \gamma_{-2} }[/math] travel asympotically towards [math]\displaystyle{ \pm \mbox{i}\infty }[/math]. [math]\displaystyle{ \gamma_1 }[/math] continues to move along the negative real axis towards [math]\displaystyle{ 0 }[/math], also in agreement with the figure.

Figure B.6: Small period behaviour of the roots of the infinite depth dispersion relation for ice-coupled waves. Demonstration of the existence of negative real roots to the infinite depth dispersion relation for ice-coupled waves. The figures show the location of the roots of the infinite depth dispersion relation when [math]\displaystyle{ \varpi }[/math] takes values of ([math]\displaystyle{ \it{a} }[/math]) [math]\displaystyle{ \varpi=\varpi_\infty/2 }[/math], ([math]\displaystyle{ \it{b} }[/math]) [math]\displaystyle{ \varpi=\varpi_\infty }[/math], and (\itc) [math]\displaystyle{ \varpi=2\varpi_\infty }[/math]. Note that the two roots in the left hand half plane move onto the real axis to form one double root as [math]\displaystyle{ \varpi\to\varpi^+_\infty }[/math], before separating to form two simple roots---one of which becomes more negative while the other moves towards zero. Their direction of motion as [math]\displaystyle{ \varpi }[/math] becomes more negative is indicated with arrows on the left hand and right hand figures. The complex roots in the right hand half plane move asymptotically towards the imaginary axis as [math]\displaystyle{ \varpi }[/math] decreases, with increasing modulus.

That [math]\displaystyle{ \gamma_{-1} }[/math] and [math]\displaystyle{ \gamma_{-2} }[/math] become closer and closer to the imaginary as [math]\displaystyle{ \varpi\to-\infty }[/math] might have been expected from the behaviour of the finite depth roots for small periods, but the behaviour of [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] could not have been guessed by considering the finite depth dispersion relation alone. In particular, the loss in symmetry of the roots is quite unexpected. However, we can start to recover the lost roots in the left hand half plane by noting that [math]\displaystyle{ \tanh\kappa\to-1 }[/math] for [math]\displaystyle{ \re[\kappa]\lt 0 }[/math] as [math]\displaystyle{ H\to\infty }[/math], which means that they will become the solutions to [math]\displaystyle{ q_\infty=-p_\infty(-\kappa)=\L\kappa+1=0 }[/math], not the roots of [math]\displaystyle{ p_\infty }[/math] itself. The remaining roots of both [math]\displaystyle{ p_\infty }[/math] and [math]\displaystyle{ q_\infty }[/math] can be recovered from the finite depth situation by also considering the equation [math]\displaystyle{ \L\kappa\tanh\kappa+1=0 }[/math]. (The finite depth dispersion relation rearranges to [math]\displaystyle{ \L\kappa\tanh\kappa-1=0 }[/math].) This equation still has infinitely many imaginary roots, but instead of having three roots in the left hand half plane it only has two, which become the infinite depth roots [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] as [math]\displaystyle{ H\to\infty }[/math]. Moreover, these two roots also move onto the real axis when [math]\displaystyle{ \varpi }[/math] takes a certain negative value (depending on [math]\displaystyle{ H }[/math]), which explains the surprising behaviour of the aforementioned infinite depth roots.

Referring to equations (3.17) and (3.18), the presence of two additional real roots complicates the calculation of [math]\displaystyle{ g_0 }[/math] and [math]\displaystyle{ g_1 }[/math], as it introduces four additional real poles into those transforms as they are written. These poles could be dealt with in the same way as the poles at [math]\displaystyle{ k=\pm\a_0 }[/math] were. However, since values of [math]\displaystyle{ \varpi\lt \varpi_\infty }[/math] only correspond to periods below about 2.1ms (for 1-m-thick ice), it was not considered worthwhile to calculate [math]\displaystyle{ g }[/math] for such values.

Reflections on Ice