Introduction - Revision history

Meylan at 09:39, 10 September 2009

2009-09-10T09:39:19Z

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	This thesis is concerned primarily with the determination of the scattering of ice-coupled waves by imperfections in large expanses of sea ice, with a possible future application being the development of a technique that is able to sense ice thickness remotely. Such ice sheets are found mainly in the Arctic Ocean but they can also be found close to the Antarctic continent, although the sheets there are of a more transient nature as they are regularly broken up by incoming waves from the surrounding Southern Ocean. They may also be dispersed by strong katabatic winds from the continent.		This thesis is concerned primarily with the determination of the scattering of ice-coupled waves by imperfections in large expanses of sea ice, with a possible future application being the development of a technique that is able to sense ice thickness remotely. Such ice sheets are found mainly in the Arctic Ocean but they can also be found close to the Antarctic continent, although the sheets there are of a more transient nature as they are regularly broken up by incoming waves from the surrounding Southern Ocean. They may also be dispersed by strong katabatic winds from the continent.

Meylan: /* Outline of Thesis */

2009-09-10T09:38:35Z

Outline of Thesis

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	This result was taken by [[Williams 2004]] to suggest that the potential for using flexural waves in remote ice-thickness-sensing could initially be considered by simply using the serial approximation. In that paper, reflection curves for a given set of ridges were compared for different ice thicknesses and were found to be fairly different over a relatively wide range of periods (from about 2s to 11s). Thus there seemed to be some hope that scattering results for different thicknesses would be sufficiently different to use those results in determining the ice thickness.		This result was taken by [[Williams 2004]] to suggest that the potential for using flexural waves in remote ice-thickness-sensing could initially be considered by simply using the serial approximation. In that paper, reflection curves for a given set of ridges were compared for different ice thicknesses and were found to be fairly different over a relatively wide range of periods (from about 2s to 11s). Thus there seemed to be some hope that scattering results for different thicknesses would be sufficiently different to use those results in determining the ice thickness.

	~~=Outline of Thesis=~~
	In [[Mathematical Formulation of the Problem \| Chapter 2]] we begin by presenting the governing equations that we wish our velocity potential to satisfy. These equations are derived and discussed in [[Governing Equations for an Elastic Plate of Variable Thickness \|Appendix A]]. Then, using the Green's functions that are discussed in [[The Green's Function \|Chapter 3]] for both finite and infinite depth, we proceed to set up an integral equation in [[Formulation of General Integral Equation \|Chapter 4]] in terms of the displacement at the ice-water interface (or at the air-water interface---such as in the problem of an open lead). This integral equation may either be over a finite or an infinite interval.

	Of particular interest in [[The Green's Function \|Chapter 3]] is the presentation of calculation techniques for the infinite depth Green's function for oblique incidence. These techniques include Fast Fourier Transform (FFT), Taylor series expansions, and numerical integration. Each of these methods have particular advantages and disadvantages, depending on the number of points that the Green's function needs to be evaluated at and the distance of the points from the singularity. They are described in more detail in [[Computation of Infinite Depth Green's Function \|Appendix C]]. Another appendix that is related to this chapter is [[Roots of the Dispersion Relation \|Appendix B]], which discusses the behaviour of the roots of the shallow water, finite depth and infinite depth dispersion relations. The small period behaviour of the latter two sets of roots has never been presented before and, although it generally only occurs at physically impractical periods, is nevertheless extremely interesting mathematically.

	In [[Formulation of General Integral Equation \|Chapter 4]] the thickness profile of the ice is allowed to vary arbitrarily over a finite strip with the ice sheets of different but constant thicknesses. The method used to include a variable thickness profile was first reported by [[Williams and Squire 2004a]], but the generalization of this approach to include different thicknesses on each side of the strip is new and is yet to be published. This chapter also presents a proof that the potential may always be represented by an eigenfunction expansion.

	[[Scattering by a Single Crack \|Chapter 5]], [[Scattering by a Single Ridge \|Chapter 6]], [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] and [[Scattering by Multiple Irregularities \|Chapter 8]] solve the integral equation derived in [[Formulation of General Integral Equation \|Chapter 4]] in a variety of different situations, and are arranged in increasing order of mathematical difficulty. [[Scattering by a Single Crack \|Chapter 5]] and [[Scattering by a Single Ridge \|Chapter 6]] can both be considered as treating special cases of the problem presented in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], both in terms of the actual complexity of the physical problems that they solve, and in the methods of solution that they employ. In physical terms, the problem solved in [[Scattering by Multiple Irregularities \|Chapter 8]] is also a special case of the one in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], but mathematically the use of the Wiener-Hopf technique in particular is more involved in [[Scattering by Multiple Irregularities \|Chapter 8]] than in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], in which the Wiener-Hopf technique is first introduced. Consequently, [[Scattering by Multiple Irregularities \|Chapter 8]] provides a more analytical solution method than [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], but to a smaller range of physical problems; although [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] reports a more numerical method, it can as a result deal with a wider range of problems.

	The simplest problem is that of a single crack in a single uniform ice sheet. As mentioned in the previous section, this problem was solved for infinitely deep water and normally incident waves by [[Squire and Dixon 2000]], obliquely incident waves by [[Williams and Squire 2002]], and finitely deep water by [[Evans and Porter 2003a]]. However, the solution follows in only one or two steps from the final result in [[Formulation of General Integral Equation \|Chapter 4]], which, as discussed above, solves a considerably more general problem. Moreover, it was also thought that the simplicity of the results obtained provide a good opportunity to investigate their behaviour with some of the more fundamental wave/ice parameters such as ice thickness, wave period, angle of incidence and water depth. For example, it is shown that the effects of the former two can be approximately combined into a single parameter, the nondimensional period, while in the discussion of the effect of water depth, criteria with regard to the accuracy of the shallow and infinite depth approximations are established.

	[[Scattering by a Single Ridge \|Chapter 6]] describes scattering by a pressure ridge. It is modelled as a thin plate of variable thickness with zero submergence ([[Williams and Squire 2004a]]; [[Williams and Squire 2004b]]; [[Williams 2004]]), and, as in those papers, the problem is solved by setting up an integral equation over the interval in which the ice thickness is varying. This equation is solved numerically.

	The results presented in this chapter are similar to the single-ridge results presented in the aforementioned papers. However, those results are adjusted to allow for the increased thickness due to the ridge keel. In light of the recent work of [[Porter and Porter 2004]], changing the thickness in this way should improve the accuracy of the [[Williams and Squire 2004a]] results for large water depths, but without significant changes to that method. [[Proposed Solution Technique for a Pressure Ridge With a Submerged Keel \|Appendix E]] also treats the topic of scattering by a pressure ridge, proposing a method by which submergence could be included in full, although no results are able to be presented yet.

	[[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] extends the variable thin plate formulation developed in [[Scattering by a Single Ridge \|Chapter 6]] by allowing the ice thickness to the right of the variable region (which in this chapter we will refer to as a ramp) to differ from the ice thickness to its left. To do this we must solve two coupled integral equations---the first is much the same as the one solved in [[Scattering by a Single Ridge \|Chapter 6]], but the second is over a semi-infinite range---over the ice-water interface to the right of the ramp. Proceeding in exactly the same way as in [[Scattering by a Single Ridge \|Chapter 6]], we proceed to solve the first integral equation numerically for the displacement in the ramp, expressing it in terms of the displacement of the ice in the semi-infinite region. The second integral equation is one of the Wiener-Hopf type, and so we are able to write an analytical expression for the displacement in the semi-infinite region in terms of the displacement within the ramp. We can now eliminate the displacement in the right hand region from the two sets of equations, solve for the displacement in the ramp, and then regenerate the displacement to the right.

	The method in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] is rather novel, as it combines a numerical integral quadrature scheme with the analytical Wiener-Hopf technique. Previously, authors have tended to favour either one approach or the other, and so this result also represents a philosophical breakthrough in that the combined weight of both methods can be brought to bear on a single problem most effectively. The numerical approach is able to deal with the arbitrariness of the ice thickness within the ramp, while the Wiener-Hopf technique is ideal for the semi-infinite interval over which the second integral equation must be solved---such intervals are extremely hard to deal with numerically.

	Unfortunately, the method does have the same drawback that our ridge model had---that submergence must be ignored. However, the method of [[Porter and Porter 2004]] also incorporated a change in ice thickness from one side of the variable region to the other, and so we could again use their result that for large water depths the effect of submergence is negligible as long as the correct total thickness is used (and as long as the mild slope assumption is valid).

	The physical problem solved in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], i.e. a ramp connecting a thinner ice sheet to a thicker one, has practical applications. Such a thickness profile could be taken to describe a sea ice/ice shelf transition such as the one observed in the Ross Sea, where the thickness of the sea ice beyond the ice shelf increases steadily to meet the thicker ice of the ice shelf itself. Alternatively, by setting the right hand ice thickness equal to zero, and by letting the incident wave arrive from the right instead of from the left, the problem could represent a breakwater shielding a VLFS from waves arriving from the open ocean.

	[[Scattering by Multiple Irregularities \|Chapter 8]] treats a special case of the problems solved in [[Scattering by a Single Ridge \|Chapter 6]] and [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]]. When the ice thickness over the variable region is constant, a more analytical approach may be taken. The problem may be written as two coupled Wiener-Hopf integral equations, producing a system of <math>M</math> linear equations in as many variables, where <math>M</math> is the number of evanescent waves generated at each end of the middle region that have not decayed sufficiently by the time they reach the other end to be able to be ignored. Since the evanescent waves decay exponentially with distance, the order of coupling <math>M</math> between the two semi-infinite regions decreases as the width of the middle region increases. If that region is wide enough, <math>M</math> becomes 1, and only the propagating waves generated need to be considered. Thus only a scalar matrix needs to be inverted, and the problem reduces to the well-known wide spacing approximation [[Evans 1990]], abbreviated by [[Williams and Squire 2004a]] to the NEW approximation (No Evanescent Waves).

	In the above special case, the approach of [[Scattering by Multiple Irregularities \|Chapter 8]] complements the numerical approach of the two earlier chapters well. When the central region is relatively wide, the method of [[Scattering by Multiple Irregularities \|Chapter 8]] becomes extremely efficient, while if <math>\alpha</math>, the width of that central region, is decreased and less evanescent modes can be neglected, less quadrature points are needed to obtain the numerical solution and that method would be preferable. Given the wave period, angle of incidence and ice thicknesses, the coupled Wiener-Hopf method can generate results for many different values of <math>\alpha</math> quite rapidly, while if <math>\alpha</math> is fixed and results are required for when the thickness of the central region is allowed to take several different values, the quadrature schemes would be more efficient.

	Practically speaking again, the situation treated in [[Scattering by Multiple Irregularities \|Chapter 8]] is a good model for a lead separating two sheets of ice. Such features are extremely common in both the Arctic and Antarctic regions. Different stages of refreezing may be represented, ranging from a freshly formed lead which is still only open water, to an older lead which has frozen over, and whose edges may have even refrozen to the two larger ice sheets.

	[[Scattering by Multiple Irregularities \|Chapter 9]] deals with the scattering by multiple features in a uniform ice sheet. After presenting some preliminary theory, it follows the structure of [[Williams and Squire 2004a]], and starts by presenting results for the scattering at normal incidence by two pressure ridges or leads, mainly investigating the effect of the feature separation on the scattering. The results are similar to results for two cracks presented in [[Scattering by Multiple Irregularities \|Chapter 8]], but can be taken a little further in that we can also look at the effect of giving the two features different properties from each other. (All cracks are identical, so this couldn't be done in [[Scattering by Multiple Irregularities \|Chapter 8]].)

	These results for two features are also compared with the NEW approximation, which is able to produce results for different feature separations much more quickly than can be obtained if an exact solution is sought. (This is also done in [[Scattering by Multiple Irregularities \|Chapter 8]].) Being a wide spacing approximation, it becomes more accurate as the separation is increased, the exact and approximate curves generally coalescing when the separation is about three quarters of one wavelength.

	[[Scattering by Multiple Irregularities \|Chapter 9]] then seeks to average out the effect of separation by using a serial approximation which can produce results even more quickly than the NEW approximation can. Ridge separations are sampled from an observed type of distribution (log-normal in that case) and the median value of <math>\|R\|</math> (calculated using the NEW method for two identical ridges) is calculated and compared with the serial result. The two curves are almost identical, although the variance of the results about the median is quite large.

	This experiment is repeated for a larger number of identical ridges with similar success; then when the ridge sail heights are also sampled from a random distribution; and it works just as well when large numbers of cracks or leads are used, and separations and lead widths are randomly sampled.

	Having established that the serial approximation summarizes the general scattering properties of a series of features for normal incidence, we also attempt to use it to predict the effect of randomising the feature orientations, and also combine the results for individual features in an attempt to model an ice field populated with cracks, pressure ridges and leads, in the manner of [[Williams and Squire 2004b]]. The scattering seems to be dominated by the effect of the leads.

	This chapter goes a little further than either [[Williams and Squire 2004a]] or [[Williams and Squire 2004b]], however, in that it also presents results for pressure ridges when the flexural effects of a keel are accounted for, or when semi-refrozen leads are also present.

	Finally, it investigates the effect of changing the background ice thickness on scattering results when either one or 100 ridges or leads are present. (The serial approximation is used in the latter case.) A similar result for ridges was given by [[Williams 2004]], but since the leads seem to dominate the scattering by an ice field, results for them are also added to complete the chapter. These last results are presented with the intention of showing that the scattering changes enough with the background ice thickness that there is potential for it to be used to determine that thickness.


	~~[[Reflections on Ice]]~~

Syan077: /* An Ice Sheet as a Thin Elastic Plate */

2007-01-02T00:52:46Z

An Ice Sheet as a Thin Elastic Plate

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	==An Ice Sheet as a Thin Elastic Plate==		==An Ice Sheet as a Thin Elastic Plate==
	An extensive amount of literature exists to support the modelling of an ice-sheet as a linear thin elastic plate, for which the governing equations are presented and discussed in [[Governing Equations for an Elastic Plate of Variable Thickness \|Appendix A]] (also cf. [[Mindlin 1951]]; [[Fung 1965]]). [[Greenhill ~~1987~~]] was the first to propose modelling a floating ice sheet by a thin elastic beam on a fluid foundation, suggesting a dispersion relation based on the Euler-Bernoulli beam theory. He also wrote a further paper, humourously entitled "Skating on thin ice" [[Greenhill 1916]], in which he elaborated further on the dispersion of waves in ice, as well as touching on other related problems. (Another noteworthy comment that was made in that paper is that ice was the first material for which an experimental value of Young's modulus was obtained.)		An extensive amount of literature exists to support the modelling of an ice-sheet as a linear thin elastic plate, for which the governing equations are presented and discussed in [[Governing Equations for an Elastic Plate of Variable Thickness \|Appendix A]] (also cf. [[Mindlin 1951]]; [[Fung 1965]]). [[Greenhill 1887]] was the first to propose modelling a floating ice sheet by a thin elastic beam on a fluid foundation, suggesting a dispersion relation based on the Euler-Bernoulli beam theory. He also wrote a further paper, humourously entitled "Skating on thin ice" [[Greenhill 1916]], in which he elaborated further on the dispersion of waves in ice, as well as touching on other related problems. (Another noteworthy comment that was made in that paper is that ice was the first material for which an experimental value of Young's modulus was obtained.)

	This elastic model for ice has since been corroborated experimentally by numerous researchers, with slight improvements to Greenhill's theory being made with the passage of time. For example, during a series of seismic experiments carried out by [[Ewing et al. 1934]], the generation of flexural waves in thin lake/canal ice was observed in addition to the longitudinal and transverse waves being investigated at the time. Extending the theoretical component of Greenhill's work to include the compressibility of the water beneath the ice sheet and to model the ice as a thin plate rather than as a beam (i.e. as a three-dimensional structure rather than as a two-dimensional one), the measured group velocities of the flexural waves corresponded quite well to those predicted by their theory ([[Ewing and Crary 1934]]).		This elastic model for ice has since been corroborated experimentally by numerous researchers, with slight improvements to Greenhill's theory being made with the passage of time. For example, during a series of seismic experiments carried out by [[Ewing et al. 1934]], the generation of flexural waves in thin lake/canal ice was observed in addition to the longitudinal and transverse waves being investigated at the time. Extending the theoretical component of Greenhill's work to include the compressibility of the water beneath the ice sheet and to model the ice as a thin plate rather than as a beam (i.e. as a three-dimensional structure rather than as a two-dimensional one), the measured group velocities of the flexural waves corresponded quite well to those predicted by their theory ([[Ewing and Crary 1934]]).

Syan077: /* Shore-fast Sea Ice */

2006-12-31T22:33:35Z

Shore-fast Sea Ice

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	where <math>\alpha_0</math> and <math>k_0</math> are the wavenumbers of the propagating waves in the water and in the ice respectively (both are real), and was pointed out by both [[Shapiro and Simpson 1953]] and [[Keller and Weitz 1953]]. (Note that although [[Wadhams 1986]] showed that there is an error in the paper by [[Shapiro and Simpson 1953]], their formula for <math>R</math> is still valid as that mistake only pertained to an energy conservation theorem and an expression for the modulus of the transmission coefficient.) Their solution was obtained by ignoring the elastic properties of the ice, and simply modelling the ice as a collection of independent point masses. This model is referred to as the "mass loading model", which, although somewhat inaccurate for ice sheets (for example it predicts that wavelengths should shorten on moving from water into ice, [[Squire et al. 1995]]), has proved reasonably successful for wave propagation through frazil or pancake ice ([[Wadhams and Holt 1991]]). It effectively reduces the order of the thin plate boundary condition from five to one, eliminating the need to apply additional conditions at the ice-edge.		where <math>\alpha_0</math> and <math>k_0</math> are the wavenumbers of the propagating waves in the water and in the ice respectively (both are real), and was pointed out by both [[Shapiro and Simpson 1953]] and [[Keller and Weitz 1953]]. (Note that although [[Wadhams 1986]] showed that there is an error in the paper by [[Shapiro and Simpson 1953]], their formula for <math>R</math> is still valid as that mistake only pertained to an energy conservation theorem and an expression for the modulus of the transmission coefficient.) Their solution was obtained by ignoring the elastic properties of the ice, and simply modelling the ice as a collection of independent point masses. This model is referred to as the "mass loading model", which, although somewhat inaccurate for ice sheets (for example it predicts that wavelengths should shorten on moving from water into ice, [[Squire et al. 1995]]), has proved reasonably successful for wave propagation through frazil or pancake ice ([[Wadhams and Holt 1991]]). It effectively reduces the order of the thin plate boundary condition from five to one, eliminating the need to apply additional conditions at the ice-edge.

	[[Evans and Davies 1968]] subsequently used the Wiener-Hopf method ([[Noble 1958]]; [[Roos 1969]]) to solve the full problem. At the time the infinite products that formed a large part of the solution proved to be too difficult to compute and so they were not able to present any results, except for certain limiting cases (such as the shallow water limit). Incomplete mode-matching approaches were then taken (e.g. [[Hendrickson and Webb 1963]], [[Wadhams 1973]],[[Squire 1978]], [[Squire 1984]], [[Wadhams 1986]]), before [[Fox and Squire 1990]] computed the solution for the full set of eigenfunctions using a conjugate gradient technique. [[Fox and Squire 1994]] used the same method to complete further studies on this problem, investigating the strain in the ice, and also the effect of shore fast ice on an incoming directional wave spectrum of specified structure.		[[Evans and Davies 1968]] subsequently used the Wiener-Hopf method ([[Noble 1958]]; [[Roos 1969]]) to solve the full problem. At the time the infinite products that formed a large part of the solution proved to be too difficult to compute and so they were not able to present any results, except for certain limiting cases (such as the shallow water limit). Incomplete mode-matching approaches were then taken (e.g. [[Hendrickson and Webb 1963]], [[Wadhams 1973]], [[Squire 1978]], [[Squire 1984]], [[Wadhams 1986]]), before [[Fox and Squire 1990]] computed the solution for the full set of eigenfunctions using a conjugate gradient technique. [[Fox and Squire 1994]] used the same method to complete further studies on this problem, investigating the strain in the ice, and also the effect of shore fast ice on an incoming directional wave spectrum of specified structure.

	[[Gol'dshtein and Marchenko 1989]] presented a later Wiener-Hopf solution for infinite depth, but again only certain limiting situations were discussed. However, in the late 1990's and early <math>21^st</math> century improvements in computing power enabled other authors to compute results using the Wiener-Hopf solution of [[Evans and Davies 1968]]. [[Balmforth and Craster 1999]] turned the required infinite products into integrals which were evaluated by quadrature, while [[Chung and Fox 2002]] showed that the products themselves could be evaluated directly with relatively little effort. Ironically [[Tkacheva 2001]] finally showed that if the inertia term in the thin plate equation was neglected for normally incident waves (as it can for most wavelengths), then <math>\|R\|</math> could be calculated by simply using the correct value for <math>k_0</math>, the wavenumber in the ice, in the formula (1.1) given by [[Keller and Weitz 1953]].		[[Gol'dshtein and Marchenko 1989]] presented a later Wiener-Hopf solution for infinite depth, but again only certain limiting situations were discussed. However, in the late 1990's and early <math>21^st</math> century improvements in computing power enabled other authors to compute results using the Wiener-Hopf solution of [[Evans and Davies 1968]]. [[Balmforth and Craster 1999]] turned the required infinite products into integrals which were evaluated by quadrature, while [[Chung and Fox 2002]] showed that the products themselves could be evaluated directly with relatively little effort. Ironically [[Tkacheva 2001]] finally showed that if the inertia term in the thin plate equation was neglected for normally incident waves (as it can for most wavelengths), then <math>\|R\|</math> could be calculated by simply using the correct value for <math>k_0</math>, the wavenumber in the ice, in the formula (1.1) given by [[Keller and Weitz 1953]].

Syan077: /* An Ice Strip of Finite Width in Open Water */

2006-12-31T22:30:10Z

An Ice Strip of Finite Width in Open Water

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	A second problem that has generated a lot of the mathematics used herein is that of the scattering of water waves by an ice strip of finite width surrounded by open water. By restricting the incoming waves to normal incidence, it was initially intended as a two-dimensional model for a single ice floe such as might occur in the MIZ ([[Meylan 1993]]; [[Meylan and Squire 1994]]). Although modelling of ice floes has since moved away from this approach towards a more three-dimensional one (e.g. [[Meylan et al. 1997]]; [[Peter and Meylan 2004),the results and techniques are still quite applicable to situations where the length of the floating object is large in comparison to other characteristic lengths of the problem. Examples include the Mega-Float discussed in Section 1.3 ([[Kanzawa et al. 2002]]).		A second problem that has generated a lot of the mathematics used herein is that of the scattering of water waves by an ice strip of finite width surrounded by open water. By restricting the incoming waves to normal incidence, it was initially intended as a two-dimensional model for a single ice floe such as might occur in the MIZ ([[Meylan 1993]]; [[Meylan and Squire 1994]]). Although modelling of ice floes has since moved away from this approach towards a more three-dimensional one (e.g. [[Meylan et al. 1997]]; [[Peter and Meylan 2004),the results and techniques are still quite applicable to situations where the length of the floating object is large in comparison to other characteristic lengths of the problem. Examples include the Mega-Float discussed in Section 1.3 ([[Kanzawa et al. 2002]]).

	The solution in the shallow water limit is relatively straightforward ([[Stoker 1957]]) and simply involves the inversion of an <math>8\times8</math> matrix. In his Ph.D. thesis, Meylan 1993 solved the problem for both finite and infinite depth by using Green's theorem and the two-dimensional open water Green's function to set up an integral equation over the floe (the infinite depth solution was also published in [[Meylan and Squire 1994]]. It is also possible to extend the mode-matching technique of [[Sahoo et al. 2001]] to treat this problem, and the author is aware of several researchers who have done this, although such a solution is still unpublished. This technique could be improved by using the residue calculus method in the same way that is it is used in [[Appendix F]] of this thesis, which treats the problem of the scattering by three adjacent ice sheets of different thicknesses. With minor adjustments, either or both of the outer ice sheets may be replaced with open water (cf. Section 1.3 of [[Scattering by a Single Lead]]; published papers on the residue calculus method include those by [[Linton and Chung 2003]], [[Chung and Linton 2005]]).		The solution in the shallow water limit is relatively straightforward ([[Stoker 1957]]) and simply involves the inversion of an <math>8\times8</math> matrix. In his Ph.D. thesis, Meylan 1993 solved the problem for both finite and infinite depth by using Green's theorem and the two-dimensional open water Green's function to set up an integral equation over the floe (the infinite depth solution was also published in [[Meylan and Squire 1994]]. It is also possible to extend the mode-matching technique of [[Sahoo et al. 2001]] to treat this problem, and the author is aware of several researchers who have done this, although such a solution is still unpublished. This technique could be improved by using the residue calculus method in the same way that is it is used in [[Comparison of Residue Calculus Solution for a Lead with Wiener-Hopf Solution \|Appendix F]] of this thesis, which treats the problem of the scattering by three adjacent ice sheets of different thicknesses. With minor adjustments, either or both of the outer ice sheets may be replaced with open water (cf. Section 1.3 of [[Scattering by a Single Lead]]; published papers on the residue calculus method include those by [[Linton and Chung 2003]], [[Chung and Linton 2005]]).

	[[Tkacheva 2002]] solved a pair of coupled Wiener-Hopf equations to produce another approach to this problem, using a similar method to the one that is used in [[Scattering by a Single Lead]] of this thesis, and [[Meylan 2002]] solved the time dependent problem by using a spectral approach.		[[Tkacheva 2002]] solved a pair of coupled Wiener-Hopf equations to produce another approach to this problem, using a similar method to the one that is used in [[Scattering by a Single Lead]] of this thesis, and [[Meylan 2002]] solved the time dependent problem by using a spectral approach.

Syan077: /* An Ice Sheet as a Thin Elastic Plate */

2006-12-28T02:52:11Z

An Ice Sheet as a Thin Elastic Plate

← Older revision		Revision as of 02:52, 28 December 2006
Line 65:		Line 65:

	==An Ice Sheet as a Thin Elastic Plate==		==An Ice Sheet as a Thin Elastic Plate==
	An extensive amount of literature exists to support the modelling of an ice-sheet as a linear thin elastic plate, for which the governing equations are presented and discussed in [[Appendix A]] (also cf. [[Mindlin 1951]]; [[Fung 1965]]). [[Greenhill 1987]] was the first to propose modelling a floating ice sheet by a thin elastic beam on a fluid foundation, suggesting a dispersion relation based on the Euler-Bernoulli beam theory. He also wrote a further paper, humourously entitled "Skating on thin ice" [[Greenhill 1916]], in which he elaborated further on the dispersion of waves in ice, as well as touching on other related problems. (Another noteworthy comment that was made in that paper is that ice was the first material for which an experimental value of Young's modulus was obtained.)		An extensive amount of literature exists to support the modelling of an ice-sheet as a linear thin elastic plate, for which the governing equations are presented and discussed in [[Governing Equations for an Elastic Plate of Variable Thickness \|Appendix A]] (also cf. [[Mindlin 1951]]; [[Fung 1965]]). [[Greenhill 1987]] was the first to propose modelling a floating ice sheet by a thin elastic beam on a fluid foundation, suggesting a dispersion relation based on the Euler-Bernoulli beam theory. He also wrote a further paper, humourously entitled "Skating on thin ice" [[Greenhill 1916]], in which he elaborated further on the dispersion of waves in ice, as well as touching on other related problems. (Another noteworthy comment that was made in that paper is that ice was the first material for which an experimental value of Young's modulus was obtained.)

	This elastic model for ice has since been corroborated experimentally by numerous researchers, with slight improvements to Greenhill's theory being made with the passage of time. For example, during a series of seismic experiments carried out by [[Ewing et al. 1934]], the generation of flexural waves in thin lake/canal ice was observed in addition to the longitudinal and transverse waves being investigated at the time. Extending the theoretical component of Greenhill's work to include the compressibility of the water beneath the ice sheet and to model the ice as a thin plate rather than as a beam (i.e. as a three-dimensional structure rather than as a two-dimensional one), the measured group velocities of the flexural waves corresponded quite well to those predicted by their theory ([[Ewing and Crary 1934]]).		This elastic model for ice has since been corroborated experimentally by numerous researchers, with slight improvements to Greenhill's theory being made with the passage of time. For example, during a series of seismic experiments carried out by [[Ewing et al. 1934]], the generation of flexural waves in thin lake/canal ice was observed in addition to the longitudinal and transverse waves being investigated at the time. Extending the theoretical component of Greenhill's work to include the compressibility of the water beneath the ice sheet and to model the ice as a thin plate rather than as a beam (i.e. as a three-dimensional structure rather than as a two-dimensional one), the measured group velocities of the flexural waves corresponded quite well to those predicted by their theory ([[Ewing and Crary 1934]]).

Syan077: /* Arctic and Antarctic Sea Ice */

2006-12-27T05:06:42Z

Arctic and Antarctic Sea Ice

← Older revision		Revision as of 05:06, 27 December 2006
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	=Arctic and Antarctic Sea Ice=		=Arctic and Antarctic Sea Ice=
	This section begins by giving an overview of the important role that polar sea ice plays in the earth's environment ([[#Environmental Role of Sea Ice \|Section 1.1.1]]). This includes the reflection of sunlight and the production of bottom water which helps to drive global currents. [[#Regions to be Modelled \|Section 1.1.2]] then proceeds to describe the specific regions we intend to model, while Section 1.3 discusses how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.		This section begins by giving an overview of the important role that polar sea ice plays in the earth's environment ([[#Environmental Role of Sea Ice \|Section 1.1.1]]). This includes the reflection of sunlight and the production of bottom water which helps to drive global currents. [[#Regions to be Modelled \|Section 1.1.2]] then proceeds to describe the specific regions we intend to model, while [[#Engineering Applications of Ice Research \|Section 1.1.3]] discusses how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.

	Given the importance of sea ice in the environment, a convenient, remote ice-thickness-sensing technique for monitoring the condition of the polar ice sheets would be a valuable asset in monitoring the overall health of the planet. Techniques currently in use are submarine sonar, most recently coupled with LIDAR---the former is able to detect the under-sea profile of the ice, while the latter uses a laser operated from an aeroplane to detect the ice's upper profile---and EMS, electromagnetic sensing from a specially-fitted helicopter. If ocean wave scattering could be used to deduce average ice thickness, it might be possible to measure it using suitably placed strain gauges or tilt meters, which would eliminate the need for aeroplanes or submarines to be in operation for extended periods of time.		Given the importance of sea ice in the environment, a convenient, remote ice-thickness-sensing technique for monitoring the condition of the polar ice sheets would be a valuable asset in monitoring the overall health of the planet. Techniques currently in use are submarine sonar, most recently coupled with LIDAR---the former is able to detect the under-sea profile of the ice, while the latter uses a laser operated from an aeroplane to detect the ice's upper profile---and EMS, electromagnetic sensing from a specially-fitted helicopter. If ocean wave scattering could be used to deduce average ice thickness, it might be possible to measure it using suitably placed strain gauges or tilt meters, which would eliminate the need for aeroplanes or submarines to be in operation for extended periods of time.

Syan077: /* Arctic and Antarctic Sea Ice */

2006-12-27T05:05:48Z

Arctic and Antarctic Sea Ice

← Older revision		Revision as of 05:05, 27 December 2006
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	=Arctic and Antarctic Sea Ice=		=Arctic and Antarctic Sea Ice=
	This section begins by giving an overview of the important role that polar sea ice plays in the earth's environment ([[#Environmental Role of Sea Ice \|Section 1.1.1]]). This includes the reflection of sunlight and the production of bottom water which helps to drive global currents. Section 1.2 then proceeds to describe the specific regions we intend to model, while Section 1.3 discusses how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.		This section begins by giving an overview of the important role that polar sea ice plays in the earth's environment ([[#Environmental Role of Sea Ice \|Section 1.1.1]]). This includes the reflection of sunlight and the production of bottom water which helps to drive global currents. [[#Regions to be Modelled \|Section 1.1.2]] then proceeds to describe the specific regions we intend to model, while Section 1.3 discusses how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.

	Given the importance of sea ice in the environment, a convenient, remote ice-thickness-sensing technique for monitoring the condition of the polar ice sheets would be a valuable asset in monitoring the overall health of the planet. Techniques currently in use are submarine sonar, most recently coupled with LIDAR---the former is able to detect the under-sea profile of the ice, while the latter uses a laser operated from an aeroplane to detect the ice's upper profile---and EMS, electromagnetic sensing from a specially-fitted helicopter. If ocean wave scattering could be used to deduce average ice thickness, it might be possible to measure it using suitably placed strain gauges or tilt meters, which would eliminate the need for aeroplanes or submarines to be in operation for extended periods of time.		Given the importance of sea ice in the environment, a convenient, remote ice-thickness-sensing technique for monitoring the condition of the polar ice sheets would be a valuable asset in monitoring the overall health of the planet. Techniques currently in use are submarine sonar, most recently coupled with LIDAR---the former is able to detect the under-sea profile of the ice, while the latter uses a laser operated from an aeroplane to detect the ice's upper profile---and EMS, electromagnetic sensing from a specially-fitted helicopter. If ocean wave scattering could be used to deduce average ice thickness, it might be possible to measure it using suitably placed strain gauges or tilt meters, which would eliminate the need for aeroplanes or submarines to be in operation for extended periods of time.

Syan077: /* Arctic and Antarctic Sea Ice */

2006-12-27T05:05:00Z

Arctic and Antarctic Sea Ice

← Older revision		Revision as of 05:05, 27 December 2006
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	=Arctic and Antarctic Sea Ice=		=Arctic and Antarctic Sea Ice=
	This section begins by giving an overview of the important role that polar sea ice plays in the earth's environment (Section 1.1). This includes the reflection of sunlight and the production of bottom water which helps to drive global currents. Section 1.2 then proceeds to describe the specific regions we intend to model, while Section 1.3 discusses how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.		This section begins by giving an overview of the important role that polar sea ice plays in the earth's environment ([[#Environmental Role of Sea Ice \|Section 1.1.1]]). This includes the reflection of sunlight and the production of bottom water which helps to drive global currents. Section 1.2 then proceeds to describe the specific regions we intend to model, while Section 1.3 discusses how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.

	Given the importance of sea ice in the environment, a convenient, remote ice-thickness-sensing technique for monitoring the condition of the polar ice sheets would be a valuable asset in monitoring the overall health of the planet. Techniques currently in use are submarine sonar, most recently coupled with LIDAR---the former is able to detect the under-sea profile of the ice, while the latter uses a laser operated from an aeroplane to detect the ice's upper profile---and EMS, electromagnetic sensing from a specially-fitted helicopter. If ocean wave scattering could be used to deduce average ice thickness, it might be possible to measure it using suitably placed strain gauges or tilt meters, which would eliminate the need for aeroplanes or submarines to be in operation for extended periods of time.		Given the importance of sea ice in the environment, a convenient, remote ice-thickness-sensing technique for monitoring the condition of the polar ice sheets would be a valuable asset in monitoring the overall health of the planet. Techniques currently in use are submarine sonar, most recently coupled with LIDAR---the former is able to detect the under-sea profile of the ice, while the latter uses a laser operated from an aeroplane to detect the ice's upper profile---and EMS, electromagnetic sensing from a specially-fitted helicopter. If ocean wave scattering could be used to deduce average ice thickness, it might be possible to measure it using suitably placed strain gauges or tilt meters, which would eliminate the need for aeroplanes or submarines to be in operation for extended periods of time.

Syan077 at 05:04, 27 December 2006

2006-12-27T05:04:06Z

← Older revision		Revision as of 05:04, 27 December 2006
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	The sea ice of both the Arctic and the Antarctic plays a very important role in the earth's climate. [[#Arctic and Antarctic Sea Ice \|Section 1.1]] below presents a general overview of this. As well as describing the specific regions we intend to model, that section also describes how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.		The sea ice of both the Arctic and the Antarctic plays a very important role in the earth's climate. [[#Arctic and Antarctic Sea Ice \|Section 1.1]] below presents a general overview of this. As well as describing the specific regions we intend to model, that section also describes how work involving wave interactions with sea ice can be generalized to provide results for certain situations that occur in marine engineering.

	The following sections describe and justify our choice of model for polar sea ice ([[#An Ice Sheet as a Thin Elastic Plate \|Section 1.2.1]]), review and discuss mathematical techniques used previously and currently in use in the wave scattering literature ([[#Existing Solution Techniques \|Section 1.2.2]]), and provide the general outline that this thesis will follow (Section 3).		The following sections describe and justify our choice of model for polar sea ice ([[#An Ice Sheet as a Thin Elastic Plate \|Section 1.2.1]]), review and discuss mathematical techniques used previously and currently in use in the wave scattering literature ([[#Existing Solution Techniques \|Section 1.2.2]]), and provide the general outline that this thesis will follow ([[#Outline of Thesis \|Section 1.3]]).

	=Arctic and Antarctic Sea Ice=		=Arctic and Antarctic Sea Ice=

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	This result was taken by [[Williams 2004]] to suggest that the potential for using flexural waves in remote ice-thickness-sensing could initially be considered by simply using the serial approximation. In that paper, reflection curves for a given set of ridges were compared for different ice thicknesses and were found to be fairly different over a relatively wide range of periods (from about 2s to 11s). Thus there seemed to be some hope that scattering results for different thicknesses would be sufficiently different to use those results in determining the ice thickness.		This result was taken by [[Williams 2004]] to suggest that the potential for using flexural waves in remote ice-thickness-sensing could initially be considered by simply using the serial approximation. In that paper, reflection curves for a given set of ridges were compared for different ice thicknesses and were found to be fairly different over a relatively wide range of periods (from about 2s to 11s). Thus there seemed to be some hope that scattering results for different thicknesses would be sufficiently different to use those results in determining the ice thickness.

	~~=Outline of Thesis=~~
	In [[Mathematical Formulation of the Problem \| Chapter 2]] we begin by presenting the governing equations that we wish our velocity potential to satisfy. These equations are derived and discussed in [[Governing Equations for an Elastic Plate of Variable Thickness \|Appendix A]]. Then, using the Green's functions that are discussed in [[The Green's Function \|Chapter 3]] for both finite and infinite depth, we proceed to set up an integral equation in [[Formulation of General Integral Equation \|Chapter 4]] in terms of the displacement at the ice-water interface (or at the air-water interface---such as in the problem of an open lead). This integral equation may either be over a finite or an infinite interval.

	Of particular interest in [[The Green's Function \|Chapter 3]] is the presentation of calculation techniques for the infinite depth Green's function for oblique incidence. These techniques include Fast Fourier Transform (FFT), Taylor series expansions, and numerical integration. Each of these methods have particular advantages and disadvantages, depending on the number of points that the Green's function needs to be evaluated at and the distance of the points from the singularity. They are described in more detail in [[Computation of Infinite Depth Green's Function \|Appendix C]]. Another appendix that is related to this chapter is [[Roots of the Dispersion Relation \|Appendix B]], which discusses the behaviour of the roots of the shallow water, finite depth and infinite depth dispersion relations. The small period behaviour of the latter two sets of roots has never been presented before and, although it generally only occurs at physically impractical periods, is nevertheless extremely interesting mathematically.

	In [[Formulation of General Integral Equation \|Chapter 4]] the thickness profile of the ice is allowed to vary arbitrarily over a finite strip with the ice sheets of different but constant thicknesses. The method used to include a variable thickness profile was first reported by [[Williams and Squire 2004a]], but the generalization of this approach to include different thicknesses on each side of the strip is new and is yet to be published. This chapter also presents a proof that the potential may always be represented by an eigenfunction expansion.

	[[Scattering by a Single Crack \|Chapter 5]], [[Scattering by a Single Ridge \|Chapter 6]], [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] and [[Scattering by Multiple Irregularities \|Chapter 8]] solve the integral equation derived in [[Formulation of General Integral Equation \|Chapter 4]] in a variety of different situations, and are arranged in increasing order of mathematical difficulty. [[Scattering by a Single Crack \|Chapter 5]] and [[Scattering by a Single Ridge \|Chapter 6]] can both be considered as treating special cases of the problem presented in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], both in terms of the actual complexity of the physical problems that they solve, and in the methods of solution that they employ. In physical terms, the problem solved in [[Scattering by Multiple Irregularities \|Chapter 8]] is also a special case of the one in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], but mathematically the use of the Wiener-Hopf technique in particular is more involved in [[Scattering by Multiple Irregularities \|Chapter 8]] than in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], in which the Wiener-Hopf technique is first introduced. Consequently, [[Scattering by Multiple Irregularities \|Chapter 8]] provides a more analytical solution method than [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], but to a smaller range of physical problems; although [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] reports a more numerical method, it can as a result deal with a wider range of problems.

	The simplest problem is that of a single crack in a single uniform ice sheet. As mentioned in the previous section, this problem was solved for infinitely deep water and normally incident waves by [[Squire and Dixon 2000]], obliquely incident waves by [[Williams and Squire 2002]], and finitely deep water by [[Evans and Porter 2003a]]. However, the solution follows in only one or two steps from the final result in [[Formulation of General Integral Equation \|Chapter 4]], which, as discussed above, solves a considerably more general problem. Moreover, it was also thought that the simplicity of the results obtained provide a good opportunity to investigate their behaviour with some of the more fundamental wave/ice parameters such as ice thickness, wave period, angle of incidence and water depth. For example, it is shown that the effects of the former two can be approximately combined into a single parameter, the nondimensional period, while in the discussion of the effect of water depth, criteria with regard to the accuracy of the shallow and infinite depth approximations are established.

	[[Scattering by a Single Ridge \|Chapter 6]] describes scattering by a pressure ridge. It is modelled as a thin plate of variable thickness with zero submergence ([[Williams and Squire 2004a]]; [[Williams and Squire 2004b]]; [[Williams 2004]]), and, as in those papers, the problem is solved by setting up an integral equation over the interval in which the ice thickness is varying. This equation is solved numerically.

	The results presented in this chapter are similar to the single-ridge results presented in the aforementioned papers. However, those results are adjusted to allow for the increased thickness due to the ridge keel. In light of the recent work of [[Porter and Porter 2004]], changing the thickness in this way should improve the accuracy of the [[Williams and Squire 2004a]] results for large water depths, but without significant changes to that method. [[Proposed Solution Technique for a Pressure Ridge With a Submerged Keel \|Appendix E]] also treats the topic of scattering by a pressure ridge, proposing a method by which submergence could be included in full, although no results are able to be presented yet.

	[[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] extends the variable thin plate formulation developed in [[Scattering by a Single Ridge \|Chapter 6]] by allowing the ice thickness to the right of the variable region (which in this chapter we will refer to as a ramp) to differ from the ice thickness to its left. To do this we must solve two coupled integral equations---the first is much the same as the one solved in [[Scattering by a Single Ridge \|Chapter 6]], but the second is over a semi-infinite range---over the ice-water interface to the right of the ramp. Proceeding in exactly the same way as in [[Scattering by a Single Ridge \|Chapter 6]], we proceed to solve the first integral equation numerically for the displacement in the ramp, expressing it in terms of the displacement of the ice in the semi-infinite region. The second integral equation is one of the Wiener-Hopf type, and so we are able to write an analytical expression for the displacement in the semi-infinite region in terms of the displacement within the ramp. We can now eliminate the displacement in the right hand region from the two sets of equations, solve for the displacement in the ramp, and then regenerate the displacement to the right.

	The method in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]] is rather novel, as it combines a numerical integral quadrature scheme with the analytical Wiener-Hopf technique. Previously, authors have tended to favour either one approach or the other, and so this result also represents a philosophical breakthrough in that the combined weight of both methods can be brought to bear on a single problem most effectively. The numerical approach is able to deal with the arbitrariness of the ice thickness within the ramp, while the Wiener-Hopf technique is ideal for the semi-infinite interval over which the second integral equation must be solved---such intervals are extremely hard to deal with numerically.

	Unfortunately, the method does have the same drawback that our ridge model had---that submergence must be ignored. However, the method of [[Porter and Porter 2004]] also incorporated a change in ice thickness from one side of the variable region to the other, and so we could again use their result that for large water depths the effect of submergence is negligible as long as the correct total thickness is used (and as long as the mild slope assumption is valid).

	The physical problem solved in [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]], i.e. a ramp connecting a thinner ice sheet to a thicker one, has practical applications. Such a thickness profile could be taken to describe a sea ice/ice shelf transition such as the one observed in the Ross Sea, where the thickness of the sea ice beyond the ice shelf increases steadily to meet the thicker ice of the ice shelf itself. Alternatively, by setting the right hand ice thickness equal to zero, and by letting the incident wave arrive from the right instead of from the left, the problem could represent a breakwater shielding a VLFS from waves arriving from the open ocean.

	[[Scattering by Multiple Irregularities \|Chapter 8]] treats a special case of the problems solved in [[Scattering by a Single Ridge \|Chapter 6]] and [[The "General Solution": Scattering by a Sea Ice/Ice Shelf Transition \|Chapter 7]]. When the ice thickness over the variable region is constant, a more analytical approach may be taken. The problem may be written as two coupled Wiener-Hopf integral equations, producing a system of <math>M</math> linear equations in as many variables, where <math>M</math> is the number of evanescent waves generated at each end of the middle region that have not decayed sufficiently by the time they reach the other end to be able to be ignored. Since the evanescent waves decay exponentially with distance, the order of coupling <math>M</math> between the two semi-infinite regions decreases as the width of the middle region increases. If that region is wide enough, <math>M</math> becomes 1, and only the propagating waves generated need to be considered. Thus only a scalar matrix needs to be inverted, and the problem reduces to the well-known wide spacing approximation [[Evans 1990]], abbreviated by [[Williams and Squire 2004a]] to the NEW approximation (No Evanescent Waves).

	In the above special case, the approach of [[Scattering by Multiple Irregularities \|Chapter 8]] complements the numerical approach of the two earlier chapters well. When the central region is relatively wide, the method of [[Scattering by Multiple Irregularities \|Chapter 8]] becomes extremely efficient, while if <math>\alpha</math>, the width of that central region, is decreased and less evanescent modes can be neglected, less quadrature points are needed to obtain the numerical solution and that method would be preferable. Given the wave period, angle of incidence and ice thicknesses, the coupled Wiener-Hopf method can generate results for many different values of <math>\alpha</math> quite rapidly, while if <math>\alpha</math> is fixed and results are required for when the thickness of the central region is allowed to take several different values, the quadrature schemes would be more efficient.

	Practically speaking again, the situation treated in [[Scattering by Multiple Irregularities \|Chapter 8]] is a good model for a lead separating two sheets of ice. Such features are extremely common in both the Arctic and Antarctic regions. Different stages of refreezing may be represented, ranging from a freshly formed lead which is still only open water, to an older lead which has frozen over, and whose edges may have even refrozen to the two larger ice sheets.

	[[Scattering by Multiple Irregularities \|Chapter 9]] deals with the scattering by multiple features in a uniform ice sheet. After presenting some preliminary theory, it follows the structure of [[Williams and Squire 2004a]], and starts by presenting results for the scattering at normal incidence by two pressure ridges or leads, mainly investigating the effect of the feature separation on the scattering. The results are similar to results for two cracks presented in [[Scattering by Multiple Irregularities \|Chapter 8]], but can be taken a little further in that we can also look at the effect of giving the two features different properties from each other. (All cracks are identical, so this couldn't be done in [[Scattering by Multiple Irregularities \|Chapter 8]].)

	These results for two features are also compared with the NEW approximation, which is able to produce results for different feature separations much more quickly than can be obtained if an exact solution is sought. (This is also done in [[Scattering by Multiple Irregularities \|Chapter 8]].) Being a wide spacing approximation, it becomes more accurate as the separation is increased, the exact and approximate curves generally coalescing when the separation is about three quarters of one wavelength.

	[[Scattering by Multiple Irregularities \|Chapter 9]] then seeks to average out the effect of separation by using a serial approximation which can produce results even more quickly than the NEW approximation can. Ridge separations are sampled from an observed type of distribution (log-normal in that case) and the median value of <math>\|R\|</math> (calculated using the NEW method for two identical ridges) is calculated and compared with the serial result. The two curves are almost identical, although the variance of the results about the median is quite large.

	This experiment is repeated for a larger number of identical ridges with similar success; then when the ridge sail heights are also sampled from a random distribution; and it works just as well when large numbers of cracks or leads are used, and separations and lead widths are randomly sampled.

	Having established that the serial approximation summarizes the general scattering properties of a series of features for normal incidence, we also attempt to use it to predict the effect of randomising the feature orientations, and also combine the results for individual features in an attempt to model an ice field populated with cracks, pressure ridges and leads, in the manner of [[Williams and Squire 2004b]]. The scattering seems to be dominated by the effect of the leads.

	This chapter goes a little further than either [[Williams and Squire 2004a]] or [[Williams and Squire 2004b]], however, in that it also presents results for pressure ridges when the flexural effects of a keel are accounted for, or when semi-refrozen leads are also present.

	Finally, it investigates the effect of changing the background ice thickness on scattering results when either one or 100 ridges or leads are present. (The serial approximation is used in the latter case.) A similar result for ridges was given by [[Williams 2004]], but since the leads seem to dominate the scattering by an ice field, results for them are also added to complete the chapter. These last results are presented with the intention of showing that the scattering changes enough with the background ice thickness that there is potential for it to be used to determine that thickness.


	~~[[Reflections on Ice]]~~