Roots of the Dispersion Relation - Revision history

Meylan at 10:34, 10 September 2009

2009-09-10T10:34:13Z

← Older revision		Revision as of 10:34, 10 September 2009
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			{{pages to be deleted}}

	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]].		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]].

Syan077: /* Finite Depth Roots */

2006-12-24T22:58:29Z

Finite Depth Roots

← Older revision		Revision as of 22:58, 24 December 2006
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	The dotted lines in Figure B.4<math>\it{b}</math> are reference lines, which for 1-m-thick ice correspond to periods of 1.9s (<math>\varpi=0</math>), 0.1s (<math>\varpi=-0.35</math>), 10ms (<math>\varpi=-0.88</math>) and 1ms (<math>\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 dotted lines in Figure B.4<math>\it{b}</math> are reference lines, which for 1-m-thick ice correspond to periods of 1.9s (<math>\varpi=0</math>), 0.1s (<math>\varpi=-0.35</math>), 10ms (<math>\varpi=-0.88</math>) and 1ms (<math>\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>\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.		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>\varpi >\sim 0.087</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>H</math>. If our minimum period was about 1.9s, so that <math>\varpi\geq0</math>, multiple roots could only occur if <math>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>\mu=0</math> and <math>\varpi=\lambda-\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>H=5</math> is used if the infinite depth results are not calculated directly.)		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>H</math>. If our minimum period was about 1.9s, so that <math>\varpi\geq0</math>, multiple roots could only occur if <math>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>\mu=0</math> and <math>\varpi=\lambda-\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>H=5</math> is used if the infinite depth results are not calculated directly.)

Syan077: /* Finite Depth Roots */

2006-12-24T22:44:32Z

Finite Depth Roots

← Older revision		Revision as of 22:44, 24 December 2006
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	<center><math>		<center><math>
	f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,		f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,\,
	</math></center>		</math></center>

Syan077: /* Shallow Water Roots */

2006-12-24T22:44:15Z

Shallow Water Roots

← Older revision		Revision as of 22:44, 24 December 2006
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	Recall that the finite depth dispersion relation for the left-hand sheet of ice is		Recall that the finite depth dispersion relation for the left-hand sheet of ice is

	<center><math>f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,</math></center>		<center><math>
			f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,\,
			</math></center>

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

Syan077 at 22:43, 24 December 2006

2006-12-24T22:43:36Z

← Older revision		Revision as of 22:43, 24 December 2006
Line 4:		Line 4:
	Recall that the finite depth dispersion relation for the left-hand sheet of ice is		Recall that the finite depth dispersion relation for the left-hand sheet of ice is

	<center><math>		<center><math>f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,</math></center>
	f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,\,
	</math></center>

	where <math>\Lambda_0=\kappa^4+\varpi</math>. Since <math>\tanh\kappa\approx\kappa H</math> for small <math>H</math>, for shallow water we can obtain the approximate dispersion relation <math>p_s(\kappa^2)=(\kappa^4+\varpi)\kappa^2H-1=0</math>, which is a cubic polynomial in <math>\kappa^2</math>.		where <math>\Lambda_0=\kappa^4+\varpi</math>. Since <math>\tanh\kappa\approx\kappa H</math> for small <math>H</math>, for shallow water we can obtain the approximate dispersion relation <math>p_s(\kappa^2)=(\kappa^4+\varpi)\kappa^2H-1=0</math>, which is a cubic polynomial in <math>\kappa^2</math>.
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	<center><math>		<center><math>
	f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,\,		f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,
	</math></center>		</math></center>

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	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>N</math>, <math>p_H</math> has <math>2N+6</math> zeros inside the square with corners <math>\pm(N+\pi/4)\times(1+\mbox{i})</math>. The <math>N</math> real <math>w_n</math> and their negatives, and <math>\pm w_0</math>, which are imaginary, account for <math>2N+2</math> of these roots, leaving four still to be located. Since it is easily shown that <math>w_0</math> is a simple root (<math>f_0'(\gamma_0)<0</math>), we have verified that <math>p_H</math> must have two complex roots, <math>-w_{-2}</math> in the first quadrant, and its complex conjugate <math>w_{-1}</math>, or else two additional real roots. In that case we will distinguish between them by choosing <math>w_{-1}</math> to have a larger modulus than <math>w_{-2}</math>.		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>N</math>, <math>p_H</math> has <math>2N+6</math> zeros inside the square with corners <math>\pm(N+\pi/4)\times(1+\mbox{i})</math>. The <math>N</math> real <math>w_n</math> and their negatives, and <math>\pm w_0</math>, which are imaginary, account for <math>2N+2</math> of these roots, leaving four still to be located. Since it is easily shown that <math>w_0</math> is a simple root (<math>f_0'(\gamma_0)<0</math>), we have verified that <math>p_H</math> must have two complex roots, <math>-w_{-2}</math> in the first quadrant, and its complex conjugate <math>w_{-1}</math>, or else two additional real roots. In that case we will distinguish between them by choosing <math>w_{-1}</math> to have a larger modulus than <math>w_{-2}</math>.

	[[Figure B.2]]: Small period behaviour of the roots of the finite depth dispersion relation for ice-coupled waves, <math>f_0(\kappa)=\coth\kappa H/\kappa-\~~L_0~~(\kappa)</math>, where <math>\~~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>\bar p_H(w)=p_H(w)/(w^5+H^5)</math> for real <math>w</math> for decreasing values of <math>\varpi</math> (shorter periods and/or larger ice thicknesses). <math>\bar p_H</math> has the same zeros as <math>p_H(w)=H^4w\sin w\times f_0(iw/H)</math> but does not grow as quickly for large <math>w</math>; if <math>w_n</math> is a root of <math>p_H</math> (or <math>\bar p_H</math>), then <math>\gamma_n=iw_n/H</math> is a root of <math>f_0</math>. Note the formation of double zeros near <math>\varpi=-1.51</math> and between <math>-60</math> and <math>-95</math>. The right hand plots show the locations of the <math>w_n</math> for the same values of <math>\varpi</math>. Arrows on two of those plots indicate the direction that the complex roots move in as <math>\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>H=1</math>.		[[Figure B.2]]: Small period behaviour of the roots of the finite depth dispersion relation for ice-coupled waves, <math>f_0(\kappa)=\coth\kappa H/\kappa-\Lambda_0(\kappa)</math>, where <math>\Lambda_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>\bar p_H(w)=p_H(w)/(w^5+H^5)</math> for real <math>w</math> for decreasing values of <math>\varpi</math> (shorter periods and/or larger ice thicknesses). <math>\bar p_H</math> has the same zeros as <math>p_H(w)=H^4w\sin w\times f_0(iw/H)</math> but does not grow as quickly for large <math>w</math>; if <math>w_n</math> is a root of <math>p_H</math> (or <math>\bar p_H</math>), then <math>\gamma_n=iw_n/H</math> is a root of <math>f_0</math>. Note the formation of double zeros near <math>\varpi=-1.51</math> and between <math>-60</math> and <math>-95</math>. The right hand plots show the locations of the <math>w_n</math> for the same values of <math>\varpi</math>. Arrows on two of those plots indicate the direction that the complex roots move in as <math>\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>H=1</math>.


Line 102:		Line 100:
	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>\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.		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>\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>H</math>. If our minimum period was about 1.9s, so that <math>\varpi\geq0</math>, multiple roots could only occur if <math>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>\mu=0</math> and <math>\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>H=5</math> is used if the infinite depth results are not calculated directly.)		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>H</math>. If our minimum period was about 1.9s, so that <math>\varpi\geq0</math>, multiple roots could only occur if <math>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>\mu=0</math> and <math>\varpi=\lambda-\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>H=5</math> is used if the infinite depth results are not calculated directly.)

	==Infinite Depth Roots==		==Infinite Depth Roots==
	Letting <math>H\to\infty</math> in the finite depth dispersion relation (for real <math>k</math> and thus positive real <math>\kappa</math>) means that the dispersion relation becomes <math>f_0(\kappa)=1/\kappa-\~~L_0~~</math>, which has zeros when the polynomial <math>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>\varpi>0</math> by [[Squire and Dixon 2000]]. Figure B.5 reproduces their figure.		Letting <math>H\to\infty</math> in the finite depth dispersion relation (for real <math>k</math> and thus positive real <math>\kappa</math>) means that the dispersion relation becomes <math>f_0(\kappa)=1/\kappa-\Lambda_0</math>, which has zeros when the polynomial <math>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>\varpi>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>1/\kappa-\kappa^4-\varpi=0</math> for positive values of the parameter <math>\varpi</math>. There is one positive real root <math>\gamma_0</math> (indicated by a circle), and four complex roots (indicated by crosses). The complex roots consist of two complex conjugate pairs, <math>\gamma_{-1}</math> and <math>\gamma_{-2}</math> in the right hand half plane (<math>\gamma_{-1}</math> is in the first quadrant), and <math>\gamma_1</math> and <math>\gamma_2</math> in the left hand half plane (<math>\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>\varpi>0</math>: <math>\pi/4<\|\Arg[\gamma_j\|\leq2\pi/5</math> for <math>j=-1,-2</math>, and <math>3\pi/4<\|\Arg[\gamma_j\|\leq4\pi/5</math> for <math>j=1,2</math>. As <math>\varpi</math> becomes negative <math>\gamma_{-1}</math> and <math>\gamma_{-2}</math> move towards the imaginary axis, while <math>\gamma_1</math> and <math>\gamma_2</math> move towards and eventually onto the negative real axis.		[[Figure B.5]] Schematic diagram showing the possible locations of the roots of the infinite depth dispersion relation <math>1/\kappa-\kappa^4-\varpi=0</math> for positive values of the parameter <math>\varpi</math>. There is one positive real root <math>\gamma_0</math> (indicated by a circle), and four complex roots (indicated by crosses). The complex roots consist of two complex conjugate pairs, <math>\gamma_{-1}</math> and <math>\gamma_{-2}</math> in the right hand half plane (<math>\gamma_{-1}</math> is in the first quadrant), and <math>\gamma_1</math> and <math>\gamma_2</math> in the left hand half plane (<math>\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>\varpi>0</math>: <math>\pi/4<\|Arg[\gamma_j\|\leq2\pi/5</math> for <math>j=-1,-2</math>, and <math>3\pi/4<\|Arg[\gamma_j\|\leq4\pi/5</math> for <math>j=1,2</math>. As <math>\varpi</math> becomes negative <math>\gamma_{-1}</math> and <math>\gamma_{-2}</math> move towards the imaginary axis, while <math>\gamma_1</math> and <math>\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>\varpi\to0</math> and <math>\varpi\to\infty</math>. In the former case the roots satisfy <math>\kappa^5-1=0</math>; in the latter the transformed roots <math>\bar\kappa=\kappa/\|\varpi\|^{1/4}</math> satisfy <math>\bar\kappa^5+\bar\kappa=\epsilon</math>, where <math>\epsilon=\|\varpi\|^{-5/4}\to0</math>. Thus, as <math>\varpi</math> decreases from <math>\infty</math> to 0, <math>\gamma_{-1}</math> moves from the line <math>\Arg[\kappa]=\pi/4</math> to <math>\Arg[\kappa]=2\pi/5</math>, while <math>\gamma_1</math> travels from the line <math>\Arg[\kappa]=3\pi/4</math> to <math>\Arg[\kappa]=4\pi/5.</math> <math>\gamma_0</math> is asymptotically zero for large <math>\varpi</math>, and it increases to 1 as <math>\varpi</math> decreases to 0. Also note that <math>\|\gamma_j\|\to\infty</math> for <math>j\neq0</math> as <math>\varpi\to\infty</math>.		The bounds shown for the arguments of the complex roots may be deduced by considering the limits as <math>\varpi\to0</math> and <math>\varpi\to\infty</math>. In the former case the roots satisfy <math>\kappa^5-1=0</math>; in the latter the transformed roots <math>\bar\kappa=\kappa/\|\varpi\|^{1/4}</math> satisfy <math>\bar\kappa^5+\bar\kappa=\epsilon</math>, where <math>\epsilon=\|\varpi\|^{-5/4}\to0</math>. Thus, as <math>\varpi</math> decreases from <math>\infty</math> to 0, <math>\gamma_{-1}</math> moves from the line <math>Arg[\kappa]=\pi/4</math> to <math>Arg[\kappa]=2\pi/5</math>, while <math>\gamma_1</math> travels from the line <math>Arg[\kappa]=3\pi/4</math> to <math>Arg[\kappa]=4\pi/5.</math> <math>\gamma_0</math> is asymptotically zero for large <math>\varpi</math>, and it increases to 1 as <math>\varpi</math> decreases to 0. Also note that <math>\|\gamma_j\|\to\infty</math> for <math>j\neq0</math> as <math>\varpi\to\infty</math>.

	In this section, we demonstrate the behaviour of the infinite depth roots as <math>\varpi</math> becomes more negative. In particular, we will show that the complex roots <math>\gamma_1</math> and <math>\gamma_2</math> actually move onto the negative real line when <math>\varpi</math> becomes less than or equal to <math>\varpi_\infty=-5/4^{4/5}\approx-1.65,</math> a proposition which follows easily by noting that <math>p_\infty</math> and its derivative <math>p_\infty'(\kappa)=5\kappa^4+\varpi</math> are simultaneously zero when <math>\kappa=\kappa_\infty=-1/4^{1/5}</math> and <math>\varpi=\varpi_\infty</math>.		In this section, we demonstrate the behaviour of the infinite depth roots as <math>\varpi</math> becomes more negative. In particular, we will show that the complex roots <math>\gamma_1</math> and <math>\gamma_2</math> actually move onto the negative real line when <math>\varpi</math> becomes less than or equal to <math>\varpi_\infty=-5/4^{4/5}\approx-1.65,</math> a proposition which follows easily by noting that <math>p_\infty</math> and its derivative <math>p_\infty'(\kappa)=5\kappa^4+\varpi</math> are simultaneously zero when <math>\kappa=\kappa_\infty=-1/4^{1/5}</math> and <math>\varpi=\varpi_\infty</math>.
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	That <math>\gamma_{-1}</math> and <math>\gamma_{-2}</math> become closer and closer to the imaginary as <math>\varpi\to-\infty</math> might have been expected from the behaviour of the finite depth roots for small periods, but the behaviour of <math>\gamma_1</math> and <math>\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>\tanh\kappa\to-1</math> for <math>\re[\kappa]<0</math> as <math>H\to\infty</math>, which means that they will become the solutions to <math>q_\infty=-p_\infty(-\kappa)=\L\kappa+1=0</math>, not the roots of <math>p_\infty</math> itself. The remaining roots of both <math>p_\infty</math> and <math>q_\infty</math> can be recovered from the finite depth situation by also considering the equation <math>\L\kappa\tanh\kappa+1=0</math>. (The finite depth dispersion relation rearranges to <math>\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>\gamma_1</math> and <math>\gamma_2</math> as <math>H\to\infty</math>. Moreover, these two roots also move onto the real axis when <math>\varpi</math> takes a certain negative value (depending on <math>H</math>), which explains the surprising behaviour of the aforementioned infinite depth roots.		That <math>\gamma_{-1}</math> and <math>\gamma_{-2}</math> become closer and closer to the imaginary as <math>\varpi\to-\infty</math> might have been expected from the behaviour of the finite depth roots for small periods, but the behaviour of <math>\gamma_1</math> and <math>\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>\tanh\kappa\to-1</math> for <math>\mbox{e}[\kappa]<0</math> as <math>H\to\infty</math>, which means that they will become the solutions to <math>q_\infty=-p_\infty(-\kappa)=\Lambda\kappa+1=0</math>, not the roots of <math>p_\infty</math> itself. The remaining roots of both <math>p_\infty</math> and <math>q_\infty</math> can be recovered from the finite depth situation by also considering the equation <math>\Lambda\kappa\tanh\kappa+1=0</math>. (The finite depth dispersion relation rearranges to <math>\Lambda\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>\gamma_1</math> and <math>\gamma_2</math> as <math>H\to\infty</math>. Moreover, these two roots also move onto the real axis when <math>\varpi</math> takes a certain negative value (depending on <math>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>g_0</math> and <math>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>k=\pm\~~a_0~~</math> were. However, since values of <math>\varpi<\varpi_\infty</math> only correspond to periods below about 2.1ms (for 1-m-thick ice), it was not considered worthwhile to calculate <math>g</math> for such values.		Referring to equations (3.17) and (3.18), the presence of two additional real roots complicates the calculation of <math>g_0</math> and <math>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>k=\pm\alpha_0</math> were. However, since values of <math>\varpi<\varpi_\infty</math> only correspond to periods below about 2.1ms (for 1-m-thick ice), it was not considered worthwhile to calculate <math>g</math> for such values.


	[[Reflections on Ice]]		[[Reflections on Ice]]

Syan077: /* Finite Depth Roots */

2006-12-24T22:31:30Z

Finite Depth Roots

← Older revision		Revision as of 22:31, 24 December 2006
Line 25:		Line 25:

	<center><math>		<center><math>
	f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,		f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,\,
	</math></center>		</math></center>

Syan077: /* Shallow Water Roots */

2006-12-24T22:30:37Z

Shallow Water Roots

← Older revision		Revision as of 22:30, 24 December 2006
Line 5:		Line 5:

	<center><math>		<center><math>
	f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,		f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,\,
	</math></center>		</math></center>

Syan077: /* Shallow Water Roots */

2006-12-24T22:30:16Z

Shallow Water Roots

← Older revision		Revision as of 22:30, 24 December 2006
Line 4:		Line 4:
	Recall that the finite depth dispersion relation for the left-hand sheet of ice is		Recall that the finite depth dispersion relation for the left-hand sheet of ice is

	<center><math>f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,</math></center>		<center><math>
			f_0(\kappa)=1/\kappa\tanh\kappa-\Lambda_0(\kappa)=0,
			</math></center>

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

Syan077 at 22:29, 24 December 2006

2006-12-24T22:29:40Z

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Syan077 at 22:25, 24 December 2006

2006-12-24T22:25:18Z

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