Dispersion Relation for a Floating Elastic Plate - Revision history

Meylan at 09:36, 20 October 2009

2009-10-20T09:36:16Z

← Older revision		Revision as of 09:36, 20 October 2009
Line 1:		Line 1:
			{{incomplete pages}}

	This is an generalisation of the [[Dispersion Relation for a Free Surface]], to the case where the		This is an generalisation of the [[Dispersion Relation for a Free Surface]], to the case where the
	surface condition is given by a [[:Category:Floating Elastic Plate\|Floating Elastic Plate]]		surface condition is given by a [[:Category:Floating Elastic Plate\|Floating Elastic Plate]]

Meylan: /* Non-dimensional form */

2008-07-06T23:44:00Z

Non-dimensional form

← Older revision		Revision as of 23:44, 6 July 2008
Line 75:		Line 75:
	</center>		</center>
	where <math>h</math> is the nondimensional water depth, <math>\alpha = L\omega^2/g,\, </math> <math>\beta = D/(\rho g L^4)\,</math>		where <math>h</math> is the nondimensional water depth, <math>\alpha = L\omega^2/g,\, </math> <math>\beta = D/(\rho g L^4)\,</math>
	and <math>\gamma = \rho_i h/(\rho L)\,</math>. This equation has four parameters		and <math>\gamma = \rho_i d/(\rho L)\,</math>. This equation has four parameters
	<math>\alpha,\beta,\gamma,H\,</math>. We can trivially remove one of these by setting the		<math>\alpha,\beta,\gamma,h\,</math>. We can trivially remove one of these by setting the
	<math>L=h\,</math>. Alternatively we can		<math>L=h\,</math>. Alternatively we can
	set the length parameter to the		set the length parameter to the
Line 92:		Line 92:
	[[Tim Williams]] noticed that the dispersion equation can be written as		[[Tim Williams]] noticed that the dispersion equation can be written as
	<center>		<center>
	<math> -\frac{D}{\rho \omega^2} k^5 \sinh(kh) - k \frac{\left(\rho g - \omega^2 \rho_i h \right)}		<math> -\frac{D}{\rho \omega^2} k^5 \sinh(kh) - k \frac{\left(\rho g - \omega^2 \rho_i d \right)}
	{\rho \omega^2} \sinh(kh) =		{\rho \omega^2} \sinh(kh) =
	- \cosh(kh) </math></center>		- \cosh(kh) </math></center>
Line 98:		Line 98:
	we obtain the following dispersion equation which depends on only two parameters		we obtain the following dispersion equation which depends on only two parameters
	<center>		<center>
	<math> -k^5 \sinh(kh) - k \frac{\rho g - \omega^2 \rho_i h }		<math> -k^5 \sinh(kh) - k \frac{\rho g - \omega^2 \rho_i d}
	{\rho \omega^2 L} \sinh(kh) =		{\rho \omega^2 L} \sinh(kh) =
	- \cosh(kh). </math></center>		- \cosh(kh). </math></center>
Line 105:		Line 105:
	<math>\cosh(k h)-(k^4+\varpi)k\sinh(k h)=0,</math></center>		<math>\cosh(k h)-(k^4+\varpi)k\sinh(k h)=0,</math></center>
	where		where
	<center><math>\varpi=\frac{\rho g - \omega^2 \rho_i h }{\rho \omega^2 L}		<center><math>\varpi=\frac{\rho g - \omega^2 \rho_i d}{\rho \omega^2 L}
	=(1-k_\infty\sigma)/(k_\infty L),\quad k_\infty=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math></center>		=(1-k_\infty\sigma)/(k_\infty L),\quad k_\infty=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math></center>
	<math> k_\infty </math> is the wave number for a wave traveling in open water of infinite depth, and		<math> k_\infty </math> is the wave number for a wave traveling in open water of infinite depth, and

Meylan: /* Separation of Variables */

2008-07-06T23:41:31Z

Separation of Variables

← Older revision		Revision as of 23:41, 6 July 2008
Line 18:		Line 18:
	<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}		<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
	+ \left(\rho g		+ \left(\rho g
	- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =		- \omega^2 \rho_i d \right) \frac{\partial \phi}{\partial z} =
	- \rho \omega^2 \phi, \, z=0</math>		- \rho \omega^2 \phi, \, z=0</math>
	</center>		</center>
Line 28:		Line 28:
	<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math></center>		<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math></center>
	where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>		where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>
	are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,D </math> are		are the densities of the plate and the water respectively, <math> \,d </math> and <math> \,D </math> are
	the thickness and flexural rigidity of the plate.		the thickness and flexural rigidity of the plate.

Line 39:		Line 39:
	(which corresponds to the wavenumber) is given by		(which corresponds to the wavenumber) is given by
	<center><math>		<center><math>
	-D k^5 \sinh(kh) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kh) =		-D k^5 \sinh(kh) - k \left(\rho g - \omega^2 \rho_i d \right) \sinh(kh) =
	-\rho \omega^2 \cosh(kh) \,\,\,(1)		-\rho \omega^2 \cosh(kh) \,\,\,(1)
	</math></center>		</math></center>

Meylan: /* Non-dimensional form */

2008-07-06T23:40:25Z

Non-dimensional form

← Older revision		Revision as of 23:40, 6 July 2008
Line 71:		Line 71:
	non-dimensional equations then become		non-dimensional equations then become
	<center>		<center>
	<math> -\beta k^5 \sinh(kH) - k \left(1 - \alpha \gamma \right) \sinh(kH) =		<math> -\beta k^5 \sinh(kh) - k \left(1 - \alpha \gamma \right) \sinh(kh) =
	-\alpha \cosh(kH) \, </math>		-\alpha \cosh(kh) \, </math>
	</center>		</center>
	where <math>H</math> is the nondimensional water depth, <math>\alpha = L\omega^2/g,\, </math> <math>\beta = D/(\rho g L^4)\,</math>		where <math>h</math> is the nondimensional water depth, <math>\alpha = L\omega^2/g,\, </math> <math>\beta = D/(\rho g L^4)\,</math>
	and <math>\gamma = \rho_i h/(\rho L)\,</math>. This equation has four parameters		and <math>\gamma = \rho_i h/(\rho L)\,</math>. This equation has four parameters
	<math>\alpha,\beta,\gamma,H\,</math>. We can trivially remove one of these by setting the		<math>\alpha,\beta,\gamma,H\,</math>. We can trivially remove one of these by setting the
	<math>L=H\,</math>. Alternatively we can		<math>L=h\,</math>. Alternatively we can
	set the length parameter to the		set the length parameter to the
	[[Characteristic Length]] <math>L = (D/(\rho g))^{1/4}\,</math> (so that in		[[Characteristic Length]] <math>L = (D/(\rho g))^{1/4}\,</math> (so that in
	the notation above <math>\beta = 1\,</math> and we also have three parameters). It should		the notation above <math>\beta = 1\,</math> and we also have three parameters). It should
	be noted that in many cases, especially for ice floes, we have infinitely deep water and		be noted that in many cases, especially for ice floes, we have infinitely deep water and
	the ~~paramter~~ <math>\gamma</math> is small so that the [[Characteristic Length]]		the parameter <math>\gamma</math> is small so that the [[Characteristic Length]]
	can be see as a single parameter which define the ice properties. It is then very useful		can be see as a single parameter which define the ice properties. It is then very useful
	to compare the value of the [[Characteristic Length]] for real ice and ice in the model		to compare the value of the [[Characteristic Length]] for real ice and ice in the model
Line 92:		Line 92:
	[[Tim Williams]] noticed that the dispersion equation can be written as		[[Tim Williams]] noticed that the dispersion equation can be written as
	<center>		<center>
	<math> -\frac{D}{\rho \omega^2} k^5 \sinh(kH) - k \frac{\left(\rho g - \omega^2 \rho_i h \right)}		<math> -\frac{D}{\rho \omega^2} k^5 \sinh(kh) - k \frac{\left(\rho g - \omega^2 \rho_i h \right)}
	{\rho \omega^2} \sinh(kH) =		{\rho \omega^2} \sinh(kh) =
	- \cosh(kH) </math></center>		- \cosh(kh) </math></center>
	and if we simply non-dimensionalise with respect to a length <math>L^5=D/(\rho\omega^2)\,</math>		and if we simply non-dimensionalise with respect to a length <math>L^5=D/(\rho\omega^2)\,</math>
	we obtain the following dispersion equation which depends on only two parameters		we obtain the following dispersion equation which depends on only two parameters
	<center>		<center>
	<math> -k^5 \sinh(kH) - k \frac{\rho g - \omega^2 \rho_i h }		<math> -k^5 \sinh(kh) - k \frac{\rho g - \omega^2 \rho_i h }
	{\rho \omega^2 L} \sinh(kH) =		{\rho \omega^2 L} \sinh(kh) =
	- \cosh(kH). </math></center>		- \cosh(kh). </math></center>
	This can be written as		This can be written as
	<center>		<center>
	<math>\cosh(k H)-(k^4+\varpi)k\sinh(k H)=0,</math></center>		<math>\cosh(k h)-(k^4+\varpi)k\sinh(k h)=0,</math></center>
	where		where
	<center><math>\varpi=\frac{\rho g - \omega^2 \rho_i h }{\rho \omega^2 L}		<center><math>\varpi=\frac{\rho g - \omega^2 \rho_i h }{\rho \omega^2 L}

Meylan: /* Separation of Variables */

2008-07-06T23:38:30Z

Separation of Variables

← Older revision		Revision as of 23:38, 6 July 2008
Line 26:		Line 26:
	</center>		</center>
	<center>		<center>
	<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math></center>		<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math></center>
	where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>		where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>
	are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,D </math> are		are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,D </math> are
Line 34:		Line 34:
	following expression for the velocity potential		following expression for the velocity potential
	<center><math>		<center><math>
	\phi(x,z) = e^{ikx} \cosh k(z+H) \,		\phi(x,z) = e^{ikx} \cosh k(z+h) \,
	</math></center>		</math></center>
	If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>		If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>
	(which corresponds to the wavenumber) is given by		(which corresponds to the wavenumber) is given by
	<center><math>		<center><math>
	-D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =		-D k^5 \sinh(kh) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kh) =
	-\rho \omega^2 \cosh(kH) \,\,\,(1)		-\rho \omega^2 \cosh(kh) \,\,\,(1)
	</math></center>		</math></center>
	This is the dispersion equation for a floating elastic plate.		This is the dispersion equation for a floating elastic plate.
Line 46:		Line 46:
	We can also separate variables as		We can also separate variables as
	<center><math>		<center><math>
	\phi(x,z) = e^{kx} \cos k(z+H) \,		\phi(x,z) = e^{kx} \cos k(z+h) \,
	</math></center>		</math></center>
	in which case the dispersion relations becomes		in which case the dispersion relations becomes
	<center><math>		<center><math>
	D k^5 \sin(kH) + k \left(\rho g - \omega^2 \rho_i h \right) \sin(kH) =		D k^5 \sin(kh) + k \left(\rho g - \omega^2 \rho_i h \right) \sin(kh) =
	-\rho \omega^2 \cos(kH)		-\rho \omega^2 \cos(kh)
	</math></center>		</math></center>

Meylan: /* Non-dimensional form */

2008-07-06T23:35:54Z

Non-dimensional form

← Older revision		Revision as of 23:35, 6 July 2008
Line 117:		Line 117:
	The dispersion relation relates the wavenumber <math> k/L\, </math> and thus wave speed <math> L\omega/k\, </math> to the above		The dispersion relation relates the wavenumber <math> k/L\, </math> and thus wave speed <math> L\omega/k\, </math> to the above

	[[Category:~~Linear Hydroelasticity~~]]		= Matlab Code =

			A program to calculate solutions to the dispersion relation for a free surface
			[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_elastic_surface.m dispersion_elastic_surface.m]


			[[Category:Floating Elastic Plate]]

Meylan at 23:34, 6 July 2008

2008-07-06T23:34:26Z

← Older revision		Revision as of 23:34, 6 July 2008
Line 1:		Line 1:
	This is an generalisation of the [[Dispersion Relation for a Free Surface]].		This is an generalisation of the [[Dispersion Relation for a Free Surface]], to the case where the
			surface condition is given by a [[:Category:Floating Elastic Plate\|Floating Elastic Plate]]

Meylan at 21:14, 28 August 2006

2006-08-28T21:14:36Z

← Older revision		Revision as of 21:14, 28 August 2006
Line 14:		Line 14:
	[[Frequency Domain Problem]] with radial frequency <math>\,\omega</math> in terms of		[[Frequency Domain Problem]] with radial frequency <math>\,\omega</math> in terms of
	the potential alone is		the potential alone is
			<center>
	<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}		<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
	+ \left(\rho g		+ \left(\rho g
	- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =		- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
	- \rho \omega^2 \phi, \, z=0</math>		- \rho \omega^2 \phi, \, z=0</math>
			</center>
	plus the equations within the fluid		plus the equations within the fluid
			<center>
	<math>\nabla^2\phi =0 </math>		<math>\nabla^2\phi =0 </math>
			</center>
	<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>		<center>
			<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math></center>
	where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>		where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>
	are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,D </math> are		are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,D </math> are
Line 90:		Line 90:
	derivation of this is as follow.		derivation of this is as follow.
	[[Tim Williams]] noticed that the dispersion equation can be written as		[[Tim Williams]] noticed that the dispersion equation can be written as
			<center>
	<math> -\frac{D}{\rho \omega^2} k^5 \sinh(kH) - k \frac{\left(\rho g - \omega^2 \rho_i h \right)}		<math> -\frac{D}{\rho \omega^2} k^5 \sinh(kH) - k \frac{\left(\rho g - \omega^2 \rho_i h \right)}
	{\rho \omega^2} \sinh(kH) =		{\rho \omega^2} \sinh(kH) =
	- \cosh(kH) </math>		- \cosh(kH) </math></center>

	and if we simply non-dimensionalise with respect to a length <math>L^5=D/(\rho\omega^2)\,</math>		and if we simply non-dimensionalise with respect to a length <math>L^5=D/(\rho\omega^2)\,</math>
	we obtain the following dispersion equation which depends on only two parameters		we obtain the following dispersion equation which depends on only two parameters
			<center>
	<math> -k^5 \sinh(kH) - k \frac{\rho g - \omega^2 \rho_i h }		<math> -k^5 \sinh(kH) - k \frac{\rho g - \omega^2 \rho_i h }
	{\rho \omega^2 L} \sinh(kH) =		{\rho \omega^2 L} \sinh(kH) =
	- \cosh(kH). </math>		- \cosh(kH). </math></center>

	This can be written as		This can be written as
			<center>
	<math>\cosh(k H)-(k^4+\varpi)k\sinh(k H)=0,</math>		<math>\cosh(k H)-(k^4+\varpi)k\sinh(k H)=0,</math></center>
			where
	~~where~~ <math>\varpi=\frac{\rho g - \omega^2 \rho_i h }{\rho \omega^2 L}		<center><math>\varpi=\frac{\rho g - \omega^2 \rho_i h }{\rho \omega^2 L}
	=(1-k_\infty\sigma)/(k_\infty L),\quad k_\infty=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>		=(1-k_\infty\sigma)/(k_\infty L),\quad k_\infty=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math></center>

	<math> k_\infty </math> is the wave number for a wave traveling in open water of infinite depth, and		<math> k_\infty </math> is the wave number for a wave traveling in open water of infinite depth, and
	<math>\sigma\,</math> is the amount that the plate would be submerged relative to a region of open water. Note that if <math>\sigma\,</math> is greater than <math> 1/k_\infty, </math> or <math>2\pi\,</math> times the infinite depth open water wavelength, then the parameter <math>\varpi</math> becomes negative. This is a highly non-physical situation		<math>\sigma\,</math> is the amount that the plate would be submerged relative to a region of open water. Note that if <math>\sigma\,</math> is greater than <math> 1/k_\infty, </math> or <math>2\pi\,</math> times the infinite depth open water wavelength, then the parameter <math>\varpi</math> becomes negative. This is a highly non-physical situation

Meylan: /* Non-dimensional form */

2006-08-23T08:17:07Z

Non-dimensional form

← Older revision		Revision as of 08:17, 23 August 2006
Line 69:		Line 69:
	to <math>L\,</math> and time with respect to <math>\sqrt{L/g}\,</math>. The		to <math>L\,</math> and time with respect to <math>\sqrt{L/g}\,</math>. The
	non-dimensional equations then become		non-dimensional equations then become
			<center>
	<math> -\beta k^5 \sinh(kH) - k \left(1 - \alpha \gamma \right) \sinh(kH) =		<math> -\beta k^5 \sinh(kH) - k \left(1 - \alpha \gamma \right) \sinh(kH) =
	-\alpha \cosh(kH) \, </math>		-\alpha \cosh(kH) \, </math>
			</center>
	where <math>H</math> is the nondimensional water depth, <math>\alpha = L\omega^2/g,\, </math> <math>\beta = D/(\rho g L^4)\,</math>		where <math>H</math> is the nondimensional water depth, <math>\alpha = L\omega^2/g,\, </math> <math>\beta = D/(\rho g L^4)\,</math>
	and <math>\gamma = \rho_i h/(\rho L)\,</math>. This equation has four parameters		and <math>\gamma = \rho_i h/(\rho L)\,</math>. This equation has four parameters

Meylan at 10:00, 28 June 2006

2006-06-28T10:00:46Z

← Older revision		Revision as of 10:00, 28 June 2006
Line 32:		Line 32:
	We then look for a separation of variables solution to [[Laplace's Equation]] and obtain the		We then look for a separation of variables solution to [[Laplace's Equation]] and obtain the
	following expression for the velocity potential		following expression for the velocity potential
			<center><math>
	<math>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>		\phi(x,z) = e^{ikx} \cosh k(z+H) \,
			</math></center>
	If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>		If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>
	(which corresponds to the wavenumber) is given by		(which corresponds to the wavenumber) is given by
			<center><math>
			-D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
			-\rho \omega^2 \cosh(kH) \,\,\,(1)
			</math></center>
			This is the dispersion equation for a floating elastic plate.

	<math> -D k^5 \~~sinh~~(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \~~sinh~~(kH) =		We can also separate variables as
	-\rho \omega^2 \~~cosh~~(kH) ~~\,\,\,(1)~~ </math>		<center><math>
			\phi(x,z) = e^{kx} \cos k(z+H) \,
			</math></center>
	~~This is the dispersion equation for a floating elastic plate.~~		in which case the dispersion relations becomes
			<center><math>
			D k^5 \sin(kH) + k \left(\rho g - \omega^2 \rho_i h \right) \sin(kH) =
			-\rho \omega^2 \cos(kH)
			</math></center>

	== Solution of the dispersion equation ==		== Solution of the dispersion equation ==