Template:Separation of variables for a floating elastic plate - Revision history

Mike smith: /* Separation of variables under the Plate */

2010-03-17T19:33:48Z

Separation of variables under the Plate

← Older revision		Revision as of 19:33, 17 March 2010
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	<center>		<center>
	<math>		<math>
	Z^{\prime\prime} + k^2 Z =0.		Z^{\prime\prime} + \kappa^2 Z =0.
	</math>		</math>
	</center>		</center>

Mike smith: /* Separation of variables under the Plate */

2010-03-16T21:32:22Z

Separation of variables under the Plate

← Older revision		Revision as of 21:32, 16 March 2010
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	</math>		</math>
	</center>		</center>
	(the first term comes from the beam eigenvalue problem, where <math>\partial_x^4 ~~X_n~~ = \~~kappa_n~~^4 ~~X_n~~</math>). We then use the boundary condition at <math>z=-h \,</math> to write		(the first term comes from the beam eigenvalue problem, where <math>\partial_x^4 X = \kappa^4 X</math>). We then use the boundary condition at <math>z=-h \,</math> to write
	<center>		<center>
	<math>		<math>
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	\kappa \tan{(\kappa h)}= -\frac{\alpha}{\beta \kappa^{4} + 1 - \alpha\gamma}		\kappa \tan{(\kappa h)}= -\frac{\alpha}{\beta \kappa^{4} + 1 - \alpha\gamma}
	</math></center>		</math></center>
	Solving for <math>\kappa</math> gives a pure imaginary root		Solving for <math>\kappa \,</math> gives a pure imaginary root
	with positive imaginary part, two complex roots (two complex conjugate paired roots		with positive imaginary part, two complex roots (two complex conjugate paired roots
	with positive imaginary part in most physical situations), an infinite number of positive real roots		with positive imaginary part in most physical situations), an infinite number of positive real roots
	which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all		which approach <math>{n\pi}/{h} \,</math> as <math>n</math> approaches infinity, and also the negative of all
	these roots ([[Dispersion Relation for a Floating Elastic Plate]]) . We denote the two complex roots with positive imaginary part		these roots ([[Dispersion Relation for a Floating Elastic Plate]]) . We denote the two complex roots with positive imaginary part
	by <math>\kappa_{-2} \,</math> and <math>\kappa_{-1} \,</math>, the purely imaginary		by <math>\kappa_{-2} \,</math> and <math>\kappa_{-1} \,</math>, the purely imaginary

Mike smith: consistent notation, correction for plate expression, subscript errors

2010-03-16T21:28:58Z

consistent notation, correction for plate expression, subscript errors

← Older revision		Revision as of 21:28, 16 March 2010
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	<center>		<center>
	<math>		<math>
	\left(k^4 + 1 - \alpha\gamma\right)Z^{\prime}(0) = \alpha Z(0)		\left(\beta \kappa^4 + 1 - \alpha\gamma\right)Z^{\prime}(0) = \alpha Z(0)
	</math>		</math>
	</center>		</center>
	We then use the boundary condition at <math>z=-h</math> to write		(the first term comes from the beam eigenvalue problem, where <math>\partial_x^4 X_n = \kappa_n^4 X_n</math>). We then use the boundary condition at <math>z=-h \,</math> to write
	<center>		<center>
	<math>		<math>
	Z = \frac{\cos k(z+h)}{\cos kh}		Z = \frac{\cos \kappa(z+h)}{\cos \kappa h}
	</math>		</math>
	</center>		</center>
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	which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all		which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all
	these roots ([[Dispersion Relation for a Floating Elastic Plate]]) . We denote the two complex roots with positive imaginary part		these roots ([[Dispersion Relation for a Floating Elastic Plate]]) . We denote the two complex roots with positive imaginary part
	by <math>\kappa_(-2)</math> and <math>\kappa_(-1)</math>, the purely imaginary		by <math>\kappa_{-2} \,</math> and <math>\kappa_{-1} \,</math>, the purely imaginary
	root with positive imaginary part by <math>\kappa_{0}</math> and the real roots with positive imaginary part		root with positive imaginary part by <math>\kappa_{0} \,</math> and the real roots with positive imaginary part
	by <math>\kappa_(n)</math> for <math>n</math> a positive integer.		by <math>\kappa_{n} \,</math> for <math>n</math> a positive integer.
	The imaginary root with positive imaginary part corresponds to a		The imaginary root with positive imaginary part corresponds to a
	reflected travelling mode propagating along the <math>x</math> axis.		reflected travelling mode propagating along the <math>x</math> axis.
	The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes.		The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes.

Meylan at 05:29, 26 August 2008

2008-08-26T05:29:29Z

← Older revision		Revision as of 05:29, 26 August 2008
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	===Separation of variables under the Plate===		==Separation of variables under the Plate==
	~~The potential velocity can be written in terms of an infinite series of separated eigenfunctions under~~		<center>
	~~each elastic plate, of~~ the ~~form~~		<math>
	<center><math>		Z^{\prime\prime} + k^2 Z =0.
	\~~psi~~ = \frac{\cos~~(\kappa~~ (z+h))}{\cos~~\kappa h~~}~~.\,~~		</math>
	</math></center>		</center>
	~~If we apply~~ the ~~boundary conditions given~~		subject to the boundary conditions
	~~we obtain~~ the [[Dispersion Relation for a Floating Elastic Plate]]		<center>
			<math>
			Z^{\prime}(-h) = 0
			</math>
			</center>
			and
			<center>
			<math>
			\left(k^4 + 1 - \alpha\gamma\right)Z^{\prime}(0) = \alpha Z(0)
			</math>
			</center>
			We then use the boundary condition at <math>z=-h</math> to write
			<center>
			<math>
			Z = \frac{\cos k(z+h)}{\cos kh}
			</math>
			</center>
			The boundary condition at the free surface (<math>z=0</math>) is
			the [[Dispersion Relation for a Floating Elastic Plate]]
	<center><math>		<center><math>
	\kappa \tan{(\kappa h)}= -\frac{\alpha}{\beta \kappa^{4} + 1 - \alpha\gamma}		\kappa \tan{(\kappa h)}= -\frac{\alpha}{\beta \kappa^{4} + 1 - \alpha\gamma}

Meylan at 05:25, 26 August 2008

2008-08-26T05:25:44Z

New page

===Separation of variables under the Plate===
The potential velocity can be written in terms of an infinite series of separated eigenfunctions under
each elastic plate, of the form
<center><math>
\psi = \frac{\cos(\kappa (z+h))}{\cos\kappa h}.\,
</math></center>
If we apply the boundary conditions given
we obtain the [[Dispersion Relation for a Floating Elastic Plate]]
<center><math>
\kappa \tan{(\kappa h)}= -\frac{\alpha}{\beta \kappa^{4} + 1 - \alpha\gamma}
</math></center>
Solving for <math>\kappa</math> gives a pure imaginary root
with positive imaginary part, two complex roots (two complex conjugate paired roots
with positive imaginary part in most physical situations), an infinite number of positive real roots
which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all
these roots ([[Dispersion Relation for a Floating Elastic Plate]]) . We denote the two complex roots with positive imaginary part
by <math>\kappa_(-2)</math> and <math>\kappa_(-1)</math>, the purely imaginary
root with positive imaginary part by <math>\kappa_{0}</math> and the real roots with positive imaginary part
by <math>\kappa_(n)</math> for <math>n</math> a positive integer.
The imaginary root with positive imaginary part corresponds to a
reflected travelling mode propagating along the <math>x</math> axis.
The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes.