Eigenfunction Matching for a Submerged Finite Dock - Revision history

Mike smith: complete page

2009-09-01T05:54:44Z

complete page

2009-09-01T03:00:48Z

Solution Method

2009-09-01T03:00:01Z

Solution Method

2009-08-14T22:10:18Z

consistent notation

2009-08-13T07:15:49Z

swapped negative signs for third phi expression

2009-08-10T23:58:36Z

Solution Method

2009-08-09T02:01:09Z

4 regions not 3

← Older revision		Revision as of 02:01, 9 August 2009
Line 39:		Line 39:
	=Solution Method=		=Solution Method=

	We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the ~~three~~ regions, <math>x<0</math>		We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the four regions, {<math>x<-L \,</math>}, {<math>x>L \,</math>}, {<math>-d<z<0,\,\,-L<x<L</math>}, and {<math>-h<z<-d,\,\,-L<x<L</math>}. The first three regions use the free-surface eigenfunction
	<math>-d<z<0,\,\,x>0</math>, and <math>-h<z<-d,\,\,x>0</math>. The first ~~two~~ regions use the free-surface eigenfunction		and the last uses dock eigenfunctions. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].
	and the ~~third~~ uses ~~the~~ dock eigenfunctions. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].

	The incident potential is a wave of amplitude <math>A</math>		The incident potential is a wave of amplitude <math>A</math>

2008-08-26T08:03:04Z

Solution Method

← Older revision		Revision as of 08:03, 26 August 2008
Line 58:		Line 58:
	<math>		<math>
	\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(		\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(
	z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<-L		z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h (x+L)}\phi_{m}^h(z), \;\;x<-L
	</math>		</math>
	</center>		</center>

2008-08-26T08:00:06Z

Solution Method

2008-08-26T07:58:20Z

Solution Method

← Older revision		Revision as of 07:58, 26 August 2008
Line 58:		Line 58:
	<math>		<math>
	\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(		\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(
	z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<-~~L_2~~		z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<-L
	</math>		</math>
	</center>		</center>
Line 67:		Line 67:
	+ \sum_{m=0}^{\infty}c_{m}		+ \sum_{m=0}^{\infty}c_{m}
	e^{k_{m}^d (x-L)}\phi_{m}^d(z)		e^{k_{m}^d (x-L)}\phi_{m}^d(z)
	, \;\;-d<z<0,\,\,-~~L_2~~<x<~~-L_1~~		, \;\;-d<z<0,\,\,-L<x<L
	</math>		</math>
	</center>		</center>
Line 73:		Line 73:
	<center>		<center>
	<math>		<math>
	\phi(x,z)= d_0 \frac{x+~~L_2~~}{~~L_2-L_1~~}\sum_{m=0}^{\infty}d_{m}		\phi(x,z)= d_0 \frac{x+L}{2 L}\sum_{m=0}^{\infty}d_{m}
	e^{\kappa_{m} (x+L)}\psi_{m}(z)		e^{\kappa_{m} (x+L)}\psi_{m}(z)
	+\sum_{m=0}^{\infty}e_{m}		+ e_0 \frac{x-L}{2 L}
			\sum_{m=1}^{\infty}e_{m}
	e^{-\kappa_{m} (x-L)}\psi_{m}(z)		e^{-\kappa_{m} (x-L)}\psi_{m}(z)
	, \;\;-h<z<-d,\,\,-~~L_2~~<x<~~L_1~~		, \;\;-h<z<-d,\,\,-L<x<L
	</math>		</math>
	</center>		</center>
	<center>		<center>
	<math>		<math>
	\phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;~~-L_1~~<x~~>L_2~~		\phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;L<x
	</math>		</math>
	</center>		</center>