Difference between revisions of "Eigenfunction Matching for a Semi-Infinite Dock"

From WikiWaves
Jump to navigationJump to search
Line 10: Line 10:
 
the region <math>x>0</math>.
 
the region <math>x>0</math>.
 
The water is assumed to have
 
The water is assumed to have
constant finite depth <math>H</math> and the <math>z</math>-direction points vertically
+
constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-H</math>. The
+
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
 
boundary value problem can therefore be expressed as
 
boundary value problem can therefore be expressed as
 
<center>
 
<center>
 
<math>
 
<math>
\Delta\phi=0, \,\, -H<z<0,
+
\Delta\phi=0, \,\, -h<z<0,
 
</math>
 
</math>
 
</center>
 
</center>
 
<center>
 
<center>
 
<math>
 
<math>
\phi_{z}=0, \,\, z=-H,
+
\phi_{z}=0, \,\, z=-h,
 
</math>
 
</math>
 
</center>
 
</center>
Line 56: Line 56:
 
</math>
 
</math>
 
</center>
 
</center>
We then use the boundary condition at <math>z=-H</math>, which is  
+
We then use the boundary condition at <math>z=-h</math>, which is  
 
the same for all <math>x</math> to write
 
the same for all <math>x</math> to write
 
<center>
 
<center>
 
<math>
 
<math>
\zeta=\cos\mu(z+H)
+
\zeta=\cos\mu(z+h)
 
</math>
 
</math>
 
</center>
 
</center>
Line 67: Line 67:
 
or <math>x<0</math>. For <math>x<0</math> the boundary condition  
 
or <math>x<0</math>. For <math>x<0</math> the boundary condition  
 
<center><math>
 
<center><math>
k\tan\left(  kH\right)  =-\alpha,\quad x<0\,\,\,(1)
+
k\tan\left(  kh\right)  =-\alpha,\quad x<0\,\,\,(1)
 
</math></center>
 
</math></center>
 
which is the [[Dispersion Relation for a Free Surface]]
 
which is the [[Dispersion Relation for a Free Surface]]
Line 73: Line 73:
 
<center>
 
<center>
 
<math>
 
<math>
\kappa\tan(\kappa H)=0,\quad
+
\kappa\tan(\kappa h)=0,\quad
 
x>0 \,\,\,(2)
 
x>0 \,\,\,(2)
 
</math>
 
</math>
Line 82: Line 82:
 
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
 
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
 
(2) are
 
(2) are
<math>\kappa_{m}=m\pi/H</math>, <math>m\geq 0</math>. We define
+
<math>\kappa_{m}=m\pi/h</math>, <math>m\geq 0</math>. We define
 
<center>
 
<center>
 
<math>
 
<math>
\phi_{m}\left(  z\right)  =\frac{\cos k_{m}(z+H)}{\cos k_{m}H},\quad m\geq0
+
\phi_{m}\left(  z\right)  =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
 
</math>
 
</math>
 
</center>
 
</center>
Line 92: Line 92:
 
<center>
 
<center>
 
<math>
 
<math>
\psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+H),\quad
+
\psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
 
m\geq 0
 
m\geq 0
 
</math>
 
</math>
Line 100: Line 100:
 
<center>
 
<center>
 
<math>
 
<math>
\int\nolimits_{-H}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
+
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
 
</math>
 
</math>
 
</center>
 
</center>
Line 106: Line 106:
 
<center>
 
<center>
 
<math>
 
<math>
A_{m}=\frac{1}{2}\left(  \frac{\cos k_{m}H\sin k_{m}H+k_{m}H}{k_{m}\cos
+
A_{m}=\frac{1}{2}\left(  \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos
^{2}k_{m}H}\right)
+
^{2}k_{m}h}\right)
 
</math>
 
</math>
 
</center>
 
</center>
Line 113: Line 113:
 
<center>
 
<center>
 
<math>
 
<math>
\int\nolimits_{-H}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
+
\int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
 
</math>
 
</math>
 
</center>
 
</center>
 
where
 
where
 
<center><math>
 
<center><math>
B_{mn}=\frac{k_{n}\sin k_{n}H\cos\kappa_{m}H-\kappa_{m}\cos k_{n}H\sin
+
B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
\kappa_{m}H}{\left(  \cos k_{n}H\cos\kappa_{m}H\right)  \left(  k_{n}
+
\kappa_{m}h}{\left(  \cos k_{n}h\cos\kappa_{m}h\right)  \left(  k_{n}
 
^{2}-\kappa_{m}^{2}\right)  }
 
^{2}-\kappa_{m}^{2}\right)  }
 
</math></center>
 
</math></center>
Line 125: Line 125:
 
<center>
 
<center>
 
<math>
 
<math>
\int\nolimits_{-H}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
+
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
 
</math>
 
</math>
 
</center>
 
</center>
Line 131: Line 131:
 
<center>
 
<center>
 
<math>
 
<math>
C_{m}=\frac{1}{2}H,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = H
+
C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h
 
</math></center>
 
</math></center>
  
Line 189: Line 189:
 
for each <math>n</math>.
 
for each <math>n</math>.
 
We solve these equations by multiplying both equations by
 
We solve these equations by multiplying both equations by
<math>\phi_{l}(z)</math> and integrating from <math>-H</math> to <math>0</math> to obtain:
+
<math>\phi_{l}(z)</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
 
<center>
 
<center>
 
<math>
 
<math>

Revision as of 09:12, 7 March 2008

Introduction

This is one of the simplest problem in eigenfunction matching. It also is an easy problem to understand the Wiener-Hopf and Residue Calculus

Governing Equations

We begin with the Frequency Domain Problem for a dock which occupies the region [math]x\gt 0[/math]. The water is assumed to have constant finite depth [math]h[/math] and the [math]z[/math]-direction points vertically upward with the water surface at [math]z=0[/math] and the sea floor at [math]z=-h[/math]. The boundary value problem can therefore be expressed as

[math] \Delta\phi=0, \,\, -h\lt z\lt 0, [/math]

[math] \phi_{z}=0, \,\, z=-h, [/math]

[math] \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt 0, [/math]

[math] \partial_z\phi=0, \,\, z=0,\,x\lt 0, [/math]

We must also apply the Sommerfeld Radiation Condition as [math]|x|\rightarrow\infty[/math]. This essentially implies that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave and a wave propagating away.

Solution Method

We use separation of variables in the two regions, [math]x\lt 0[/math] and [math]x\gt 0[/math].

Separation of variables

We now separate variables and write the potential as

[math] \phi(x,z)=\zeta(z)\rho(x) [/math]

Applying Laplace's equation we obtain

[math] \zeta_{zz}+\mu^{2}\zeta=0. [/math]

We then use the boundary condition at [math]z=-h[/math], which is the same for all [math]x[/math] to write

[math] \zeta=\cos\mu(z+h) [/math]

where the separation constant [math]\mu^{2}[/math] must satisfy different equations depending on whether [math]x\gt 0[/math] or [math]x\lt 0[/math]. For [math]x\lt 0[/math] the boundary condition

[math] k\tan\left( kh\right) =-\alpha,\quad x\lt 0\,\,\,(1) [/math]

which is the Dispersion Relation for a Free Surface and for [math]x\gt 0[/math]

[math] \kappa\tan(\kappa h)=0,\quad x\gt 0 \,\,\,(2) [/math]

Note that we have set [math]\mu=k[/math] under the free surface and [math]\mu=\kappa[/math] under the plate. We denote the positive imaginary solution of (1) by [math]k_{0}[/math] and the positive real solutions by [math]k_{m}[/math], [math]m\geq1[/math]. The solutions of (2) are [math]\kappa_{m}=m\pi/h[/math], [math]m\geq 0[/math]. We define

[math] \phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 [/math]

as the vertical eigenfunction of the potential in the open water region and

[math] \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0 [/math]

as the vertical eigenfunction of the potential in the dock covered region. For later reference, we note that:

[math] \int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn} [/math]

where

[math] A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos ^{2}k_{m}h}\right) [/math]

and

[math] \int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn} [/math]

where

[math] B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin \kappa_{m}h}{\left( \cos k_{n}h\cos\kappa_{m}h\right) \left( k_{n} ^{2}-\kappa_{m}^{2}\right) } [/math]

and

[math] \int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn} [/math]

where

[math] C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h [/math]

Therefore the potential can be expanded as

[math] \phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x\lt 0 [/math]

and

[math] \phi(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-\kappa_{m}x}\psi_{m}(z), \;\;x\gt 0 [/math]

where [math]a_{m}[/math] and [math]b_{m}[/math] are the coefficients of the potential in the open water and the plate covered region respectively.

Incident potential

The incident potential is a wave of amplitude [math]A[/math] in displacement travelling in the positive [math]x[/math]-direction. The incident potential can therefore be written as

[math] \phi^{\mathrm{I}} =e^{-k_{0}x}\phi_{0}\left( z\right) [/math]

An infinite dimensional system of equations

The potential and its derivative must be continuous across the transition from open water to the plate covered region. Therefore, the potentials and their derivatives at [math]x=0[/math] have to be equal. We obtain

[math] \phi_{0}\left( z\right) + \sum_{m=0}^{\infty} a_{m} \phi_{m}\left( z\right) =\sum_{m=0}^{\infty}b_{m}\psi_{m}(z) [/math]

and

[math] -k_{0}\phi_{0}\left( z\right) +\sum _{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi _{m}(z) [/math]

for each [math]n[/math]. We solve these equations by multiplying both equations by [math]\phi_{l}(z)[/math] and integrating from [math]-h[/math] to [math]0[/math] to obtain:

[math] A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{n=0}^{\infty}b_{m}B_{ml}\,\,\,(8) [/math]

and

[math] -k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l =-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(9) [/math]

If we mutiple equation (8) by [math]-k_l[/math] and add this to equation (9) we obtain

[math] -2k_{0}A_{0}\delta_{0l} =-\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(10) [/math]

Equation (10) gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

Numerical Solution

To solve the system of equations (10) we set the upper limit of [math]l[/math] to be [math]M[/math].