Difference between revisions of "Waves reflecting off a vertical wall"

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Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.
 
Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.
Let the incident wave elevation be
 
<center><math> \zeta_I = \mathrm{Re} \left \{ A_I e^{-\mathrm{i} k x+\mathrm{i}\omega t} \right \} </math></center>
 
and the reflected is
 
<center><math> \zeta_R = \mathrm{Re} \left \{ A_R e^{\mathrm{i}k x+\mathrm{i}\omega t} \right \} </math></center>
 
where the sign of the <math> e^{\mathrm{i} k x}\, </math> term has been reversed to denote a wave propagating to the left. <math> A_R\,</math> can be complex in order to allow for phase differences.
 
  
The corresponding velocity potentials are
+
{{incident plane wave}}
<center><math> \Phi_I = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_I}{\omega} \frac{\cosh k k h (z+h)}{\cosh k h}
+
 
e^{-\mathrm{i} k x + \mathrm{i}\omega t} \right \} </math></center>
 
<center><math> \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i} gA_R}{\omega} \frac{\cosh k (z+h)}{\cosh k h}
 
e^{\mathrm{i} k x +\mathrm{i}\omega t} \right \} </math></center>
 
 
On <math> x = 0 \, </math>:
 
On <math> x = 0 \, </math>:
<center><math> \frac{\partial}{\partial x} \left( \Phi_I + \Phi_R \right) = 0 </math></center>
+
<center><math> \partial_n\phi = \frac{\partial}{\partial x}  
or
+
\left( \phi^{\mathrm{I}} + \phi^{\mathrm{D}} \right) = 0 </math></center>
<center><math> - \mathrm{i} k A_I + \mathrm{i} k A_R \Longrightarrow A_R = A_I \equiv A </math></center>
+
where <math>\phi^{\mathrm{D}}</math> is the diffraction potential.
  
The resulting total velocity potential is
+
Therefore the total potential is  
<center><math> \Phi = \Phi_I + \Phi_R = \mathrm{Re} \left \{ \frac{\mathrm{i}gA_R}{\omega}
+
<center><math>
\frac{\cosh k (z+h}{\cosh kh} e^{\mathrm{i}\omega t} \left( e^{\mathrm{i}kx} + e^{-\mathrm{i}kx} \right) \right \} </math></center>
+
\phi = \phi_0(z) e^{-k_0 x} + A \phi_0(z) e^{k_0 x}
<center><math> = 2 A \mathrm{Re} \left \{ \frac{\mathrm{i}g}{\omega}
+
</math></center>
\frac{\cosh k (z+h)}{\cosh kh} \cos kx e^{\mathrm{i}\omega t} \right \} </math></center>
+
Note that in this case the reflected wave is particularly simple.
  
The resulting standing-wave elevation is:
 
<center><math> \zeta = \zeta_I + \zeta_R = 2 A \cos k x \cos \omega t \, </math></center>
 
  
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Complete Pages]]
 
[[Category:Complete Pages]]

Latest revision as of 21:53, 3 April 2010

Waves reflecting off a vertical wall is one of the few very important analytical solutions of regular waves interacting with solid boundaries seen in practice. The other is wavemaker theory.

The equation is subject to some radiation conditions at infinity. We assume the following. [math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,

[math]\displaystyle{ \phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \, }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \mathrm{i} k }[/math] is the positive imaginary solution of the Dispersion Relation for a Free Surface (note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math]) and

[math]\displaystyle{ \phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h} }[/math]

On [math]\displaystyle{ x = 0 \, }[/math]:

[math]\displaystyle{ \partial_n\phi = \frac{\partial}{\partial x} \left( \phi^{\mathrm{I}} + \phi^{\mathrm{D}} \right) = 0 }[/math]

where [math]\displaystyle{ \phi^{\mathrm{D}} }[/math] is the diffraction potential.

Therefore the total potential is

[math]\displaystyle{ \phi = A \phi_0(z) e^{-k_0 x} + A \phi_0(z) e^{k_0 x} }[/math]

Note that in this case the reflected wave is particularly simple.