Wave Scattering By A Vertical Circular Cylinder: Difference between revisions
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The total complex potential, incident and scattered was derived above. The hydrodynamic pressure follows from Bernoulli: | The total complex potential, incident and scattered was derived above. The hydrodynamic pressure follows from Bernoulli: | ||
<center><math> P = \ | <center><math> P = \mathbf{Re} \left\{ \mathbf{P} e^{i\omega t} \right\} \,</math></center> | ||
<center><math> \mathbf{P} = - i\omega \rho \left( \phi_I + \phi_7 \right) \, </math></center> | <center><math> \mathbf{P} = - i\omega \rho \left( \phi_I + \phi_7 \right) \, </math></center> |
Revision as of 09:07, 6 March 2007
- Mccamy-Fuchs analytical solution of the scattering of regular waves by a vertical circular cylinder.
This important flow accepts a closed-form analytical solution for arbitrary values of the wavelength [math]\displaystyle{ \lambda\, }[/math]. This was shown to be the case by Mccamy-Fuchs using separation of variables
Let the diffraction potential be:
- For [math]\displaystyle{ \phi_7\, }[/math] to satisfy the 3D Laplace equation, it is easy to show that [math]\displaystyle{ \psi\, }[/math] must satisfy the Helmholtz equation:
In polar coordinates:
The Helmholtz equation takes the form:
On the cylinder:
or
Here we make use of the familiar identity:
Try:
Upon substitution in Helmholtz's equation we obtain:
This is the Bessel equation of order m accepting as solutions linear combinations of the Bessel functions
The proper linear combination in the present problem is suggested by the radiation condition that [math]\displaystyle{ \psi\, }[/math] must satisfy:
As [math]\displaystyle{ R \to \infty\, }[/math]:
Also as [math]\displaystyle{ R \to \infty\, }[/math]:
Hence the Hankel function:
Satisfies the far field condition required by [math]\displaystyle{ \psi(R,\theta) \, }[/math]. So we set:
With the constants [math]\displaystyle{ A_m \, }[/math] to be determined. The cylinder condition requires:
It follows that:
or:
where [math]\displaystyle{ (')\, }[/math] denotes derivatives with respect to the argument. The solution for the total velocity potential follows in the form
And the total original potential follows:
In the limit as [math]\displaystyle{ H \to \infty \quad \frac{\cosh K (Z+H)}{K H} \longrightarrow e^{K Z} \, }[/math] and the series expansion solution survives.
Surge exciting force
The total complex potential, incident and scattered was derived above. The hydrodynamic pressure follows from Bernoulli:
The Surge exciting force is given by
Simple algebra in this case of water of infinite depth leads to the expression.
Ocean Wave Interaction with Ships and Offshore Energy Systems