Wave Scattering By A Vertical Circular Cylinder
This important flow accepts a closed-form analytical solution for arbitrary values of the wavelength . This was shown to be the case by McCamy and Fuchs 1954 using separation of variables.
The incident potential is given as
Let the diffraction potential be
For to satisfy the 3D Laplace equation, it is easy to show that must satisfy the Helmholtz equation:
In polar coordinates:
The Helmholtz equation takes the form:
On the cylinder:
Here we make use of the familiar identity:
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 must satisfy:
Also as :
Hence the Hankel function:
Satisfies the far field condition required by . So we set:
with the constants to be determined. The cylinder condition requires:
It follows that:
where 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 and the series expansion solution survives.
The total complex potential, incident and scattered, was derived above.
The hydrodynamic pressure follows from Bernoulli:
Surge exciting force
The surge exciting force is given by
Simple algebra in this case of water of infinite depth leads to the expression.