Difference between revisions of "Category:Multipole Methods for Linear Water Waves"

Multipole Expansions

Multipole expansions are a technique used in linear water wave theory where the potential is represented as a sum of singularities (multipoles) placed within any structures that are present. Multipoles can be constructed with their singularity submerged or on the free surface and these are treated separately.

Multipoles satisfy:

(note that the last expression can be obtained from combining the expressions:

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math])

In two-dimensions the Sommerfeld Radiation Condition is

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

A linear combination of these multipoles can then be made to satisfy the body boundary conditions.

2D Exterior Neumann Problem

We start by developing the theory for Laplace's equation without the free surface for a disk of radius [math]\displaystyle{ a }[/math] centered at the origin. The equation in polar coordinates is

Using separation of variables we write [math]\displaystyle{ \phi = R(r)\Theta(\theta).\, }[/math] Substituting into Laplace's equations gives

The equation for [math]\displaystyle{ \Theta\, }[/math] is

When m=0 the solution is

The equation for R is

This is a standard DE, to solve we substitute R = r^n giving:

This gives two independent solutions for all m except m = 0, where we only get the constant solution.

Noting that:

So along with the obvious constant function we have two independant sotutions. Thus for m=0 we can write:

and for [math]\displaystyle{ m\neq 0: }[/math]

The derivatives of the radial eigenfunctions must go to zero indicating zero fluid flow at infinity:

Hence the general solution can be expressed as:

We don't need to include the negative m terms because of the symmetry of the sine and cosine eigenfunctions means these solutions are not independent.

The solution can also be expressed simply in terms of complex exponentials.

With the Free Surface BC

When we introduce the free surface BC we need to find a new set of eigenfunctions. We can use eigenfunctions from the infinite domain problem to construct them.

Note that the origin is in the free surface directly above the center of the disc for cartesian coordinates. However the origin for the polar coordinates is still at the center of the disk.

We add a solution of Laplace's equation to the eigenfunctions for the infinite domain problem and require that these new functions(multipoles) satisfy the free surface condition.

For m>0

and for m=0

Where

We want

For m=0

Using

For m>0

Using the expansion (valid for z>-f)

So our psi functions satisfy

Taking a Fourier transform in x

Hence

and

Example: 2D Wave Scattering for a Submerged Horizontal Cylinder (in Infinite Depth)

• • • Picture to show problem set up***

The potential for the scattering problem splits as incident and diffracted potentials and also into symmetric and antisymmetric parts.

[math]\displaystyle{ \phi = \phi_I + \phi_D = \phi^+ + \phi^- }[/math]

where

[math]\displaystyle{ \partial_r \phi_I |_{r=a} = - \partial_r \phi_D |_{r=a} }[/math]

and [math]\displaystyle{ \phi^+ }[/math], [math]\displaystyle{ \phi^- }[/math] are the symmetric and antisymmetric parts of [math]\displaystyle{ \phi }[/math] respectively.

For a wave incident from the left we have

[math]\displaystyle{ \phi_I = e^{Kz} e^{iKx} = e^{-Kf}\sum_{n=0}^\infty \frac{(-Kr)^n }{n!} e^{-in\theta} }[/math],

[math]\displaystyle{ \phi_D = \sum_{n=1}^{\infty}a^n \alpha_n \phi_n^+ + \sum_{n=1}^{\infty}a^n \beta_n \phi_n^- }[/math],

[math]\displaystyle{ \phi^+ = e^{Kz} \cos(Kx) + \sum_{n=1}^{\infty}a^n \alpha_n \phi_n^+ }[/math],

and [math]\displaystyle{ \phi^- = ie^{Kz} \sin(Kx) + \sum_{n=1}^{\infty}a^n \beta_n \phi_n^- }[/math].

Converting [math]\displaystyle{ \phi^+ }[/math] and [math]\displaystyle{ \phi^- }[/math] into the polar coordinate system:

[math]\displaystyle{ \phi^+ = e^{-Kf}\sum_{n=0}^\infty \frac{(-Kr)^n }{n!} \cos(n\theta) + \sum_{n=1}^{\infty}a^n \alpha_n \left(\frac{\cos(n\theta)}{r^n} + \sum_{m=0}^\infty A_{mn}^+ r^m \cos(m\theta)\right) }[/math]

[math]\displaystyle{ \phi^- = ie^{-Kf}\sum_{n=0}^\infty \frac{(-Kr)^n }{n!} \sin(n\theta) + \sum_{n=1}^{\infty}a^n \alpha_n \left(\frac{\sin(n\theta)}{r^n} + \sum_{m=0}^\infty A_{mn}^+ r^m \sin(m\theta)\right) }[/math]

The above power series converge for r < 2f and the coefficients are given by

[math]\displaystyle{ A_{mn} = \frac{(-1)^{m+n}}{m!(n-1)!} \quad SingularIntegral_0^\infty \quad \frac{\mu + K}{\mu - K} \mu ^{m+n-1} e^{-2\mu f}d\mu }[/math]

Applying the body boundary condition [math]\displaystyle{ \partial_r \phi^\pm |_{r=a} = 0 }[/math] and noting that the cosines are orthogonal over [math]\displaystyle{ \theta \in (-\pi,\pi] }[/math] gives the results

[math]\displaystyle{ \alpha_m - \sum_{n=1}^{\infty}a^{n+m} A_{mn} \alpha_n = e^{-Kf}\frac {(-Ka)^m}{m!}; m=1,2,3,\ldots }[/math]

[math]\displaystyle{ \beta_m - \sum_{n=1}^{\infty}a^{n+m} A_{mn} \beta_n = -ie^{-Kf}\frac {(-Ka)^m}{m!}; m=1,2,3,\ldots }[/math]

From which we can see that [math]\displaystyle{ \beta_n = -i\alpha_n }[/math].

The above sums can be truncated at the Nth term yielding NxN system of linear equations that can be solved for the coefficients [math]\displaystyle{ \alpha_n }[/math]

Hence we can calculate [math]\displaystyle{ \phi }[/math] from:

[math]\displaystyle{ \phi = \phi^+ + \phi^- = \phi_I + \sum_{n=1}^{\infty}a^n \alpha_n ( \phi^+_n + \phi^-_n ) }[/math]

This expression also gives the values of the reflection and transmission coefficients when we substitute the far field expressions for the symmetric and antisymmetric multipoles. In the far field:

[math]\displaystyle{ \phi^+_n \sim \frac{2\pi i (-K)^n}{(n-1)!} e^{K(z-f)} e^{\pm Kx} }[/math]

[math]\displaystyle{ \phi^-_n \sim \mp \frac{2\pi (-K)^n}{(n-1)!} e^{K(z-f)} e^{\pm Kx} }[/math]

Giving:

[math]\displaystyle{ R=0, \quad T = 1 + 4 \pi i e^{-Kf} \sum_{n=1}^{\infty} \alpha_n \frac{(-Ka)^n}{(n-1)!} }[/math]

Hydrodynamic Forces