Helmholtz's Equation

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Indroduction

This is a very well known equation given by

[math]\displaystyle{ \nabla^2 \phi + k^2 \phi = 0 }[/math].

It applies to a wide variety of situations such as electromagnetics and acoustics. In water waves it arises when we Remove The Depth Dependence. Often there is then a cross over from the study of water waves to the study of scattering problems more generally. Also, if we perform a Cylindrical Eigenfunction Expansion we find that the modes all decay rapidly as distance goes to infinity except the solutions which satisfy Helmholtz's equation. This means that many asymptotic results in linear water waves can be derived from results in acoustic or electromagnetic scattering.

Solution for a Circle

We can solve for the scattering by a circle using separation of variables. This is the basis of the method used in

Separation of Variable for the [math]\displaystyle{ r }[/math] and [math]\displaystyle{ \theta }[/math] coordinates

For the solution of

[math]\displaystyle{ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial Y}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 Y}{\partial \theta^2} = k_m^2 Y(r,\theta), }[/math]

we use the separation

[math]\displaystyle{ \,\!Y(r,\theta) =: R(r) \Theta(\theta). }[/math]

Substituting this into Laplace's equation yields

[math]\displaystyle{ \frac{r^2}{R(r)} \left[ \frac{1}{r} \frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d} R}{\mathrm{d}r} \right) - k_m^2 R(r) \right] = - \frac{1}{\Theta (\theta)} \frac{\mathrm{d}^2 \Theta}{\mathrm{d} \theta^2} = \eta^2, }[/math]

where the separation constant [math]\displaystyle{ \eta }[/math] must be an integer, say [math]\displaystyle{ \nu }[/math], in order for the potential to be continuous. [math]\displaystyle{ \Theta (\theta) }[/math] can therefore be expressed as

[math]\displaystyle{ \Theta (\theta) = C \, \mathrm{e}^{\mathrm{i} \nu \theta}, \quad \nu \in \mathbb{Z}. }[/math]

We also obtain the following expression

[math]\displaystyle{ r \frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d} R}{\mathrm{d} r} \right) - (\nu^2 + k_m^2 r^2) R(r) = 0, \quad \nu \in \mathbb{Z}. }[/math]

Substituting [math]\displaystyle{ \tilde{r}:=k_m r }[/math] and writing [math]\displaystyle{ \tilde{R} (\tilde{r}) := R(\tilde{r}/k_m) = R(r) }[/math], this can be rewritten as

[math]\displaystyle{ \tilde{r}^2 \frac{\mathrm{d}^2 \tilde{R}}{\mathrm{d} \tilde{r}^2} + \tilde{r} \frac{\mathrm{d} \tilde{R}}{\mathrm{d} \tilde{r}} - (\nu^2 + \tilde{r}^2)\, \tilde{R} = 0, \quad \nu \in \mathbb{Z}, }[/math]

which is the modified version of Bessel's equation. Substituting back, the general solution is given by

[math]\displaystyle{ R(r) = D_\nu \, I_\nu(k_m r) + E_\nu \, K_\nu(k_m r),\ \nu \in \mathbb{Z}, }[/math]

where [math]\displaystyle{ I_\nu }[/math] and [math]\displaystyle{ K_\nu }[/math] are the modified Bessel functions of the first and second kind, respectively, of order [math]\displaystyle{ \nu }[/math].

Note that [math]\displaystyle{ K_\nu (\mathrm{i} x) = \pi / 2\,\, \mathrm{i}^{\nu+1} H_\nu^{(2)}(x) }[/math] with [math]\displaystyle{ H_\nu^{(2)} }[/math] denoting the Hankel function of the second kind of order [math]\displaystyle{ \nu }[/math]. Also, [math]\displaystyle{ I_\nu }[/math] does not satisfy the Sommerfeld Radiation Condition since it becomes unbounded for increasing real argument and it represents incoming waves.

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