Wave Equation for Spherically Symmetric Systems

Lecture 47

The formulation of linear wave phenomenon in terms of the wave equation, the Helmholtz equation, and its solutions in terms of orthonormal function on the Euclidean plane can be extended readily to three dimensional Euclidean space. For this space the wave equation

$$\nabla^2\psi -\frac{1}{c^2}~\frac{\partial^2\psi}{\partial t^2} = 0$$

can be solved relative to various orthogonal coordinate systems (there are at least eleven of them). The choice of coordinates is invariably dictated by symmetry and boundary conditions. This means that the coordinates are usually chosen so one or more of the coordinate surfaces mold themselves onto boundaries where the boundary conditions are specified. In terms of ubiquity, the three most important coordinate systems are the rectangular, cylindrical, and the spherical coordinates.

We shall now consider the wave equation relative to spherical coordinates is

$$\begin{array}{lll} x = r\sin\theta\cos\varphi &0\le\theta\le\... ...arphi &0\le r<\infty ~&~~\\ z = r\cos\theta &~~&~~ \end{array}$$

The angles $\varphi$ and $\theta$ are called the azimuthal and the polar angle respectively. Relative to these coordinates, the Laplace operator has the form

$$\nabla^2\psi = \frac{1}{r^2}~\frac{\partial}{\partial r} r^2\frac... ... +\frac{1}{\sin^2\theta}~ \frac{\partial^2}{\partial\varphi^2} \right] \psi\,.$$

A useful observation, rather valuable as we shall see momentarily, is the fact that the first term can be written in the form

$$\frac{1}{r^2}~\frac{\partial}{\partial r} r^2\frac{\partial}{\par... ...} r\right)^2 \psi \equiv \frac{1}{r} ~\frac{\partial^2}{\partial r^2} r \psi~.$$

Another useful observation is that the second term is easy to remember. Indeed, for small $\theta$ ( $\theta \ll 1$ ) one has

$$\frac{1}{\sin\theta}~ \frac{\partial}{\partial\theta}\sin\theta\f... ...}{\partial\theta} +\frac{1}{\theta^2}~ \frac{\partial^2}{\partial\varphi^2} ~,$$

the familiar two-dimensional Laplacian, Eq.(5.1) on P, for the Euclidean plane. This is as it should be: around the north pole of a sphere the spherical coordinates reduce to the polar coordinates of the Euclidean plane around the origin.

The physically and mathematically most revealing solutions are the normal modes. They are time translation eigenfunctions and, as we have already learned from Section 5.1.4, they satisfy the equation

$$\frac{\partial}{\partial t}\Psi = i\omega\Psi\,.$$

A normal mode has the form

$$\Psi (r,\theta ,\varphi ,t)=\psi (r,\theta ,\varphi )e^{i\omega t}\,.$$

Here $\psi (r,\theta ,\varphi )$ is the spatial amplitude profile which satisfies the Helmholtz equation

$$(\nabla^2+k^2)\psi =0\,,~~\qquad~~k^2=\frac{\omega^2}{c^2}$$

or

$$\left\{\frac{1}{r}\frac{\partial^2}{\partial r^2} r+\frac{1}{r^2}... ...n^2\theta}~\frac{\partial^2}{\partial\varphi^2} \right] +k^2\right\} \psi=0\,,$$

relative to spherical coordinates. This partial differential equation lends itself to being separated into a system of ordinary differential equations. Letting $\psi =j(r)Y(\theta ,\varphi)$ , dividing by $j(r)$ , multiplying by $r^2$ , and tranferring the $r$ -dependent term to the right hand side, one finds that the r.h.s. is independent of $\theta$ and $\phi$ , while the l.h.s. is independent of $r$ . But these two sides are equal and hence are independent of all three variables. Thus both the l.h.s. and the r.h.s. are equal to the same constant, the separation constant, say $-\lambda$ . This yields two equations for $Y(\theta,\phi)$ and $j(r)$ respectively. Explicitly one has

$$\frac{1}{\sin\theta}~\frac{\partial}{\partial\theta}\sin\theta\fr... ...\partial\theta} +\frac{1}{\sin^2\theta}~\frac{\partial^2 Y}{\partial \varphi^2} = -\lambda Y\,;$$ (573)

and

$$\left\{\frac{d^2}{dr^2} +\frac{2}{r}~\frac{d}{dr}+\left(k^2-\frac{\lambda} {r^2}\right)\right\} j(r)=0$$ (574)

These are two eigenvalue equations. The separation constant $\lambda$ is the eigenvalue determined by the boundary conditions on the angular function, and $k^2$ is the eigenvalue determined by the boundary conditions on the radial function $j(r)$ . One of the allowed eigenvalues for $\lambda$ expresses the circumstance where the amplitude profile is spherically symmetric, i.e. is independent of the angles $\theta$ and $\phi$ . For this circumstance the solutions to the Helmholtz equation can be found immediately.