6: Problems in Higher Dimensions

"Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition that determines the equations completely or at least almost completely."

"What I have to say about this book can be found inside this book."

~ Albert Einstein (1879-1955)

In this chapter we will explore several examples of the solution of initial-boundary value problems involving higher spatial dimensions. These are described by higher dimensional partial differential equations, such as the ones presented in Table 2.1.1 in Chapter 2. The spatial domains of the problems span many different geometries, which will necessitate the use of rectangular, polar, cylindrical, or spherical coordinates.

We will solve many of these problems using the method of separation of variables, which we first saw in Chapter 2. Using separation of variables will result in a system of ordinary differential equations for each problem. Adding the boundary conditions, we will need to solve a variety of eigenvalue problems. The product solutions that result will involve trigonometric or some of the special functions that we had encountered in Chapter 5. These methods are used in solving the hydrogen atom and other problems in quantum mechanics and in electrostatic problems in electrodynamics. We will bring to this discussion many of the tools from earlier in this book showing how much of what we have seen can be used to solve some generic partial differential equations which describe oscillation and diffusion type problems.

As we proceed through the examples in this chapter, we will see some common features. For example, the two key equations that we have studied are the heat equation and the wave equation. For higher dimensional problems these take the form

\[\begin{align} u_{t} &=k \nabla^{2} u,\label{eq:1} \\ u_{t t} &=c^{2} \nabla^{2} u .\label{eq:2} \end{align} \]

We can separate out the time dependence in each equation. Inserting a guess of \(u(\mathbf{r}, t)=\phi(\mathbf{r}) T(t)\) into the heat and wave equations, we obtain

\[T^{\prime} \phi=k T \nabla^{2} \phi \text {, }\label{eq:3} \]

\[T^{\prime \prime} \phi=c^{2} T \nabla^{2} \phi .\label{eq:4} \]

Dividing each equation by \(\phi(\mathbf{r}) T(t)\), we can separate the time and space dependence just as we had in Chapter ??. In each case we find that a function of time equals a function of the spatial variables. Thus, these functions must be constant functions. We set these equal to the constant \(-\lambda\) and find the respective equations

\[\begin{align} \frac{1}{k} \frac{T^{\prime}}{T} &=\frac{\nabla^{2} \phi}{\phi}=-\lambda\label{eq:5} \\ \frac{1}{c^{2}} \frac{T^{\prime \prime}}{T} &=\frac{\nabla^{2} \phi}{\phi}=-\lambda\label{eq:6} \end{align} \]

The sign of \(\lambda\) is chosen because we expect decaying solutions in time for the heat equation and oscillations in time for the wave equation and will pick \(\lambda>0\).

The respective equations for the temporal functions \(T(t)\) are given by

\[\begin{align} T^{\prime} &=-\lambda k T,\label{eq:7} \\ T^{\prime \prime}+c^{2} \lambda T &=0 .\label{eq:8} \end{align} \]

These are easily solved as we had seen in Chapter ??. We have

\[\begin{align} &T(t)=T(0) e^{-\lambda k t},\label{eq:9} \\ &T(t)=a \cos \omega t+b \sin \omega t, \quad \omega=c \sqrt{\lambda},\label{eq:10} \end{align} \]

where \(T(0), a\), and \(b\) are integration constants and \(\omega\) is the angular frequency of vibration.

In both cases the spatial equation is of the same form,

\[\nabla^{2} \phi+\lambda \phi=0 \text {. }\label{eq:11} \]

This equation is called the Helmholtz equation. For one dimensional problems, which we have already solved, the Helmholtz equation takes the form \(\phi^{\prime \prime}+\lambda \phi=0\). We had to impose the boundary conditions and found that there were a discrete set of eigenvalues, \(\lambda_{n}\), and associated eigenfunctions, \(\phi_{n}\).

In higher dimensional problems we need to further separate out the spatial dependence. We will again use the boundary conditions to find the eigenvalues, \(\lambda\), and eigenfunctions, \(\phi(\mathbf{r})\), for the Helmholtz equation, though the eigenfunctions will be labeled with more than one index. The resulting boundary value problems are often second order ordinary differential equations, which can be set up as Sturm-Liouville problems. We know from Chapter 5 that such problems possess an orthogonal set of eigenfunctions. These can then be used to construct a general solution from the product solutions which may involve elementary, or special, functions, such as Legendre polynomials and Bessel functions.

We will begin our study of higher dimensional problems by considering the vibrations of two dimensional membranes. First we will solve the problem of a vibrating rectangular membrane and then we will turn our attention to a vibrating circular membrane. The rest of the chapter will be devoted to the study of other two and three dimensional problems possessing cylindrical or spherical symmetry.

Thumbnail: A three dimensional view of the vibrating annular membrane. (CC BY-NC-SA 3.0 Unported; Russell Herman)