11.1 Introduction

The Cauchy problem or initial value problem (IVP) for the homogeneous wave equation in the free space ℝn is given by

Here, n ⩾ 2 is an integer, the (spatial) dimension; c > 0 is a constant, the speed of propagation and u0 and u1 are given smooth functions, the initial values.

Spherical Mean Function: Given a C2 function ℎ defined on ℝn, define its spherical mean function, denoted by Mℎ, by

for x ∈ ℝn and r > 0. The integration is over the sphere of radius r, centred at x and is the surface measure of this sphere with denoting the surface measure of the unit sphere in ℝn; Γ is the Euler gamma function. By a change of variable, equation (11.1.3) can be written as

The form of equation (11.1.4) enables us to define Mℎ for all real r, and it is readily seen that Mℎ(x, −r) = Mℎ(x, r), that is, Mℎ is an even function of r. This property is used repeatedly in the computations below.

A computation using the divergence theorem yields the Darboux equation:

The notation Δx in the above expressions means the Laplacian taken with respect to the x variables. Note that in the above equation, x is a parameter, and the equation (11.1.5) is a second-order ordinary differential equation (ODE) in the variable r.

Using the Darboux equation and some manipulation gives us the solution of IVP (11.1.1) and (11.1.2), for n = 3:

The representation (11.1.6) is known as the Kirchhoff's formula. By carrying out the t differentiation, we can also write the Kirchoff's formula as follows:

Thus, the Kirchoff's formula (11.1.6) is rewritten as

The above formula brings out the essential features of the solution in the case n = 3. Thus, any C2 solution of the Cauchy problems (11.1.1) and (11.1.2) is given by equation (11.1.6) and hence unique.

For n ⩾ 3 odd, we now write down a formula for the solution of the homogeneous wave equation, similar to the Kirchhoff's formula for n = 3; the formula for the solution for n even is obtained by the method of descent from dimension n + 1, which is discussed in the next section.