The Fourier Transform and the Wave Equation

We solve the Cauchy problem for the n-dimensional wave equation, where n is an odd integer, using elementary properties of the Fourier transform. We also show that the solution can be obtained by invoking Bessel functions. The wave equation has a rich history fraught with controversy, both mathematical, see [9], and physical, see [16]. It all started in 1747 when d’Alembert, in a memoir presented to the Berlin Academy, introduced the one dimensional wave equation as a model of a vibrating string his traveling waves solution is a source of enjoyment up to this day, see [1] and [2]. Various methods have been utilized since to obtain the solution to the Cauchy problem for the wave equation in higher dimensions see [3] for a nice coverage of this -, notably the Poisson spherical means, see [10] and [8], and Fourier transform, see [4], [13], and [14]. As for the latter, the results for the wave equation lack the simplicity one would expect from the corresponding results for the Laplace and heat equations. It is our purpose here is to provide an elementary solution to the n dimensional wave equation via the Fourier transform. Now, by Hadamard’s method of descent, if the solution to the wave equation is known in dimension n, the solution for all dimensions less than n can be derived, see [6]. Also, the solution for higher dimensions can be expressed compactly in terms of spherical means, see [7] and [5]. We propose thus to construct the spherical means in n-dimensional space, where n is an odd integer greater than or equal to 3, by simple Fourier analysis techniques. Specifically, we will prove Proposition 1. Let n be an odd integer greater than or equal to 3. Then, sin(R|ξ|) |ξ| = cn ( 1 R ∂ ∂R )(n−3)/2( 1 ωnR ∫