The Generalized Radially Symmetric Wave Equation

• Published in: Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences Mathematical and physical sciences, 1956-08, Vol.236 (1205), p.265-277
• Main Author: Mackie, A. G.
• Format: Article
• Language: English
• Subjects: Bessel functions; Boundary conditions; Cauchy problem; Gas dynamics; Integers; Integrands; Mathematical functions; Mathematical integrals; Uniqueness; Wave equations
• ISSN: 1364-5021; 0080-4630; 1471-2946; 2053-9169
• DOI: 10.1098/rspa.1956.0134

In this paper a detailed study is made of solutions of the differential equation ∂2ϕ/∂R2 + k/R ∂ϕ/∂R – ∂2ϕ/∂T2 = 0 in the quarter plane R ≽ 0, T ≽ 0. The boundary value problem considered is that of finding a solution which satisfies Cauchy data on T = 0. The contour integral solutions developed for an equation occurring in gas dynamics, shown to be equivalent to that considered here, are the main aid in the investigation. The solution is obtained first for values of T ≼ R but is continued into the whole quarter plane. This continuation follows from a fundamental uniqueness result that knowledge of ϕ on the characteristic R = T specifies its value in the domain R ≼ T. A point emphasized is that the continuation is not in general the analytic continuation from the domain T ≼ R, even for analytic initial data. Different interpretations of the solutions found are examined. When k is a positive integer the equation is that satisfied by radially symmetric solutions of the wave equation in k + 1 space dimensions, and this leads to the solution of the full wave equation for given initial conditions on T = 0. The Huygens principle is clearly illustrated. For general positive values of k the discussion clarifies a problem in gas dynamics in the study of which the original contour integral solutions were first devised. The general solution is also compared with a solution by separation of variables, and some conclusions are drawn regarding certain infinite integrals involving Bessel functions. In the final section negative values of k are considered. The contour integral representation solves in a concise form the singular initial value problem of finding a solution which takes prescribed values on R = 0, thus generalizing a result well known for positive value of k.