The mean curvature of the influence surface of wave equation with sources on a moving surface

Abstract

The mean curvature of the influence surface of the space-time point (x,t) appears in linear supersonic propeller noise theory and in the Kirchhoff formula for a supersonic surface. Both these problems are governed by the linear wave equation with sources on a moving surface. The influence surface is also called the Sigma-surface in the aeroacoustic literature. This surface is the locus, in a frame fixed to the quiescent medium, of all the points of a radiating surface f(x,t) = 0 whose acoustic signals arrive simultaneously to an observer at position x and at the time t. Mathematically, the Sigma-surface is produced by the intersection of the characteristic conoid of the space-time point (x,t) and the moving surface. In this paper, we derive the expression for the local mean curvature of the Sigma-surface of the space-time point (x,t) for a moving rigid or deformable surcace f(x,t) = 0. This expression is a complicated function of the geometric and kinematic parameters of the surface f(x,t) = 0. Using the results of this paper, the solution of the governing wave equation of high speed propeller noise radiation as well as the Kirchhoff formula for a supersonic surface can be written as very compact analytic expressions.