Analytical solutions to two-dimensional acoustic fields with two-dimensional temperature variation

This paper presents exact solutions for two-dimensional acoustic fields in the presence of two-dimensional mean temperature gradients, in the absence of mean flows. Using similarity transforms, the wave equations in two dimensions is reduced to ordinary differential equation for quadratic and product temperature profiles The resultant ODE is solved by assuming a power series solution and applying the Frobenius method. A method for reducing the two-dimensional Helmholtz equation into two ODE's is developed for temperature profiles of the form H(ax+by+c) where H is any arbitrary function of ax+by+c, by using conformal mapping and separation of variables. Examples involving linear and exponential temperature profiles are then presented. Exact solution for the linear temperature profile is obtained by reducing the resultant ODE to Confluent hypergeometric differential equation by applying suitable transformations. For the exponential temperature profile the ODE is solved by assuming a power series solution and by applying the Frobenius method.