Transient evolution of steady‐state shock profiles for sonic booms in a relaxing atmosphere

Abstract

When a shock wave propagates through a uniform medium over a long distance, there is an asymptotic tendency for the portion of the waveform that includes a sudden pressure rise (shock) to approach an invariant form F(x−Vl), where Vis the shock speed and where the function F, in addition to the indicated argument, depends only on the net pressure jump Psh and the local properties of the atmosphere (including relaxation times). The present paper is concerned with the broad question of how long it takes (or how far the shock must propagate) before this asymptotic limit is realized, given some other nonasymptotic profile at some initial time. Although this question has in principle been answered by Benton and Platzman [Q. Appl. Math. (1972)] for the one‐dimensional Burgers' equation, new techniques are required when internal relaxation is to be taken into account. A perturbation theory is used here, with the shock profile taken as the steady‐state profile plus a perturbation, which must asymptotically decay to zero. This perturbation, to lowest order, is found to satisfy a set of linear homogeneous partial differential equations with nonconstant coefficients. With the choice of l and ξ = x − Vl as the independent variables, the coefficients depend only on ξ. Consequently, the equations can be solved for any given initial value problem with the use of Laplace or Fourier transforms. The asymptotic decay of the inverse Fourier transforms to zero is developed by well‐known techniques such as described in the text by Bleistein and Handelsman [Dover, New York (1986)]. [Work supported by NASA‐LRC and by the William E. Leonhard endowment to Penn State Univ. The author acknowledges the advice of A. D. Pierce.]When a shock wave propagates through a uniform medium over a long distance, there is an asymptotic tendency for the portion of the waveform that includes a sudden pressure rise (shock) to approach an invariant form F(x−Vl), where Vis the shock speed and where the function F, in addition to the indicated argument, depends only on the net pressure jump Psh and the local properties of the atmosphere (including relaxation times). The present paper is concerned with the broad question of how long it takes (or how far the shock must propagate) before this asymptotic limit is realized, given some other nonasymptotic profile at some initial time. Although this question has in principle been answered by Benton and Platzman [Q. Appl. Math. (1972)] for the one‐dimensional Burgers' equation, new techniques are required when internal relaxation is to be taken into account. A perturbation theory is used here, with the shock profile taken as the steady‐state profile plus a perturbation, which must asymptotically decay t...