Exact N-wave solutions for the non-planar Burgers equation

Abstract

An exact representation of N-wave solutions for the non-planar Burgers equation u t + uu x + ½ju/t = ½δu xx , j = m/n, m < 2n , where m and n are positive integers with no common factors, is given. This solution is asymptotic to the inviscid solution for | x | < √(2 Q 0 t ), where Q 0 is a function of the initial lobe area, as lobe Reynolds number tends to infinity, and is also asymptotic to the old age linear solution, as t tends to infinity; the formulae for the lobe Reynolds numbers are shown to have the correct behaviour in these limits. The general results apply to all j = m/n, m < 2n , and are rather involved; explicit results are written out for j = 0, 1, ½, 1/3 and 1/4. The case of spherical symmetry j = 2 is found to be ‘singular’ and the general approach set forth here does not work; an alternative approach for this case gives the large time behaviour in two different time regimes. The results of this study are com pared with those of Crighton & Scott (1979).

Analytic and Numerical Study of N‐Waves Governed by the Nonplanar Burgers Equation

“…The analysis is fortified by an accurate numerical solution of the problem. This study is brought in close conjunction with the earlier work of Crighton and Scott [13] and Sachdev, Joseph and Nair [3]. …”

Large-time behaviour of solutions of the inviscid non-planar Burgers equation

“…1.4 with j = 0) embeds the solution of the inviscid Burgers equation and the anti-symmetric solution of the heat equation. Inspired by the form of the exact N -wave solution of the Burgers equation, Sachdev et al [22] constructed the exact N -wave solution of the viscous non-planar Burgers eqution (1.4) by making use of the behaviour of solutions of the inviscid eqution (1.1) and the linearized equation. They assumed that 0 < j = m/n < 2, m and n are positive integers with no common factor.…”

“…They proved that positive self-similar solutions of (1.4) vanishing at x = ±∞ exist only for the parametric regime 1/( j + 1) ≤ α < 1/j (see also [21]). It is also interesting to refer to the work of Sachdev et al [22], wherein the constructed N -wave solution of (1.4) embeds both inviscid solution and the old age solution of (1.4). The organization of this paper is as follows.…”

Development of a matched filter detector for acoustic signals at local distances from small explosions

“…Because there is no true analytical expression for a spherical shock wave from a differential equation (because the particle velocity and stress at the shock arrival time are discontinuous), only approximations can be made (e.g., Sachdev et al, 1994). Furthermore, our interest is more on the detection of small explosions at distances of <10 km, which typically have a lower initial rise time than does a true shock wave.…”