ABSTRACT
Consider Euler equations (2.3.16 - 18) governing the motion of a polytropic gas. In section 4.3.2 we derived the equations of WNLRT for an upstream propagating wave (i.e., the wave corresponding to the eigenvalue c1 = 〈n,q〉−a) on a given study solution (ρ0(x),q0(x), p0(x)). In this section we shall derive the equations of WNLRT for a downstream propagating wave Ωt (corresponding to the eigenvalue c5 = 〈n,q〉+ a) running into a uniform state at rest (ρ0 = constant, q = 0 and p0 = constant). These equations can be deduced from those in the section 4.3.2 by taking ρ0 = constant, q0 = 0 and p0 = constant and then changing the sign of a0 everywhere. This gives the ray equations
ρ− ρ0 = (ρ0/a0)w, q = nw, p− p0 = ρ0a0w (6.1.1)
dx dt
= ( a0 +
γ + 1 2
w
) n,
dn dt
= − γ + 1 2
Lw (6.1.2)
and the transport equation for the amplitude w (which is assumed to be small)
dw
dt = Ωa0w (6.1.3)
where d
dt =
∂
∂t + ( a0 +
γ + 1 2
w
) 〈n,∇〉 (6.1.4)
Ω = −1 2 〈∇,n〉 = mean curvature of Ωt (6.1.5)
and L = ∇− n〈n,∇〉 (6.1.6)
We note the expression (2.4.20) for the components of L in terms of the tangential derivatives ∂∂ηα
β defined by (2.4.8). Another expression
for the mean curvature of Ωt is
Ω = − ( ∂n1 ∂η13
) (6.1.7)
We would like to emphasize once more (see derivation of 5.2.13) that the system (6.1.2 - 4) is a true generalization of Burgers’ equation (1.1.6) to multi-dimensions for the propagation of a multidimensional nonlinear wavefront. Some other equations (Hunter (1995) or section 10.4.2) have been called two-dimensional Burgers’ equations but they are valid neither for an arbitrarily curved wavefront nor do they account for nonlinearity in directions transversal to the direction of propagation (Prasad and Ravindran (1977)).
