ABSTRACT
We have already noticed in Chapter 2 that the solutions of conservation law (1.1.1) with m = 1, which are piecewise smooth, admit discontinuities across certain smooth curves in the (x, t) plane. In this chapter, we study systems (1.1.1) in 1-and 3-space dimensions. We consider only piecewise smooth solutions so that there exists a smooth orientable surface
∑ : φ(x, t) = 0, with
outward space-time normal (φ,t, φ,1, φ,2, φ,3), across which u suffers a jump discontinuity satisfying the R-H condition
G[u] = [f,j ]nj ,
and outside of which u is a C1 function; here G = −φ,t/(φ,jφ,j)1/2 is the propagation speed of
∑ in the direction nj = φ,j/(φ,iφ,i)
1/2, 1 ≤ i, j ≤ 3. Explicit formulae for the discontinuities in the first and higher order derivatives of u across
∑ , which are of the nature of compatibility conditions for
the existence of discontinuities, have been derived by Thomas [196] as well as Truesdell and Toupin [201]. A surface
∑ across which the field variable u or
its derivative is discontinuous is called the singular surface or a wave-front; it is only such a wave that is being studied here. Indeed, the use of compatibility conditions that hold on
∑ enable us to obtain some interesting results in the
general theory of surfaces of discontinuity in continuum mechanics.
