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4. 1 33 Examples In this subsection, a few of the results seen above will be derived as examples. After studying these examples, the reader should be able to derive any of the results given above. For convenience, this subsection will use numbered indices (1,2,3) rather than (0, +, - ) . 17). Solution The characteristic values of C are the solutions of det (XI — C) = 0. But 0 X— u —p 0 - p a - C)= 2 X - u = ( A - u) [(X - u)2 - a2} . 17). 40). Solution Tofindthe right characteristic vector associated with X\ = u, solve the following system of equations: (uI-C)r= ro - 0 0 0 -i -pa 2 0 "0" ~r\ rri2 == 0 .
The wavefronts dx/dt = a are sometimes also called characteristics curves or simply characteristics. There are infinitely many wavefronts coating the entire x-t plane. 1. The space-time vectors (1, a) are also shown in a few instances to emphasize that the wavefronts are always parallel to (1, a). 1, are known as wave diagrams. 1 Solve the following quasi-linear partial differential equation: du du ir + ^- = 0 . dt dx Suppose the domain is x > 0, t > 0. Also, suppose the initial conditions are u(x, 0) = c(x) and the boundary conditions are w(0, t) = b(t).
2 never intersect. 3, this is not always the case. A conflict occurs when two wavefronts with different signals meet. 3, u = 1 and u = 2 both cannot be true. 3. Shock waves are different from other waves. 3). Instead, shock waves are governed by jump relations and the theory of weak solutions. 3 2 Wave diagram for a shock wave in a scalar model problem. 26 3 I Waves absorb any waves they meet, effectively destroying them and the signals they carry. Notice that shocks can occur any time wavefronts converge, despite the fact that the boundary and initial conditions may be completely smooth and continuous.