Advanced Calculus fi..

682 Advanced Calculus, Fifth EditionFigure 10.8f (x) versus f (x + at), f (x - at).Figure 10.9 Solution u(x, t) of the wave equation. The curves show thewave forms for t = 0, 1, . .., 10. The units are chosen so that the wavevelocity a is 0.2 L per unit of time.. This representation is at first valid only forHowever, if we extend the definition of f (x) to all x by (10.81), then (10.82) hasmeaning for all x and t and, under the assumptions of the theorem above, representsa solution of the wave equation for all x and t.The term f (x +at) represents the initial displacement translated at units to theleft; the second term represents this displacement translated at units to the right; thisis suggested in Fig. 10.8. In Fig. 10.9, f (x) is chosen as a displacement confinedalmost entirely to an interval in - 6 < x < in + 6, where 6 is small; the solutioncan then be plotted as a function of x and t. The disturbance is seen to split intotwo disturbances that travel in opposite directions until they reach the walls, wherethey are reflected, with a change in sign, and move back together. This can bedemonstrated experimentally in various ways-by displacing and releasing a violinstring or by sound echoes, for example.If the initial displacement f (x) has a jump discontinuity, for example, at xo, butis piecewise very smooth, then (10.82) continues to define a solution of the waveequation except for x f at = xo f kn. These lines are the paths of "propagation ofdiscontinuities"; they are called characteristics.