The partial differential equation 2u x 2 + 2u y2 + 2u z2 = c 2u t 2 is known as the wave

Chapter 2, Problem 36

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The partial differential equation 2u x 2 + 2u y2 + 2u z2 = c 2u t 2 is known as the wave equation. It models the motion of a wave u(x, y,z, t) in R3 and was originally derived by Johann Bernoulli in 1727. In this equation, c is a positive constant, the variables x, y, and z represent spatial coordinates, and the variable t represents time. (a) Let u = cos(x t) + sin(x + t) 2ez+t (y t) 3. Show that u satisfies the wave equation with c = 1. (b) More generally, show that if f1, f2, g1, g2, h1, and h2 are any twice differentiable functions of a single variable, then u(x, y,z, t) = f1(x t) + f2(x + t) + g1(y t) + g2(y + t) + h1(z t) + h2(z + t) satisfies the wave equation with c = 1.