CALC Wave Equation and Standing Waves. (a) Prove by direct substitution that is a solution of the wave equation, Eq. (15.12), for u = ?/k (b) Explain why the relationship u = ?/k for ?traveling ?waves also applies to ?standing ?waves.

Solution 38E Problem (a) Step 1: Displacement of the wave at any time t is given by y(x,t) = (A SW sin kx)sint ------(1) Velocity of wave v = ------(2) k Where - angular frequency k - wave number Step 2: Partially differentiating the equation (1) with respect to x y(x,t) = A SW k.cos kx.sint -----(2) x y(x,t) 2 2 = -A SW k .sin kx.sint -----(3) x Step 3: Partially differentiating the equation (1) with respect to t y(x,t) t = A SW .sin kx.cost -----(4) y(x,t) 2 t2 = -A SW .sin kx.sint -----(5) Step...