CALC ?Which of the following wave functions satisfies the wave equation, Eq. (15.12)? (a) ; (b) (d) For the wave of part (b), write the equations for the transverse velocity and transverse acceleration of a particle at point ?x?.

Solution 9E Step 1 of 10: The wave equation is given by, 2 2 y2x,t= 12 y(2,t)………...1 x v t and the given relation k = 2 …………..2 v2 Where k is wavenumber, v is wave speed and is the angular speed. In the given problem, we need to check whether the given wave functions satisfies wave equation. Step 2 of 10: (a) y(x,t)= A cos(kx+ t) y(x,t) To calculate 2 x 2 2 y(x,t= (A cos(kx+t)) x2 x2 y(x,t) (A cos(kx+t)) 2 = ( ) x x x On differentiating y(x,t) x2 = x ( Ak sin(kx + t)) On differentiating y(x,t) 2 x2 = Ak cos(kx + t)………...3 Step 3 of 10: 2 To calculate 2 y(x,t) v t2 2 2 1 y(x,t) 1 (A cos(kx+t)) v2 t2 = v2 t2 1 y(x,t) 1 (A cos(kx+t)) v 2 2 = v2 ( ) t t t On differentiating 2 12 y(x,t)= 2 ( A sin(kx + t)) v t2 v t On differentiating 2 1 y(x,t) 2 v2 t2 = A v2 cos(kx + t) 2 2 Using k = v2 2 1 y(x,t)= Ak cos(kx + t)…………..4 v 2 t2 Comparing equation 3 and 4 we can conclude that they are equal,hence given wave function satisfies wave equation. Step 4 of 10: (b) y(x,t)= A sin (kx+ t) y(x,t) To calculate x2 2 2 y(x,t) (A sin(kx+t)) x 2 = x2 2 y(x,t) = ((A sin(kx+t))) x 2 x x On differentiating y(x,t) 2 = (Ak cos(kx + t)) x x On differentiating y(x,t) 2 2 = Ak sin(kx + t)………...5 x Step 5 of 10: 1 y(x,t) To calculate v2 2 t 2 2 12 y(2,t) = 12 (A sin2kx+t)) v t v t y(x,t) (A sin(kx+t)) 12 2 = 12 ( ) v t v t t On differentiating 2 1 y(x,t) 1 v2 t = v2t (A cos(kx + t)) On differentiating 2 2 2 y(x,t)= A 2 sin(kx + t) v t2 v 2 Using k = 2 2 v y(x,t) 2 12 2 = Ak sin(kx + t)…………..6 v t Comparing equation 5 and 6 we can conclude that they are equal,hence given wave function satisfies wave equation.