Question

PrOBleM: SoLuTiONS To THE WAvE EQuATION a) By direct substitution determine which of the following functions satisfy the wave equation 1. g(z, t)-A cos(kr - wt) where A, k, w are positive constants 2. h(z,t)-Ae-(kz-wt)2 where A, k, ω are positive constants 3. p(x, t) A sinh(kx-wt) where A, k,w are positive constants 4. q(z, t) - Ae(atut) where A,a, w are positive constants 5. An arbitrary function: f(x, t) - f(kx -wt) where k and w are positive constants. (Hint: Be careful with your derivative rules.) b) What similarities among functions that satisfy the wave equation do you observe? Do you no- tice any differences between functions that satisfy the wave equation and those that do not? c) Verify by direct substitution that an arbitrary linear combination of two functions that satisfy the wave equation will also satisfy the wave equation. That is, if f(z, t) and g(x,t) both satisfy the wave equation you are to show that the function F(x, t)-αυζ,t)+Ag(x, t) also satisfies the wave equation provided a and ß are constants

Answer 1

WAVE EQUATION partial differential^2 psi/partial differential x^2 = 1/v^2 partial differential^2 psi/partial differential t^2 V is WAVE VELOCITY q(x, t) = A cos(kx - wt) partial differential^2 q/partial differential x^2 = partial differential/partial differential x [partial differential/partial differential x A cos(kx - wt)] = partial differential/partial differential x [-A sin(kx - wt) k] = -k^2 A cos(kx - wt) equation (1) partial differential^2 q/partial differential t^2 = partial differential/partial differential t [partial differential/partial differential t A cos(kx - wt)] = partial differential/partial differential t [-A sin(kx - wt)(-w)] = -w^2 A cos(kx - wt) equation (2) From (1) AND (2) partial differential^2 q/partial differential x^2 = k^2/w^2 partial differential^2 q/partial differential t^2 COMPARE WITH WAVE EQ v = w/k partial differential^2 q/partial differential x^2 = 1/v^2 partial differential^2 q/partial differential t^2 therefore q SATISFY WAVE EQUATIONS h(x, t) = Ae^-(kx - wt)^2] partial differential^2 h/partial differential x^2 = partial differential/partial differential x partial differential/partial differential x [Ae^-(kx - wt)^2] = partial differential/partial differential x [Ae^-(kx - wt)^2 (-2) (kx - wt) k] = -2Ak partial differential/partial differential x [(kx - wt) e^-(kx - wt)^2] = -2Ak [(kx - wt) e^-(kx - wt)^2 (-2) (kx - wt) k +

I will be glad to see your comment if you have any query and thumb up if you are satisfied. Thanks...