Vibrating string deflection

Let the string vibrate according to the wave equation u_tt = c² u_xx, with

• u(0, t) = u(π, t) = 0 (fixed ends),
• u(x, 0) = 0 (initially in equilibrium position),
• u_t(x, 0) = 0.03 sin x − 0.04 sin 3x.

General solution. Separation of variables with the end conditions gives p = n:

u = Σ (Bₙ cos nct + Cₙ sin nct) sin nx

Initial displacement. u(x, 0) = 0 gives Σ Bₙ sin nx = 0, so every Bₙ = 0.

Initial velocity. Differentiating with respect to t and putting t = 0:

u_t(x, 0) = Σ nc Cₙ sin nx = 0.03 sin x − 0.04 sin 3x

Comparing coefficients of sin x and sin 3x (all others are zero):

• n = 1: c C₁ = 0.03, so C₁ = 0.03/c
• n = 3: 3c C₃ = −0.04, so C₃ = −0.04/(3c)

Answer:

u(x, t) = (0.03/c) sin ct sin x − (0.04/3c) sin 3ct sin 3x

Here c² = T/ρ, where c is the wave speed on the string (with c = 1 this is 0.03 sin t sin x − 0.0133 sin 3t sin 3x).