Derive or solve wave equation

Separate variables in u_tt = c² u_xx with u = X(x)T(t): X″/X = T″/(c²T) = −p².

X″ + p²X = 0     →  X = A cos px + B sin px
T″ + c²p²T = 0   →  T = C cos cpt + D sin cpt

Boundary conditions. u(0, t) = 0 gives A = 0. u(π, t) = 0 gives B sin pπ = 0, so p = n (an integer).

u = sin nx (C cos nct + D sin nct)

Initial velocity. ∂u/∂t = 0 at t = 0. The t-derivative is sin nx · nc(−C sin nct + D cos nct); at t = 0 it gives D = 0.

So u = Σ bₙ sin nx cos nct.

Initial displacement. u(x, 0) = x on (0, π):

x = Σ bₙ sin nx
bₙ = (2/π)∫₀^π x sin nx dx = (2/π)[−x cos nx/n + sin nx/n²]₀^π
   = −2 cos nπ / n = 2(−1)ⁿ⁺¹/n

Answer:

u(x, t) = 2 Σ (n=1 to ∞) [(−1)ⁿ⁺¹/n] sin nx cos nct

= 2( sin x cos ct − sin 2x cos 2ct / 2 + sin 3x cos 3ct / 3 − … )

(Note that the initial shape x jumps to 0 at x = π, so the series converges to x only for 0 ≤ x < π.)