Step 1: A string of length L = 1 m is fixed at both ends and plucked so its initial shape is u(x,0) = sin(πx/L). Find u(x,t) using separation of variables. Step 2: Assume u(x,t) = X(x)T(t). Substituting into the wave equation ∂²u/∂t² = c²∂²u/∂x² and dividing by X·T gives T''/(c²T) = X''/X = −k². Step 3: The spatial equation is X'' + k²X = 0 with X(0) = X(1) = 0. The solution is X = A sin(kx). The boundary condition at x = 1 forces k = nπ for n = 1, 2, 3, … . Step 4: The time equation is T'' + (ck)²T = 0, giving T = cos(ck·t) (no initial velocity, so the sine term drops out). Step 5: The initial shape sin(πx/L) already matches the n = 1 mode, so only the n = 1 term survives: u(x,t) = sin(πx)cos(πct). Result: The string vibrates as a single standing wave of frequency f = c/(2L), oscillating in place.