The vibrating string: modes and energy

For a string fixed at both ends, \( u_{tt} = c^2u_{xx} \), \( u(0,t) = u(L,t) = 0 \), separation of variables gives the same spatial modes as the heat equation, but the time factor now solves \( T'' + c^2\lambda_n T = 0 \) and oscillates: \[ u(x,t) = \sum_{n=1}^\infty\Big(A_n\cos\frac{n\pi ct}{L} + B_n\sin\frac{n\pi ct}{L}\Big)\sin\frac{n\pi x}{L}. \] The initial position fixes \( A_n = \frac2L\int_0^L f\sin\frac{n\pi x}L\,dx \), and the initial velocity fixes \( \frac{n\pi c}{L}B_n = \frac2L\int_0^L g\sin\frac{n\pi x}L\,dx \). The frequencies \( n\pi c/L \) are whole-number multiples of the fundamental: the harmonics of a musical string.

The energy of the string is kinetic plus potential, \[ E(t) = \frac12\int_0^L\big(u_t^2 + c^2u_x^2\big)\,dx. \] Differentiate and integrate the second term by parts: \( E'(t) = \int_0^L u_t\big(u_{tt} - c^2u_{xx}\big)\,dx + \big[c^2u_xu_t\big]_0^L \). The integrand vanishes by the equation, and the boundary term vanishes because \( u_t = 0 \) at fixed ends. So \( E \) is constant. For the single mode \( u = \sin x\cos t \) on \( [0, \pi] \) with \( c = 1 \), \( E = \frac12\int_0^\pi(\sin^2x\sin^2t + \cos^2x\cos^2t)\,dx = \pi/4 \), using the integral the example computes.

Energy gives uniqueness with no formula at all. If \( u \) and \( v \) solve the same problem, \( w = u - v \) solves it with zero data, so \( E_w(0) = 0 \), hence \( E_w(t) = 0 \), hence \( w_t = w_x = 0 \) everywhere, so \( w \) is constant and the boundary condition makes it zero.

Picture it: a plucked string is a sum of standing waves, each node pinned, each loop swinging at its own frequency. Energy sloshes between stretching (at maximum displacement) and motion (as it passes through equilibrium), and the total never changes.

Think it: the energy argument used only integration by parts and the sign of each term, so it survives variable coefficients, curved domains and higher dimensions where no series exists. That portability is why energy methods, not formulas, are the working tools for nonlinear equations.