The Wave Equation, Normal Modes, and d'Alembert's Formula

Definition The one-dimensional wave equation with wave speed $c > 0$ is

For small transverse vibrations of a string with tension $\tau$ and mass per unit length $\rho$, it follows from $\rho\,u_{tt} = \tau\,u_{xx}$, so $c = \sqrt{\tau/\rho}$. The equation is second order in time, so an initial-value problem needs two initial functions, the initial displacement $u(x, 0) = f(x)$ and the initial velocity $u_t(x, 0) = g(x)$. On a bounded interval it also needs one boundary condition at each end.

Definition On $0 \le x \le L$ with fixed ends $u(0, t) = u(L, t) = 0$, the product $u = X(x)\,T(t)$ separates the equation into $X'' + \lambda X = 0$ with $X(0) = X(L) = 0$ and $T'' + c^2\lambda T = 0$, where primes are derivatives in each factor’s own variable. With $\lambda_n = (n\pi/L)^2$ and the angular frequencies $\omega_n = n\pi c/L$, the standing modes are the products $X_n(x)\,T_n(t)$ with

for $n = 1, 2, 3, \dots$. The solution with displacement $f$ and velocity $g$ is the sum of these modes with

where $f_n$ and $g_n$ are the half-range sine coefficients of $f$ and $g$. When $f$ and $g$ are smooth enough and compatible with the fixed ends, this series converges to the classical solution. The division by $\omega_n$ is needed because the derivative of $\sin\omega_n t$ is $\omega_n\cos\omega_n t$. Each mode oscillates with the angular frequency $\omega_n$ and stays at rest at its nodes $x = kL/n$ for $k = 0, 1, \dots, n$. For a smooth solution with fixed ends the energy

is constant in time.

Definition A fixed end holds the displacement at zero, so $u = 0$ there, a Dirichlet condition. A free end, such as a ring that slides on a frictionless vertical rod, feels no vertical force from its support, so $u_x = 0$ there, a Neumann condition. With free ends at $x = 0$ and $x = L$, the modes are $\cos(n\pi x/L)$ for $n = 0, 1, 2, \dots$, with the same $\omega_n = n\pi c/L$. The zero mode $n = 0$ has $\lambda_0 = 0$ and the time equation $T_0'' = 0$, so

where $A_0$ and $B_0$ are the mean values of $f$ and $g$ over the interval. The mean displacement of the string is $A_0 + B_0 t$, a uniform drift. A fixed end at $0$ and a free end at $L$ give the modes $\sin\big((2n - 1)\pi x/(2L)\big)$ for $n = 1, 2, 3, \dots$.

Definition For any twice continuously differentiable functions $F$ and $G$ on $\mathbb{R}$, the function

solves the wave equation. The term $F(x - ct)$ is a profile moving right at speed $c$, and $G(x + ct)$ is one moving left at speed $c$. Conversely, every twice continuously differentiable solution for $x \in \mathbb{R}$ and $t \ge 0$ has this form. To see it, set $\xi = x - ct$ and $\eta = x + ct$ and write $u(x, t) = U(\xi, \eta)$. The chain rule gives

so the equation says $U_{\xi\eta} = 0$. Then $U_\xi$ depends on $\xi$ alone, and integrating once more gives $U = F(\xi) + G(\eta)$.

Definition Let $c > 0$, let $f$ be twice continuously differentiable and let $g$ be continuously differentiable on $\mathbb{R}$. The problem $u_{tt} = c^2u_{xx}$ for $x \in \mathbb{R}$ and $t \ge 0$, with $u(x, 0) = f(x)$ and $u_t(x, 0) = g(x)$, has exactly one twice continuously differentiable solution. Its value at each point $(x, t)$ is given by d’Alembert’s formula

It comes from the traveling form $u = F(x - ct) + G(x + ct)$. The data give $F + G = f$ and $-cF' + cG' = g$, and together with $F' + G' = f'$ this yields

With an antiderivative $H$ of $g$, this gives $F = \tfrac{1}{2}f - H/(2c) + k$ and $G = \tfrac{1}{2}f + H/(2c) - k$ for a constant $k$, which cancels in the sum, and $H(x + ct) - H(x - ct)$ is the integral of $g$. On a string $0 \le x \le L$ with fixed ends, the same formula holds after $f$ and $g$ are replaced by their odd $2L$-periodic extensions, and with free ends by their even $2L$-periodic extensions. For the odd extensions of $f$ and $g$ to be twice and once continuously differentiable, the values $f(0)$, $f''(0)$ and $g(0)$ and the same values at $L$ must vanish.

Definition By d’Alembert’s formula, the value $u(x_0, t_0)$ with $t_0 > 0$ depends only on $f$ at the two points $x_0 \pm ct_0$ and on $g$ over the interval

which is the domain of dependence of the point $(x_0, t_0)$. In the $(x, t)$ plane it is the base of the triangle cut out by the two characteristic lines $x - ct = x_0 - ct_0$ and $x + ct = x_0 + ct_0$. Conversely, changing the data only on an interval $[a, b]$ changes $u$ only in the region $a - ct \le x \le b + ct$, the domain of influence of that interval. This is finite propagation speed. In particular, if $f$ and $g$ vanish outside $[a, b]$, then $u(x, t) = 0$ whenever $x < a - ct$ or $x > b + ct$. The heat equation behaves differently. On the whole line, the heat kernel solution with a bounded continuous initial temperature that is nonnegative and positive somewhere is positive everywhere at every time $t > 0$.