3.5 Standing waves and modes

A string of length $L$ clamped at both ends. Apply boundary conditions $y(0, t) = y(L, t) = 0$ to the general d’Alembert solution. The only consistent steady oscillations are sinusoidal, with spatial profile

y_n(x, t) \;=\; A_n \sin\!\left( \frac{n \pi x}{L} \right) \cos(\omega_n t + \varphi_n),

with $n = 1, 2, 3, \ldots$ and

\omega_n \;=\; \frac{n \pi c}{L}, \qquad f_n \;=\; \frac{\omega_n}{2\pi} \;=\; \frac{n c}{2 L}.

These are the modes (or normal modes, or harmonics) of the clamped string. The lowest is the fundamental ( $n = 1$ ); higher $n$ are overtones.

▶ Why the modes are sinusoidal and the frequencies are $n c / 2L$

Look for a solution by separation of variables: $y(x, t) = X(x) T(t)$ . The wave equation becomes

\frac{\ddot T}{c^2 T} = \frac{X''}{X} = -k^2,

a separation constant we name $-k^2$ . The spatial equation $X'' + k^2 X = 0$ has solution $X(x) = A \sin(k x) + B \cos(k x)$ . Boundary condition $X(0) = 0$ kills the cosine ( $B = 0$ ); $X(L) = 0$ forces $\sin(kL) = 0$ , i.e. $k L = n\pi$ for $n = 1, 2, 3, \ldots$ . So $k_n = n\pi/L$ and $\omega_n = c k_n = n\pi c / L$ .

The time equation $\ddot T + \omega_n^2 T = 0$ gives $T(t) = A_n \cos(\omega_n t) + B_n \sin(\omega_n t)$ .

Standing waves as superposed travelling waves

A mode can be written as a sum of two travelling waves of equal amplitude going opposite ways:

\sin(k x) \cos(\omega t) \;=\; \tfrac12 \big[\sin(k x - \omega t) + \sin(k x + \omega t)\big].

A standing wave is what you get when an equal-amplitude right-goer and left-goer interfere everywhere. Where they’re always in phase you get an antinode (maximum amplitude); where they’re always out of phase you get a node (zero amplitude). For mode $n$ on a clamped string, there are $n + 1$ nodes (including the two endpoints) and $n$ antinodes between them.

The general motion

Because the wave equation is linear, any motion of the clamped string is a superposition of modes:

y(x, t) \;=\; \sum_{n=1}^{\infty} A_n \sin\!\left( \frac{n\pi x}{L} \right) \cos(\omega_n t + \varphi_n).

This is a Fourier series in space at each instant. The amplitudes $A_n$ and phases $\varphi_n$ are fixed by initial conditions and are projections of the initial profile onto each mode shape — an integral with $\sin(n\pi x / L)$ as the weighting function.

Why this matters

Two things to carry forward:

Sound in a tube is going to be the same story. The wave equation governs the acoustic pressure; clamped/free boundaries at the tube ends pick a discrete set of allowed frequencies; the general motion is a Fourier sum. Chapter 7 lesson 6 makes the swap explicit.
A musical string plays a fundamental at $f_1 = c/2L$ plus harmonics at integer multiples. The integer-harmonic spacing is the reason plucked-string and bowed-string instruments sound “musical”: their overtone series is consonant. The same integer spacing appears in any 1-D medium with equal boundary conditions on both ends.

Looking ahead

We have, in this chapter, derived a real wave equation from coupled oscillators, solved it explicitly, handled boundaries, and built standing waves. The wave equation we have is for transverse displacement on a string. The wave equation we want — for pressure perturbations in air — has exactly the same mathematical form but a different physical content. Chapter 4 derives that equation, four different ways, and the rest of the book is its consequences.