Pluck a guitar string and something strange happens. The disturbance you make does not fly off the end of the string like a thrown ball — it sits there, humming, the string bulging up and down in a fixed shape while the whole room fills with a single steady note. Yet the very same string, tapped at one end, will send a visible pulse racing to the other end. One string, two utterly different behaviours: a wave that travels, and a wave that stands still. This page is about both, and — the punchline worth holding onto from the start — the standing one is secretly built out of two travelling ones running in opposite directions.

We will write down the tidy piece of mathematics that describes a travelling wave, learn to read every symbol in it, and use it to compute real numbers. Then we will add two such waves nose to nose and watch a standing wave fall out of the algebra. Finally we will see why a string clamped at both ends can only sing certain notes — the harmonics — and why that is the reason a guitar sounds like a guitar.

The sinusoidal travelling wave

The simplest travelling wave is a single pure sine shape sliding along the x axis without changing form. If y(x,t) is the displacement of the medium at position x and time t, that wave is

Four symbols carry all the meaning, and each one deserves a moment:

• Amplitude A — the maximum displacement from rest, how "tall" the wave is. Louder sound, bigger ripple: larger A.
• Wavenumber k — how much phase the wave packs into each metre of space. It is set by the wavelength \lambda (the distance between two crests) through k = \dfrac{2\pi}{\lambda}, measured in radians per metre. Short wavelength means large k: many wiggles crammed into a short distance.
• Angular frequency \omega — how much phase the wave sweeps through each second. It is set by the frequency f (oscillations per second) through \omega = 2\pi f, in radians per second.
• Phase \varphi = kx - \omega t — the angle fed to the sine. Everything that matters about "where in its cycle" a point is lives here.

Why does the combination kx - \omega t make the shape move? Pick out one crest — a point of fixed phase, say kx - \omega t = \tfrac{\pi}{2}. As time t ticks up, x must grow to keep the phase constant, so that crest slides steadily towards larger x: the wave moves in the +x direction. Flip the sign to kx + \omega t and the same argument sends the crest the other way, towards −x. Sign of the \omega t term = direction of travel. Remember that; the standing wave depends on it.

Watch it move

Drag the Time slider and the whole shape marches to the right, exactly as the constant-phase argument promised — but its form never changes. Push the Wavenumber slider up and the crests crowd together (shorter wavelength); push Amplitude and it grows taller. This one curve is the entire behaviour of an ideal travelling wave in a single picture.

The relations that tie it together

A crest of fixed phase advances at the wave speed v. Setting kx - \omega t = \text{const} and differentiating gives k\,\dfrac{dx}{dt} = \omega, so

Example 1 — from wavelength and frequency to everything. A wave on a rope has wavelength \lambda = 0.50\ \text{m} and frequency f = 10\ \text{Hz}. Then

Example 2 — reading a formula. A wave is written as y = (0.02\ \text{m})\sin\!\big(4x - 200t\big) in SI units. Match term by term: A = 0.02\ \text{m}, k = 4\ \text{rad/m}, \omega = 200\ \text{rad/s}. Hence

Notice you never needed the wavelength to get the speed — v = \omega/k reads it straight off the two coefficients.

Two waves collide: the standing wave

Now send two identical waves at each other — same amplitude, same k, same \omega — one going right, one going left. This is exactly what happens when a wave on a string reflects off a fixed end and runs back into the wave still arriving. By the principle of superposition, the total displacement is just their sum:

Use the identity \sin(P) + \sin(Q) = 2\sin\!\big(\tfrac{P+Q}{2}\big)\cos\!\big(\tfrac{P-Q}{2}\big) with P = kx - \omega t and Q = kx + \omega t. The averages come out clean: \tfrac{P+Q}{2} = kx and \tfrac{P-Q}{2} = -\omega t. Since cosine is even,

Stare at what just happened. The x and the t have separated. There is no longer any kx - \omega t travelling combination — nothing moves along. Instead the wave has a fixed spatial shape, \sin(kx), whose height is simply scaled up and down in time by the factor \cos(\omega t). Every point oscillates in place. This is a standing wave.

In the picture below, drag Time. The whole curve breathes up and down between the two dashed envelope curves \pm 2A\sin(kx) — but the points where it crosses zero, the nodes, never budge. That is the signature of a standing wave: fixed nodes, pulsing antinodes.

A string fixed at both ends: the normal modes

Clamp a string of length L at both ends — a guitar, a violin, a piano wire. The ends can never move, so they are forced to be nodes. That single boundary condition is enormously restrictive: only standing waves whose shape \sin(kx) is zero at both x = 0 and x = L can survive. The first is automatic; the second demands \sin(kL) = 0, i.e. kL = n\pi for a whole number n = 1, 2, 3, \dots

Translating k = n\pi/L back into wavelength and frequency (using k = 2\pi/\lambda and f = v/\lambda) gives the allowed normal modes:

Step through the modes with the selector. Mode n has exactly n humps (antinodes) and n-1 interior nodes, with both clamped ends forced to zero. These are the only pure shapes the string will hold; every real pluck is a mixture of them.

Example 3 — tuning a guitar string. A guitar's high E string has vibrating length L = 0.65\ \text{m} and the wave speed along it is v = 429\ \text{m/s}. Its fundamental is

which is indeed the pitch of E4. The harmonics ring at f_2 \approx 660\ \text{Hz}, f_3 \approx 990\ \text{Hz}, and so on — the integer stack that gives the string its rich, musical timbre. Press the string down at a fret to shorten L, and every f_n rises: shorter string, higher note.