Standing Wave Simulator

Two clamped ends is the whole story

A string fixed at both ends can carry a wave only if that wave has a node at each end. There is no other constraint, and everything about harmonics follows from it. A node at each end means the length must be a whole number of half wavelengths, so the only allowed wavelengths are 2L, L, 2L/3, L/2 and so on: λₙ = 2L/n.

The speed is fixed by the string rather than by the wave, v = √(T/μ), so the allowed frequencies are fₙ = nv/2L. Every one of them is an exact integer multiple of the lowest. That is why a plucked string sounds like a note and a struck saucepan does not: the saucepan’s allowed frequencies are not in whole-number ratios.

A standing wave really is two travelling waves

This is usually asserted and rarely shown, so the simulator draws it. Add a wave travelling right to an identical wave travelling left:

A sin(kx − ωt) + A sin(kx + ωt) = 2A sin(kx) cos(ωt)

Position and time have separated. The sin(kx) factor fixes an amplitude for each point on the string, and the cos(ωt) factor makes every point oscillate in step with every other. That separation is exactly what “standing” means, and it is why nothing appears to move along the string even though both components are moving at 141 m/s.

Watch the two dashed components slide in opposite directions while their sum stays put. The nodes are where sin(kx) = 0, so the two components are equal and opposite there at every instant, forever.

The test that distinguishes standing from travelling

Pause a quarter of a period after the extreme and the entire string is flat at once. Every point crosses zero simultaneously, and at that instant all the energy is kinetic. A travelling wave is never flat: there is always a crest somewhere. If you can flatten it, it was standing.

Counting nodes, and the off-by-one

Harmonic n has n + 1 nodes and n antinodes. The plus one catches almost everyone, because the two clamped ends are nodes and are easy to overlook when you are counting the visible bumps. The fundamental has two nodes and one antinode. The third harmonic has four nodes and three antinodes.

What changes the pitch, and what does not

• Tension. Pitch goes as √T, so quadrupling the tension doubles the frequency. This is what a tuning peg does.
• Length. Pitch goes as 1/L. Halving the vibrating length doubles the frequency, which is what a finger on a fretboard does.
• Mass per unit length. Pitch goes as 1/√μ, which is why the bass strings on a guitar are the thick ones.
• Amplitude. Nothing. Watch the frequency readout while you drag the amplitude slider; it does not move. This is why a plucked string holds its pitch as it fades instead of sliding downwards.

What this model leaves out

The string is perfectly flexible, so it has no stiffness of its own. Real strings, and especially thick piano bass strings, are slightly stiff, which raises the upper harmonics above exact integer multiples. Piano tuners deal with the consequences of that every day.

There is also no damping here, so the wave rings forever. And only one harmonic is present at a time. A real plucked string vibrates in many harmonics at once, and the mixture is what makes a guitar sound like a guitar rather than like a flute playing the same note.

Common mistakes

• Forgetting the end nodes. Harmonic n has n+1 nodes, not n.
• Thinking the wavelength is the length of the string. It is 2L/n. Only the second harmonic has a wavelength equal to L.
• Expecting frequency to scale with tension. It scales with the square root of tension. Doubling the pitch needs four times the tension.
• Assuming nothing is moving. The string is moving fast; it is the pattern that is stationary. Two travelling waves are crossing at the full wave speed the whole time.
• Mixing grams per metre with kilograms per metre. A factor of a thousand inside a square root is a factor of about 32 in the speed. The slider here is in g/m and the conversion is done for you.
• Believing amplitude affects pitch. It does not, for any wave in a linear medium.