The medium decides the speed, and the source only decides the note

Drive two strings with the same vibrator, at the same frequency, at the same tension. One is thin and one is thick. The waves that run along them travel at different speeds, and since both are being shaken exactly as often, the two waves have different wavelengths.

That is a division of labour worth stating explicitly, because it is easy to spend a long time with waves without noticing it. The source fixes the frequency. The medium fixes the speed. The wavelength is not chosen by either of them — it is the quotient,

$$\lambda = \frac{v}{f},$$

and it is the only one of the three quantities that nothing is directly responsible for.

What decides a speed #

For a string, the answer is

$$v = \sqrt{\frac{T}{\mu}},$$

the tension divided by the mass per unit length, square-rooted. The structure of that expression is more useful than the expression.

Every mechanical wave speed is the square root of a restoring quantity divided by an inertial one. On a string, tension pulls a displaced element back and the mass per length resists it being moved. In air, the bulk modulus provides the restoring push and the density resists: $v = \sqrt{K/\rho}$, which gives 343 m/s at room temperature without any new physics. In a solid rod it is Young’s modulus over density; for deep water waves it is gravity that restores and the answer depends on wavelength, which is the case that breaks the pattern and is dealt with below.

The rule generalises because the underlying situation always does: something displaced is pulled back, something has to be accelerated to displace it, and the competition between the two sets the pace. It is the same competition that sets the period of a pendulum and the frequency of any oscillator, and it is why the same square root appears in all of them.

A dimensional check makes the form nearly inevitable. Tension is a force, so it has units of kg·m/s²; $\mu$ is kg/m. Their quotient is m²/s², whose square root is a speed. There is no other combination of a tension and a linear density that has the units of a speed, so the formula could have been guessed up to a dimensionless factor — which turns out to be exactly one.

The square root, and why a guitar stays in tune #

The flatness of the square root has a practical meaning that anyone who has tuned an instrument has felt.

At 80 N a string carries a wave at 283 m/s. Adding a further 8 N — a ten per cent increase — takes it to 297 m/s, a rise of 4.9 per cent, which is most of a semitone. Near the working tension, a percentage change in tension produces half that percentage change in pitch, because differentiating a square root halves the exponent.

That is why tuning is done by small turns and why an instrument holds its tuning at all. A one per cent relaxation of the string overnight — from creep, from the wood moving, from the temperature — is half a per cent in frequency, which is about eight cents and is at the edge of audibility. Had the speed been proportional to the tension rather than to its square root, every instrument would need retuning continuously.

The same halving works against the maker. Raising a string an octave means doubling the frequency, which needs four times the tension, which is why the difference in tension across a piano’s compass is handled by changing the mass of the strings rather than by tensioning the high ones sixteen times as hard. The bass strings are overwound with copper to raise $\mu$ without raising their stiffness, and that choice is the formula on this page being solved for the variable that is easiest to change.

Why frequency survives a boundary and wavelength does not #

When a wave crosses from one medium into another, one of its three quantities is preserved and it is always the same one.

The frequency is fixed by continuity at the join: the last element of the first medium and the first element of the second are attached, so they move together, and whatever rate the first oscillates at, the second is driven at. Nothing can arrive at a boundary a thousand times a second and leave nine hundred times a second without material accumulating there.

The speed changes, because the speed is a property of the new medium. So the wavelength must change, and it changes in exactly the same ratio: $\lambda_2/\lambda_1 = v_2/v_1$.

What crosses a boundary unchanged is how often the wave repeats in a second; what changes is the distance between repeats. That is the whole content of “frequency survives and wavelength does not”: the far side is being driven by the near side at the driving frequency, so the frequency is imposed rather than intrinsic — and the wavelength is then whatever the new medium’s speed makes it.

That single fact is the whole of Snell’s law once it is put with a geometric argument about wavefronts, and it is why the refractive index of a material is a statement about the speed of light in it. It is also why a diver hears a sound with the pitch it had in the air, and not the pitch its wavelength would suggest: the wavelength has changed by a factor of four and a half on entering water, and the pitch has not moved at all.

What the medium fixes about which notes exist #

A string fixed at both ends can only carry standing waves that fit, which means an integer number of half-wavelengths in its length. The length is a fact about the instrument. The wavelengths are therefore fixed by geometry, and it is the speed that turns them into frequencies:

$$f_n = \frac{nv}{2L} = \frac{n}{2L}\sqrt{\frac{T}{\mu}}.$$

The modes a fixed length allows are decided by the boundary conditions and are the same on every string; the pitch of each is decided by the speed. So a guitar and a bass have the same pattern of harmonics and different notes, and retuning a string changes every one of its modes together — which is what makes the medium’s speed the single quantity a player is adjusting.

So the whole difference between a guitar and a double bass is contained in $v$ and $L$. The set of allowed shapes is identical. The frequencies attached to those shapes come from the speed, which is why the fat strings are the low ones and why shortening a string with a finger raises the pitch in exact inverse proportion to the length that remains.

Reading it backwards makes it an instrument. Measure the pitch of a wire of known length and known mass per length, and the tension follows — which is how the tension in the cables of a suspension bridge is checked, by striking one and listening. The method is good to a per cent or two and requires no load cell, and it is the formula above solved for $T$.

Where a single speed stops existing #

Everything on this page assumes the medium has a speed — one number, the same for every wave in it. That assumption is the exception rather than the rule, and where it fails is the next rung’s subject.

Dispersion. In deep water, gravity restores the surface and the speed comes out as $\sqrt{g\lambda/2\pi}$: long waves travel faster than short ones. A stone dropped in a pond makes a spreading pattern in which the long waves lead and the short ones trail, and there is no single number to call the speed of water waves. Light in glass does the same thing more weakly, which is the whole cause of a rainbow.

Stiffness. A real string resists being bent as well as being stretched, and that adds a restoring effect that grows with curvature — so short wavelengths travel faster than the formula says. The consequence is audible: a piano’s upper harmonics are sharper than exact multiples of the fundamental, tuners stretch the octaves to match, and a stretched-octave piano is a piece of tuning built around a correction term to $v = \sqrt{T/\mu}$.

Amplitude. The derivation assumes small displacements, so that the tension does not change as the string moves. Shake a string hard enough and the tension varies through the cycle, the speed varies with it, and the wave stops obeying a linear equation — the waveform steepens, and in air the same effect turns a large enough pressure wave into a shock, where the crest catches up with the trough and the Mach cone is what results.

A source moving faster than the medium’s own wave speed is the sharpest statement of this page’s claim. The speed belongs to the air and not to the source, so nothing prevents a source from outrunning it — and what happens then is a cone rather than a contradiction. A wave speed is a property of what is being disturbed, and the thing doing the disturbing is not bound by it.

More than one kind of wave. A solid carries both transverse and longitudinal waves and they travel at different speeds, because the restoring quantity is different for each. Seismology is built on that: the longitudinal P wave arrives before the transverse S wave, the gap between them gives the distance to the earthquake, and the fact that S waves do not cross the outer core is the evidence that it is liquid. One medium, two speeds, and a planetary interior read from the difference.

Mersenne’s laws, a century before the equation #

The formula on this page was known as three separate experimental laws long before anyone could derive it, and the way they were stated is a good illustration of what a derivation buys.

Mersenne published them in 1636. The frequency of a stretched string is inversely proportional to its length; proportional to the square root of its tension; and inversely proportional to the square root of its thickness. Three statements, each obtained by holding two things fixed and varying the third, and each correct.

What they do not contain is any suggestion that the three are one statement. There is no reason within Mersenne’s laws why the tension should enter as a square root at all, no reason why the same square root should appear in the thickness law with the opposite sign, and no way to predict what happens in a medium that is not a string. The derivation — a free-body diagram of one element, the restoring force from the curvature, and Newton’s second law — produces all three at once and produces the speed of sound in air as well, from a completely different pair of quantities.

That is the difference between a correlation and a mechanism, and it is worth noticing that the correlations were good enough to build instruments with for two centuries. Mersenne’s laws are exactly right for every question a luthier asks. They are useless for asking what a wave is.

Newton’s twenty per cent #

The speed of sound is the same argument applied to air, and its history contains the clearest example on this site of an approximation that is wrong in a stated, findable way.

Newton derived $v = \sqrt{K/\rho}$ in the Principia and evaluated it using Boyle’s law, which relates pressure and volume at constant temperature. The answer came out around 290 m/s. The measured value, which he knew, was about 350. A twenty per cent discrepancy in a result derived from first principles is not a rounding matter, and Newton attempted to close it by arguing about the finite size of air molecules and the water vapour between them — corrections that were invented to fit the gap and that do not survive inspection.

Laplace resolved it a century later by identifying the wrong assumption, which was the isothermal one. A sound wave compresses air far too quickly for heat to leave the compressed region, so the compressions are warmer and the rarefactions cooler than the surrounding air; the correct modulus is the adiabatic one, larger than the isothermal by the ratio of specific heats $\gamma = 1.4$. The speed rises by $\sqrt{1.4} = 1.18$, and the discrepancy closes to within measurement error.

The lesson is the site’s standing one, and it is why naming the approximation is an invariant here rather than a courtesy. Newton’s derivation was correct. Its inputs contained a physical assumption that was never stated, because it did not look like an assumption — and the twenty per cent sat in the literature for a hundred years being blamed on the properties of the air rather than on the properties of the argument.

Helium does not raise the pitch of a voice #

The best demonstration of the division of labour on this page is one that almost everybody has seen and almost everybody describes wrongly.

Breathe helium and speak. Helium is light and springy — a molar mass of four against air’s twenty-nine, and a ratio of specific heats of 5/3 against 1.4 — so the speed of sound in it is about 965 metres per second, nearly three times air’s. The voice that comes out is unmistakably altered, and it is always said that the helium has raised its pitch.

It has not. Pitch is the rate at which the vocal folds open and close, and the folds are a source: they are driven by air pressure and their own tension and mass, not by what the sound they emit will travel through. Measure the fundamental and it is essentially where it was.

What changes is the vocal tract. The tract is a resonant tube of fixed length, so the frequencies it emphasises are set by the speed in the gas filling it, and every one of them moves up by the same factor of nearly three. Those resonances are the formants, and they are what makes a vowel a particular vowel. So the note is unchanged and the timbre is transformed, which the ear — trained to hear formants as vowel identity and as the size of the speaker — reports as a small squeaky person.

Sulphur hexafluoride does the reverse, being five times heavier than air, and produces a voice of the same pitch with the formants dropped by a factor of nearly three. Two gases, two shifts in opposite directions, and in neither case has anything touched the frequency the source is producing.

The one an orchestra has to correct for #

The speed of sound in a gas depends on its temperature and not on its pressure, which is a slightly surprising consequence of the formula. The modulus is proportional to the pressure and so is the density, so the pressure divides out and what remains is temperature over molar mass. A wind instrument on a mountain plays the note it plays at sea level; the same instrument in a cold hall does not.

The dependence is about 0.6 metres per second per degree, so a hall warming by six degrees over the first half of a concert raises the speed of sound by one per cent — which for every wind and brass instrument in the room is one per cent in pitch, or about seventeen cents. Their pitch is set by the standing waves in a tube of fixed length, and it is the speed that converts those wavelengths into frequencies, exactly as it does for the string above.

The strings, meanwhile, drift the other way. Warming lengthens the string and softens the wood, tension falls slightly, and the pitch falls with the square root of it. So the two families of instrument in an orchestra respond to the same temperature change in opposite directions, which is why players warm their instruments before tuning and why the tuning is checked again after an interval.

What the figure cannot show #

The figures draw a wave along a string at one instant, with the wavelength marked on the drawing. Two things are outside them.

The first is time. A still picture of a travelling wave is identical to a still picture of a standing one, and the difference — whether the pattern moves or merely breathes in place — is exactly what a snapshot discards. The ghost curve in the third figure is the minimum device for putting it back.

The second is what is actually moving. The figure shows a transverse displacement against position, and it is easy to read the peak of the curve as an object travelling to the right. Nothing travels to the right; each element of the string moves only up and down, and the shape is what propagates. The speed on this page is the speed of the shape, and the string’s own material moves at a completely different speed — one that depends on the amplitude and the frequency and not at all on the tension.

Measuring it without measuring a speed #

The speed on this page is a few hundred metres per second, which is awkward to time directly over a length of string. It is measured instead by not measuring it.

Melde’s arrangement drives one end of a horizontal string with a vibrator of known frequency and hangs a weight over a pulley at the other. The tension is the weight. The frequency is the vibrator’s. Adjusting the length until the string settles into a clean standing pattern makes the wavelength readable with a ruler — the distance between adjacent nodes is half of it — and the speed is then the product of two things that were both read off apparatus at rest.

That is a recurring move worth naming. A quantity too fast to time is converted into a standing pattern, whose spacing is a length, and lengths are the easiest thing in physics to measure well. The same substitution underlies the interferometer, the diffraction grating and every determination of the speed of light after Fizeau’s — and each of them replaces “how long did it take” with “how far apart are the fringes”.

The experiment also produces the three of Mersenne’s laws in an afternoon, which is why it survives in teaching laboratories two centuries after it stopped being research.

Where the ladder goes next #

The rungs from here: the wave equation derived from a free-body diagram of one element of the string, in which $\sqrt{T/\mu}$ falls out as the coefficient rather than being quoted; reflection and transmission at a boundary, and why a wave reflects inverted from a heavier medium; impedance, which is the quantity that decides how much crosses and is not the speed; group and phase velocity, once a medium has more than one speed; the speed of sound from the properties of a gas, where the answer comes out wrong until the compression is treated as adiabatic; and waves in a moving medium, where the speed and the wind add and everything on this page acquires a frame.

The claim to carry forward is the division of labour. The wave’s frequency belongs to whatever made it, and its speed belongs to what it is crossing — and because those two are set independently by unrelated parties, the wavelength is a consequence rather than a property.