Waves as travelling disturbances

Observation

Stretch a rope across a room, tie a short ribbon to its midpoint, and snap the near end once. A hump races down the rope and reaches the far wall a moment later. Now watch the ribbon: it jumps up, drops back, and stays where it was tied. Something plainly travelled the length of the rope, because the far end jerks when the hump arrives, and it delivered enough energy there to do that work. Yet no piece of rope made the trip, since the ribbon that rode one piece of it went nowhere.

The question this entry answers: what is the thing that travels, and what sets how fast it goes?

Variables

The displacement of any one piece of the medium is its distance from rest, and the amplitude A is the largest value that displacement reaches. The frequency f is the number of full cycles a single piece completes per second, and the period T = 1/f is the time one cycle takes. The wavelength lambda is a distance measured along the direction of travel, from one crest to the next. The wave speed v is the rate at which the pattern itself advances, a different quantity from the velocity of any piece of rope, which points sideways and changes throughout every cycle. For a stretched string the medium has two relevant properties: the tension F_T, a force, and the mass per unit length mu.

One division of authority is worth fixing before any equation. The source, the hand doing the shaking, chooses the frequency and the amplitude. The medium, the rope under its tension, sets the speed. Whatever remains gets forced, and the derivation below shows that the wavelength is what remains.

Model

Treat the rope as a chain of small masses, each coupled to its neighbors. Displace one piece and the coupling drags its neighbor after it a moment later, and that neighbor drags the next. Each piece behaves as an oscillator whose push arrives from the piece before it. A wave, in this model, is a disturbance handed down the chain: every piece repeats the motion of its neighbor after a short delay, and the pattern those staggered repetitions trace out is the thing that moves. A stadium crowd doing the wave is the household version, each person standing and sitting in place while the shape sweeps around the stands. The analogy breaks at the mechanism: spectators move on a visual cue they choose to follow, while pieces of rope are dragged by forces from their neighbors, so the rope's hand-off rate is set by tension and mass rather than by convention.

Now derive the relation the pattern quantities must obey. Sit at one fixed point and watch crests pass. Each piece completes one full cycle in one period T, and in exactly that time the pattern advances by one full wavelength: after one cycle every piece is back to doing what it did at the start, so the shape has shifted along by one repeat length lambda. A thing that moves a distance lambda in a time T has speed

v = lambda / T = f lambda

This is the wave relation. Notice what it is: bookkeeping, exact for any repeating wave in any medium, because it says nothing about forces. It cannot by itself tell you the speed; what it does is lock the three quantities together, so that fixing any two forces the third.

Derivation

What does set the speed? For the stretched string the candidates are the two properties the medium has. Tension F_T carries the units of force, kilogram metre per second squared; mass per length mu carries kilograms per metre. Divide them: F_T / mu has units of metres squared per second squared, which is a speed squared. So sqrt(F_T / mu) is the one combination of the string's properties with the units of a speed, and dimensional analysis fixes the wave speed up to a dimensionless factor:

v = sqrt(F_T / mu)

The full force analysis on a small curved element of string, which belongs to the calculus treatment and is deferred, shows that the dimensionless factor is 1.

The causal account behind the formula: tension is the restoring agent. A displaced piece of string is pulled back toward the line by the tension on either side of it, and the harder that pull, the faster the piece snaps back and yanks its neighbor into motion. Mass per length is the inertia; heavier pieces respond more sluggishly to the same pull. A stiffer and lighter medium therefore hands the disturbance from neighbor to neighbor faster, which is exactly the grouping the formula reports: restoring agent on top, inertia underneath. The division generalizes beyond strings; sound in air runs on pressure as the restoring agent over density as the inertia, with the same shape.

Combine the two results. The medium fixes v. The source fixes f, since every piece of the chain repeats the source's cycle at the source's rate. The wavelength then has no freedom left:

lambda = v / f

A fast medium stretches the pattern out; a high driving frequency packs it tight.

One classification falls straight out of the model. The direction of travel and the direction of the local oscillation need not agree. On the rope each piece moves perpendicular to the direction of travel, and such a wave is called transverse. Send a compression pulse down a long soft spring and each coil oscillates along the direction of travel; that wave is called longitudinal, and sound in air is the standard case. The relation v = f lambda holds for both, since the bookkeeping above did not ask which way the pieces move.

Limits and boundary conditions

The account above assumes a linear, non-dispersive medium: displacements small enough that the coupling forces stay proportional and the tension is effectively unchanged by the wave, and a speed that comes out the same for every frequency. The named failure case is dispersion, a medium in which speed depends on frequency. Deep-water waves are the standard example: long-wavelength swell outruns short chop, which is why the smooth long-period swell from a distant storm reaches a coast well ahead of the short choppy waves the same storm made. In a dispersive medium v = f lambda still holds for each frequency separately, but there is no single speed of the medium left to plug in.

The larger scope limit: this entry's mechanism, coupled pieces of matter handing a disturbance along, covers mechanical waves, which need a medium. Light and radio obey the same v = f lambda bookkeeping with no medium at all. That is a boundary of the medium framing used here, rather than of the kinematic relation, which survives intact.

Common mistakes

First, believing the medium travels with the wave. The ribbon answers this: the pattern crossed the room and the rope stayed put. Ocean waves invite the mistake especially strongly, since the crests visibly march ashore, yet a gull floating beyond the breakers bobs in place as they pass.

Second, expecting a bigger or louder wave to move faster. Amplitude does not appear in sqrt(F_T / mu). The speed is the medium's property; a violent shake and a gentle one send their patterns down the same rope at the same rate.

Third, assuming the frequency changes when a wave crosses into a new medium. At the junction between a light string and a heavy one, the two sides share the junction point, and that one point can complete cycles at one rate at a time, so the frequency carries across unchanged. The speed changes because the medium changed, and the wavelength is what adjusts, by lambda = v / f in each medium separately.

Build with it

Verify lambda = v / f on a real rope or a long soft spring, using two independent measurements. First the speed: stretch the rope to a steady tension over a measured length, snap one end, and time a pulse over that length, letting it make several round trips within one timing so reaction error averages out; v is distance over time. Then the wavelength: drive the same end at a steady frequency, set by a metronome or by counting cycles against a stopwatch, and freeze the travelling wave in a photograph or a single video frame. Read the wavelength directly off the frame as the crest-to-crest distance, scaled against a metre stick lying in the shot. Compute the prediction lambda = v / f from the measured speed and the driving frequency, and compare. Success is a directly measured wavelength that matches the prediction within a stated tolerance; ten percent is honest for a hand-driven rope. One deliberate exclusion: do not read the wavelength off the nodes of a standing pattern, tempting as that shortcut is. Node spacing rests on a half-wavelength fact that belongs to superposition, the next entry, and this measurement should stand on the travelling wave alone. If the check misses, diagnose before re-driving: a tension that drifted between the two measurements, a miscounted driving frequency, or a frame blurred enough to smear the crests.