Waves in Fluids

Count the seconds between a flash of lightning and its thunder, divide by three, and you have your distance to the storm in kilometres. That little trick works because a sound wave — a travelling ripple of pressure — crawls through air at a fixed, knowable speed, roughly a million times slower than the light that outran it. Down at the beach a different family of waves is at work: the long, smooth swell arriving from a distant storm has sorted itself out on the journey, the longest waves running ahead and reaching you first. And once in a lifetime the same water carries a wave that crosses an entire ocean in a matter of hours — a tsunami, sprinting over the deep Pacific as fast as a jetliner.

These are all waves in fluids, and remarkably they are governed by just a handful of formulas. This page builds the three great cases side by side: the sound wave, a small compressible squeeze whose speed is set by how stiff the fluid is; and the surface gravity wave on water, whose speed depends — surprisingly — on both its wavelength and the depth of the water beneath it. From that one dispersion relation fall the two limits that explain the sorted swell and the racing tsunami.

Sound: a wave of compression

A sound wave is a tiny compressible disturbance: a region where the fluid is squeezed slightly denser and higher-pressure, followed by one where it is rarefied, marching forward as neighbouring parcels shove one another along. The restoring "spring" is the fluid's resistance to being compressed, measured by its bulk modulus K = \rho\,\partial p/\partial\rho; the "mass" is its density \rho. Balancing the two gives the wave speed:

c = \sqrt{\frac{K}{\rho}} = \sqrt{\frac{\partial p}{\partial \rho}}.

A stiffer fluid (large K) transmits sound faster; a denser one, slower. Water, far stiffer than air, carries sound at about 1480\ \text{m/s} versus air's 340\ \text{m/s}. For a gas the compression happens too fast for heat to leak away, so it is adiabatic and p \propto \rho^{\gamma}, giving the classic result

c = \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\gamma R T}.

The last form is striking: for an ideal gas the sound speed depends only on the temperature (through RT) — not on the pressure, and not on how far up the mountain you are. Warm air carries sound faster than cold, which is why the pitch of an orchestra and the tuning of a wind instrument drift as the hall heats up.

Surface waves: the water goes round, not along

The waves you watch roll toward a beach are surface gravity waves: gravity is the restoring force, pulling raised crests down and letting the surface overshoot into troughs. Their most important — and most counter-intuitive — feature is that the water itself hardly goes anywhere. A parcel near the surface travels in a small, nearly circular orbit: forward under a crest, up, backward under the trough, down, and back roughly to where it started. The shape races forward at the phase speed, but the water only bobs in place. A gull sitting on the swell rides up and down and drifts barely at all; it is the pattern, not the sea, that is travelling.

As the orbit diagram shows, the orbital motion decays with depth. In deep water it dies away exponentially and the sea floor never feels the wave; in shallow water the orbits are squashed into flat ellipses because the bottom is in the way. That single difference — whether the wave "feels the bottom" — is what splits surface waves into two dramatically different regimes.

The dispersion relation

Solving the fluid equations for a small-amplitude wave of wavenumber k = 2\pi/\lambda on water of undisturbed depth h gives one master formula linking the angular frequency \omega to k. It is the beating heart of the whole subject.

Full surface wave. For depth h, \omega^2 = g\,k\,\tanh(kh). The phase speed is c = \omega/k = \sqrt{\dfrac{g}{k}\tanh(kh)}.
Deep water (kh \gg 1, so \tanh(kh)\to 1): \omega^2 = gk, giving phase speed c = \sqrt{g/k} = \sqrt{g\lambda/2\pi} — dispersive.
Shallow water (kh \ll 1, so \tanh(kh)\to kh): \omega^2 = g h\,k^2, giving phase speed c = \sqrt{gh} — non-dispersive.
Sound. For comparison, a compressive (sound) wave has the simple non-dispersive \omega = c\,k with c = \sqrt{K/\rho} = \sqrt{\gamma p/\rho}.

The function \tanh(kh) is the whole story. When the water is deep compared with the wavelength (kh large) it saturates at 1 and drops out; when the water is shallow (kh small) it becomes just kh. Everything between is a smooth interpolation, plotted below.

Deep water: long waves win the race

In deep water the phase speed c = \sqrt{g/k} = \sqrt{g\lambda/2\pi} grows with wavelength: longer waves travel faster. A wave is called dispersive when its speed depends on its wavelength, and deep-water waves are the textbook example. This is exactly why an ocean swell sorts itself out: a distant storm churns up a jumble of wavelengths all at once, but on the long trip across the ocean the fastest, longest waves pull ahead and arrive first, while the short choppy stuff trails behind. By the time the swell reaches your beach it has combed itself into clean, long-period sets — the surfer's "groundswell".

Dispersion forces a distinction between two speeds. The phase speed c_p = \omega/k is the speed of an individual crest; the group velocity c_g = d\omega/dk is the speed of the overall wave packet — and it is the group velocity that carries the energy. For deep-water waves the group velocity is exactly half the phase speed:

c_g = \frac{d\omega}{dk} = \frac{1}{2}\sqrt{\frac{g}{k}} = \frac{1}{2}c_p.

Watch a group of ripples closely and you can see it: individual crests appear at the back of the packet, race forward through it, and vanish off the front, because each crest moves at twice the speed of the group it belongs to.

Shallow water: why tsunamis race across oceans

Now go the other way. When the wavelength is far longer than the depth (kh \ll 1) the phase speed collapses to

c = \sqrt{gh},

which depends only on the depth — not on the wavelength. Shallow-water waves are non-dispersive: every wavelength travels at the same speed, so a shallow-water pulse keeps its shape as it goes. "Shallow" here is relative to the wavelength, and this is the secret of the tsunami. A tsunami has a wavelength of hundreds of kilometres, so even the 4\ \text{km}-deep open Pacific counts as "shallow water" to it. Its speed is

c = \sqrt{g h} = \sqrt{(9.8)(4000)} \approx 200\ \text{m/s} \approx 700\ \text{km/h},

as fast as an airliner — which is how a tsunami crosses an entire ocean in half a day. As it runs into shallowing water near the coast, h drops, so c drops; the back of the wave catches up on the front, the wave bunches up and rears into the towering wall that does the damage. Out in the deep ocean, though, the same wave is barely a metre high and a passing ship would never notice it.

Worked examples

Example 1 — the speed of sound in air. Take air at room temperature with \gamma = 1.4, pressure p = 1.01\times 10^5\ \text{Pa} and density \rho = 1.20\ \text{kg/m}^3:

c = \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\frac{(1.4)(1.01\times 10^5)}{1.20}} \approx \sqrt{1.18\times 10^5} \approx 343\ \text{m/s}.

Bang on the familiar value — and notice it climbs with temperature, since c = \sqrt{\gamma R T}.

Example 2 — a tsunami's speed and crossing time. Over ocean of depth h = 4000\ \text{m} (take g = 9.8\ \text{m/s}^2):

c = \sqrt{gh} = \sqrt{(9.8)(4000)} \approx 198\ \text{m/s}.

To cross d = 8000\ \text{km} of the Pacific takes

t = \frac{d}{c} = \frac{8.0\times 10^6}{198} \approx 4.0\times 10^4\ \text{s} \approx 11\ \text{hours}.

Example 3 — a deep-water ocean swell. A swell of wavelength \lambda = 200\ \text{m} in deep water has k = 2\pi/\lambda \approx 0.0314\ \text{rad/m}, so its phase speed is

c_p = \sqrt{\frac{g}{k}} = \sqrt{\frac{9.8}{0.0314}} \approx 17.7\ \text{m/s},

and the energy of the group travels at only c_g = \tfrac12 c_p \approx 8.8\ \text{m/s}. A 100\ \text{m} swell, being shorter, would run slower — c_p = \sqrt{g\lambda/2\pi} \approx 12.5\ \text{m/s} — which is exactly why the 200\ \text{m} waves reach the beach first.

No — and this is the classic trap. A surface wave carries energy and shape across the sea, but almost no water. Each parcel just loops around a small near-circular orbit and returns essentially to where it began; the crest that seems to "rush toward the shore" is a moving pattern, like the bump that runs along a shaken rope while the rope stays put. This is why a cork or a gull bobs up and down and drifts hardly at all as swell after swell rolls under it. Only right at the breaking point in the shallows does the orbit stop closing and real water get thrown forward onto the sand.

Both are real, and they answer different questions. The phase speed c_p = \omega/k is how fast a single crest travels. The group velocity c_g = d\omega/dk is how fast the wave packet — the envelope you would point at and call "the wave" — travels, and it is what carries the energy and the information. For deep-water waves c_g = \tfrac12 c_p, so crests outrun the group they live in: look closely at a spreading ring of ripples and you will see crests being born at the trailing edge, sliding forward through the packet, and dissolving at the leading edge. For a non-dispersive wave (sound, or shallow water) the two speeds are equal and this vanishes — the packet and its crests march in lockstep.