The Gold Standard of Physical Oceanography | Fixed Wave Period, Changing Depth and a Group-Speed Reversal

Phase speed describes the movement of a particular wave phase, such as a crest. Group speed describes the motion of a slowly varying envelope made from a narrow range of nearby wave frequencies and wavenumbers.

In a dispersive system the two speeds need not match. Individual crests can move through a packet while the packet itself travels more slowly. Neither speed is the same quantity as a water particle’s oscillatory velocity.

This distinction matters when an observer follows one visible crest but the question concerns the movement of a wave packet or energy. First decide what feature is being tracked, then choose the speed that describes it.

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Assume small-amplitude, two-dimensional surface gravity waves over a horizontal bed, with uniform depth h, no background current and negligible surface tension. Use gravitational acceleration g = 9.81 m/s².

Let period be T, angular frequency ω = 2π/T, wavelength λ and wavenumber k = 2π/λ. The dispersion relation is ω² = gk tanh(kh). Here tanh is the hyperbolic tangent, which approaches its argument for small arguments and approaches one for large arguments.

Depth is shallow or deep relative to wavelength. The controlling quantity is kh, not whether a site looks coastal on a map. A numerical depth without a wavelength or period is insufficient to identify the regime.

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Combine two waves with nearly equal frequencies and wavenumbers. Their rapidly oscillating crests are modulated by a more slowly varying envelope.

The envelope’s phase contains the differences Δk and Δω. Following a fixed envelope feature gives a speed Δω/Δk. In the limit of a narrow band, this becomes c_g = dω/dk at the local depth.

Phase speed instead is c = ω/k. A ratio of values and a derivative need not be equal. They agree for a linear dispersion relation through the origin, but not for every relationship between frequency and wavenumber.

This is why applying a remembered deep-water ratio everywhere is unreliable. The derivative must come from the dispersion relation appropriate to the wave model.

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Differentiate ω² = gk tanh(kh) with respect to k while holding h fixed. The result is 2ω(dω/dk) = g[tanh(kh) + kh sech²(kh)], where sech is the reciprocal of the hyperbolic cosine.

Use the original dispersion relation to simplify. The group speed is c_g = (c/2)[1 + 2kh/sinh(2kh)], with sinh the hyperbolic sine.

For positive finite kh, the ratio c_g/c lies between one-half and one. In deep water it approaches one-half. In shallow water it approaches one, and both speeds approach √(gh).

Those are local model relationships. Differentiating at fixed depth does not mean wavelength stays fixed when we later compare separate basins at different depths.

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Set T = 8 s. Then ω = π/4 rad/s, approximately 0.7854 rad/s. At every depth we must solve the dispersion relation for a compatible k.

A convenient dimensionless form uses q = kh. It becomes q tanh(q) = ω²h/g. For q > 0 the left-hand side increases continuously, so a positive target gives a unique positive solution.

A numerical bracket is easy to check: choose a lower and upper q whose left-hand sides straddle the target, halve the interval and retain the half that still contains the solution. Repeat until the desired precision is reached.

Then recover k = q/h, wavelength λ = 2π/k, phase speed ω/k and group speed from the derivative formula. Do not change h while retaining an old k and still claim to have kept the same period.

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The following rounded results come from solving the same dispersion relation at four depths. Each row describes its own uniform basin; the table does not model a packet crossing an abrupt step.

The group travels faster in the 20 m basin than in the 100 m basin even though its crests travel more slowly. At 1 m depth, the group has slowed substantially and nearly catches up with its much slower phase speed.

Thus decreasing depth does not make group speed fall monotonically across this comparison. This is a calculated counterexample to that simple claim, not an assertion that 20 m is a universal depth of maximum group speed.

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At 100 m depth, kh is approximately 6.288 and c_g/c is approximately 0.5000. At 20 m depth, kh is approximately 1.415 and the ratio rises to approximately 0.6675.

The phase speed has fallen, but the fraction multiplying it has risen enough for group speed to increase. With still shallower water, phase speed falls far enough that the rising ratio no longer produces a faster group.

At 1 m depth, kh is approximately 0.2534 and c_g/c is approximately 0.9792. Being nearly equal to phase speed does not make the group fast: both speeds are now around three metres per second.

Whenever a quantity is the product of two changing factors, inspect both. A rising ratio can coexist with a falling absolute result.

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For a fictional 1,000 m path wholly inside a uniform basin, a narrow-band packet travel time is approximately distance divided by group speed. Retain unrounded values during the calculation.

The 100 m basin gives approximately 160 s, while the 20 m basin gives approximately 135 s. The 1 m basin gives approximately 330 s.

Using the deep-basin phase speed would give approximately 80 s there, roughly half the group travel time. It would answer a crest-propagation question instead.

For a real varying-depth route, a single average depth does not generally produce the full travel time. Direction changes, currents, reflection, bandwidth and the validity of the local approximation require additional treatment.

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Within the stated linear gravity-wave model without a background current, the period-averaged energy flux per unit crest width is the energy per unit horizontal area multiplied by group speed. MIT’s wave-energy notes derive that relationship.

This does not mean a marked water parcel travels with the envelope. Linear wave motion contains local oscillation; higher-order drift and background currents introduce further distinctions.

Nor does a higher group speed alone establish a larger energy flux. The energy density may differ. To compare fluxes, state both the energy being transported and the speed appropriate to its model.

Our table stipulates periods and depths, not wave heights or a complete energy budget. It therefore does not supply a coastal impact or hazard ranking.

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Is group speed always half of phase speed?

No. One-half is the deep-water limit for the gravity-wave model here. Finite-depth ratios lie between one-half and one and approach one in shallow water.

Why not hold wavelength fixed while changing depth?

You can make that comparison, but then the dispersion relation generally changes the period. This article deliberately holds period fixed; different controlled quantities define different questions.

Does shallow mean a fixed depth such as ten metres?

No. It means shallow relative to the relevant wavelength. Calculate kh or the depth-to-wavelength ratio before choosing a limiting formula.

Can I use these numbers for a real wave arrival?

Not as an operational prediction. Real bathymetry, currents, spectra, nonlinearity, breaking and other processes can matter. The uniform basins are a controlled learning comparison.

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In the deep-water limit with T = 8 s, use k = ω²/g. The phase speed becomes g/ω = gT/(2π), approximately 12.49 m/s, and group speed is approximately 6.25 m/s.

Now consider the shallow-water approximation at h = 1 m. Both speeds would be √9.81, approximately 3.13 m/s. Compare that with the full model’s phase speed 3.10 m/s and group speed 3.03 m/s.

The limit is a useful approximation, not an exact replacement at finite kh. State which error matters for the purpose rather than labelling a basin shallow and discarding the comparison.

Finally, double the period while staying in sufficiently deep water. The deep-water phase and group speeds double. That result comes from holding the regime valid; at a fixed finite depth, the longer wavelength may require checking the regime again.

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Return to The Gold Standard of Physical Oceanography for density, pressure, currents, heat and waves. Its companion Geostrophic Shear and the Missing Reference Velocity addresses a current problem with different governing assumptions.

For nearby wave questions, continue with capillary waves and the effect of ripple size and Rossby waves in the atmosphere. Surface tension and planetary rotation create different dispersion relationships; the gravity-wave formula here should not be transferred unchanged.

Use the primary MIT OpenCourseWare notes on linear water-wave dispersion and wave groups, energy and group velocity to follow the model in more detail. The fixed-period basin comparison and calculations here are original teaching examples.

Connect this work to The Gold Standard of Education and the How X Works hub. The transferable question is precise: which quantity is fixed, which speed is being measured and which assumptions make the calculation valid?

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