Dispersion (water waves)

For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths.

Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves.

Frequency dispersion for surface gravity waves

This section is about frequency dispersion for waves on a fluid layer forced by gravity, and according to linear theory. For surface tension effects on frequency dispersion, see surface tension effects in Airy wave theory and capillary wave.

Wave propagation and dispersion

The simplest propagating wave of unchanging form is a sine wave. A sine wave with water surface elevation ${\displaystyle \eta (x,t)}$ is given by:

${\displaystyle \eta (x,t)=a\sin \left(\theta (x,t)\right),}$

where a is the amplitude (in metres) and ${\displaystyle \theta =\theta (x,t)}$ is the phase function (in radians), depending on the horizontal position (x, in metres) and time (t, in seconds):

${\displaystyle \theta =2\pi \left(\frac{x}{\lambda }-\frac{t}{T}\right)=kx-\omega t,}$ with ${\displaystyle k=\frac{2\pi }{\lambda }}$ and ${\displaystyle \omega =\frac{2\pi }{T},}$

where:

• λ is the wavelength (in metres),
• T is the period (in seconds),
• k is the wavenumber (in radians per metre) and
• ω is the angular frequency (in radians per second).

Characteristic phases of a water wave are:

• the upward zero-crossing at θ = 0,
• the wave crest at θ = ⁠1/2⁠ π,
• the downward zero-crossing at θ = π and
• the wave trough at θ = ⁠3/2⁠ π.

A certain phase repeats itself after an integer m multiple of ${\displaystyle 2\pi }$. ${\displaystyle \sin (\theta )=\sin (\theta +2\pi m)}$

Essential for water waves, and other wave phenomena in physics, is that free propagating waves of non-zero amplitude only exist when the angular frequency ω and wavenumber k (or equivalently the wavelength λ and period T) satisfy a functional relationship: the frequency dispersion relation

${\displaystyle {\omega }^2={\mathrm{\Omega }}^2(k).}$

The dispersion relation has two solutions: ${\displaystyle \omega =+\mathrm{\Omega }(k)}$ and ${\displaystyle \omega =-\mathrm{\Omega }(k)}$, corresponding to waves travelling in the positive or negative x–direction. The dispersion relation will in general depend on several other parameters in addition to the wavenumber k. For gravity waves, according to linear theory, these are the acceleration by gravity g and the water depth h. The dispersion relation for these waves is:

${\displaystyle {\omega }^2=gk\tanh (kh)}$ or ${\displaystyle {\displaystyle \lambda =\frac{g}{2\pi }{T}^2\tanh \left(2\pi \frac{h}{\lambda }\right),}}$

an implicit equation with tanh denoting the hyperbolic tangent function.

An initial wave phase ${\displaystyle \theta ={\theta }_0}$ propagates as a function of space and time. Its subsequent position is given by:

${\displaystyle x=\frac{\lambda }{T}t+\frac{\lambda }{2\pi }{\theta }_0=\frac{\omega }{k}t+\frac{{\theta }_0}{k}.}$

This shows that the phase moves with the velocity:

${\displaystyle c_p=\frac{\lambda }{T}=\frac{\omega }{k}=\frac{\mathrm{\Omega }(k)}{k},}$

which is called the phase velocity.

Phase velocity

A sinusoidal wave, of small surface-elevation amplitude and with a constant wavelength, propagates with the phase velocity, also called celerity or phase speed. While the phase velocity is a vector and has an associated direction, celerity or phase speed refer only to the magnitude of the phase velocity. According to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth. For a fixed water depth, long waves (with large wavelength) propagate faster than shorter waves.

In the left figure, it can be seen that shallow water waves, with wavelengths λ much larger than the water depth h, travel with the phase velocity

${\displaystyle c_p=\sqrt{gh}\text{(shallow water)}}$

with g the acceleration by gravity and c_p the phase speed. Since this shallow-water phase speed is independent of the wavelength, shallow water waves do not have frequency dispersion.

Using another normalization for the same frequency dispersion relation, the figure on the right shows that for a fixed wavelength λ the phase speed c_p increases with increasing water depth. Until, in deep water with water depth h larger than half the wavelength λ (so for h/λ > 0.5), the phase velocity c_p is independent of the water depth:

${\displaystyle c_p=\sqrt{\frac{g\lambda }{2\pi }}=\frac{g}{2\pi }T\text{(deep water),}}$

with T the wave period (the reciprocal of the frequency f, T=1/f ). So in deep water the phase speed increases with the wavelength, and with the period.

Since the phase speed satisfies c_p = λ/T = λf, wavelength and period (or frequency) are related. For instance in deep water:

${\displaystyle \lambda =\frac{g}{2\pi }{T}^2\text{(deep water).}}$

The dispersion characteristics for intermediate depth are given below.

Group velocity

Interference of two sinusoidal waves with slightly different wavelengths, but the same amplitude and propagation direction, results in a beat pattern, called a wave group. As can be seen in the animation, the group moves with a group velocity c_g different from the phase velocity c_p, due to frequency dispersion.

The group velocity is depicted by the red lines (marked B) in the two figures above. In shallow water, the group velocity is equal to the shallow-water phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: ${\displaystyle c_g=\frac{1}{2}{c}_p}$.

The group velocity also turns out to be the energy transport velocity. This is the velocity with which the mean wave energy is transported horizontally in a narrow-band wave field.

In the case of a group velocity different from the phase velocity, a consequence is that the number of waves counted in a wave group is different when counted from a snapshot in space at a certain moment, from when counted in time from the measured surface elevation at a fixed position. Consider a wave group of length Λ_g and group duration of τ_g. The group velocity is:

${\displaystyle c_g=\frac{{\mathrm{\Lambda }}_g}{{\tau }_g}.}$

The number of waves in a wave group, measured in space at a certain moment is: Λ_g / λ. While measured at a fixed location in time, the number of waves in a group is: τ_g / T. So the ratio of the number of waves measured in space to those measured in time is:

${\displaystyle \frac{\text{No. of waves in space}}{\text{No. of waves in time}}=\frac{{\mathrm{\Lambda }}_g/\lambda }{{\tau }_g/T}=\frac{{\mathrm{\Lambda }}_g}{{\tau }_g}\cdot \frac{T}{\lambda }=\frac{c_g}{c_p}.}$

So in deep water, with ${\displaystyle c_g=\frac{1}{2}{c}_p}$, a wave group has twice as many waves in time as it has in space.

The water surface elevation η(x,t), as a function of horizontal position x and time t, for a bichromatic wave group of full modulation can be mathematically formulated as:

${\displaystyle \eta =a\sin \left(k_{1}x-{\omega }_{1}t\right)+a\sin \left(k_{2}x-{\omega }_{2}t\right),}$

with:

• a the wave amplitude of each frequency component in metres,
• k₁ and k₂ the wave number of each wave component, in radians per metre, and
• ω₁ and ω₂ the angular frequency of each wave component, in radians per second.

Both ω₁ and k₁, as well as ω₂ and k₂, have to satisfy the dispersion relation:

${\displaystyle {\omega }_1^2={\mathrm{\Omega }}^2(k_1)}$ and ${\displaystyle {\omega }_2^2={\mathrm{\Omega }}^2(k_2).}$

Using trigonometric identities, the surface elevation is written as:

${\displaystyle \eta =\left[2a\cos \left(\frac{k_1-k_2}{2}x-\frac{{\omega }_1-{\omega }_2}{2}t\right)\right]\cdot \sin \left(\frac{k_1+k_2}{2}x-\frac{{\omega }_1+{\omega }_2}{2}t\right).}$

The part between square brackets is the slowly varying amplitude of the group, with group wave number ⁠1/2⁠ ( k₁ − k₂ ) and group angular frequency ⁠1/2⁠ ( ω₁ − ω₂ ). As a result, the group velocity is, for the limit k₁ → k₂ :

${\displaystyle c_g=\underset{k_1\to k_2}{lim}\frac{{\omega }_1-{\omega }_2}{k_1-k_2}=\underset{k_1\to k_2}{lim}\frac{\mathrm{\Omega }(k_1)-\mathrm{\Omega }(k_2)}{k_1-k_2}=\frac{\text{d}\mathrm{\Omega }(k)}{\text{d}k}.}$

Wave groups can only be discerned in case of a narrow-banded signal, with the wave-number difference k₁ − k₂ small compared to the mean wave number ⁠1/2⁠ (k₁ + k₂).

Multi-component wave patterns

The effect of frequency dispersion is that the waves travel as a function of wavelength, so that spatial and temporal phase properties of the propagating wave are constantly changing. For example, under the action of gravity, water waves with a longer wavelength travel faster than those with a shorter wavelength.

While two superimposed sinusoidal waves, called a bichromatic wave, have an envelope which travels unchanged, three or more sinusoidal wave components result in a changing pattern of the waves and their envelope. A sea state – that is: real waves on the sea or ocean – can be described as a superposition of many sinusoidal waves with different wavelengths, amplitudes, initial phases and propagation directions. Each of these components travels with its own phase velocity, in accordance with the dispersion relation. The statistics of such a surface can be described by its power spectrum.

Dispersion relation

In the table below, the dispersion relation ${\displaystyle {\omega }^2=[\mathrm{\Omega }(k){]}^2}$ between angular frequency ω = 2π / T and wave number k = 2π / λ is given, as well as the phase and group speeds.

Deep water corresponds with water depths larger than half the wavelength, which is the common situation in the ocean. In deep water, longer period waves propagate faster and transport their energy faster. The deep-water group velocity is half the phase velocity. In shallow water, for wavelengths larger than twenty times the water depth, as found quite often near the coast, the group velocity is equal to the phase velocity.