For simplicity we deal with waves in ideal water, that is, in a fluid that is

1. incompressible
2. inviscid.

The first approximation is very good for real water (see the discussion of the speed of mechanical waves). Mathematically it is equivalent to taking the density ρ of water as constant. The second approximation is equivalent to neglecting the effect of friction forces and, for example, rules out the formation of vortices. Mathematically it is equivalent to taking the velocity field u(x,y,z,t) as conservative, that is, there exists φ(x,y,z,t) such that

${\displaystyle \mathbf{𝐮}=-\mathrm{\nabla }\varphi }$.

Consider a body of water of depth h in which waves of wavelength ${\displaystyle \lambda }$ propagate. Depending on the ratio between depth and wavelength, two cases are distinguished.

Long waves in shallow water (tidal waves)

These are, for example, the waves produced by the tide, for which the depth of the water is much smaller than the wavelength

${\displaystyle h\ll \lambda }$.

The only forces to consider in this case are the force of gravity and the pressure forces in the water. The motion of a small volume of water is governed by the equations of motion of fluids (Euler's equations), from which Bernoulli's equation, expressing the conservation of mechanical energy per unit volume, is also derived.

Since the water is shallow, we can neglect the vertical acceleration of its molecules, and the exact equation of motion in the vertical direction

${\displaystyle \frac{\partial u_z}{\partial t}+\mathbf{𝐮}\cdot \mathrm{\nabla }{u}_z=-g-\frac{1}{\rho }\frac{\partial p}{\partial z}}$

can be approximated by

${\displaystyle 0=-g-\frac{1}{\rho }\frac{\partial p}{\partial z}}$,

which gives Stevin's law

${\displaystyle p(z)=p_0+\rho g(\zeta (x,y,t)-z)}$,

where ζ(x,y,t) is the height of the liquid at the point (x,y) with respect to the plane ${\displaystyle z=0}$.

and substituting into the equations of horizontal motion

${\displaystyle \frac{\partial u_x}{\partial t}+\mathbf{𝐮}\cdot \mathrm{\nabla }{u}_x=-g\frac{\partial \zeta }{\partial x}}$

${\displaystyle \frac{\partial u_y}{\partial t}+\mathbf{𝐮}\cdot \mathrm{\nabla }{u}_y=-g\frac{\partial \zeta }{\partial y}}$

from which, noting that neither right-hand side depends on z, it follows that the molecules lying in a vertical plane at a given instant always remain aligned during the motion.

To find the speed of the waves, we consider small-amplitude waves travelling along x over a flat bottom, and we measure ζ from the bottom, so that at rest ζ = h. For small amplitudes the term ${\displaystyle \mathbf{𝐮}\cdot \mathrm{\nabla }{u}_x}$ is negligible and the horizontal equation of motion becomes

${\displaystyle \frac{\partial u_x}{\partial t}=-g\frac{\partial \zeta }{\partial x}}$.

Since water is incompressible, if more water flows into a stretch of channel than flows out, the level rises (conservation of mass):

${\displaystyle \frac{\partial \zeta }{\partial t}=-h\frac{\partial u_x}{\partial x}}$.

Differentiating the second equation with respect to time and using the first, we obtain the wave equation

${\displaystyle \frac{{\partial }^2\zeta }{\partial t^2}=gh\frac{{\partial }^2\zeta }{\partial x^2}}$,

with propagation speed

${\displaystyle c=\sqrt{gh}}$.

In shallow water the speed of the waves therefore depends only on the depth and not on the wavelength: these waves are not dispersive. This is the case of tides and tsunamis, whose wavelengths of hundreds of kilometres are much larger than the depth of the oceans.

Waves in deep water (surface waves)

These are, for example, the waves produced by the wind on the open sea. For very short waves (with a wavelength below about 2 cm, the ripples) surface tension, which the formulas on this page do not include, is the dominant effect: these are the so-called capillary waves. The depth of the water is much greater than the wavelength

${\displaystyle h\gg \lambda }$.

In this case the vertical acceleration of the particles of the liquid can no longer be neglected, and the equations of motion become more complicated. Leaving out the details, we find that the propagation speed is

${\displaystyle c=\sqrt{\frac{g\lambda }{2\pi }\tanh \left(\frac{2\pi h}{\lambda }\right)}}$,

where the hyperbolic tangent function is defined as follows

${\displaystyle \tanh (x)=\frac{\exp (2x)-1}{\exp (2x)+1}}$.

This formula holds for any depth. If the water is shallow (${\displaystyle h\ll \lambda }$) we have ${\displaystyle \tanh (x)\approx x}$ and we recover the speed ${\displaystyle c=\sqrt{gh}}$ of waves in shallow water.

Note that, if the water is deep enough, that is, for

${\displaystyle h\ge \frac{\lambda }{2}}$,

the hyperbolic tangent is close to 1, so the speed can be written as

${\displaystyle \sqrt{\frac{g\lambda }{2\pi }}}$

That is, for example, if ${\displaystyle \lambda }$ is expressed in metres

${\displaystyle 1.25\sqrt{\lambda }}$.

This shows the phenomenon of dispersion for surface waves: the propagation speed of the wave depends on (the square root of) the wavelength.

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