For linear surface gravity waves with wavelength much larger than the depth, . It follows from when , and the wave speed is .
The rigid-lid approximation suppresses free surface displacement in a fluid layer while retaining a variable pressure that enforces the mass constraint. At constant depth it gives horizontally nondivergent motion. In a one-layer quasi-geostrophic approximation, removing free-surface stretching eliminates the term from shallow-water quasi-geostrophic potential vorticity. This is also the small-horizontal-scale limit compared with the Rossby deformation radius, when that term is negligible rather than exactly absent.
The rigid-lid approximation removes external surface displacement from volume conservation while retaining its pressure as a constraint multiplier. It does not mean zero horizontal pressure gradient. In two-layer geostrophic flow, and , so the upper-layer depth anomaly is . This pressure difference supplies internal potential-vorticity coupling.
A vertically mixed gravity current with horizontally varying reduced gravity obeys and . Consequently the material acceleration is . Advected density variables add repeated contact characteristic curves of speed to the gravity-wave characteristic curves of speeds .
The shallow-water approximation applies when a fluid layer's vertical scale is much smaller than its horizontal scale. Vertical acceleration is then negligible to leading order, pressure is hydrostatic pressure, and the horizontal velocity can be represented by a depth average.
On a beta plane with , set . Exact elimination of the depth-integrated shallow-water transports givesThe essential commutator is . Freezing undifferentiated factors to is a local approximation. The slow-wave limit gives the shallow-water Rossby-wave dispersion relation with a negative zonal phase velocity for .
For a depth-independent horizontal velocity in a layer of reference depth , the linear transports are , . They have dimensions of area per time and satisfy the surface continuity equation . Their momentum equations are obtained by multiplying the depth-independent velocity equations by .
In a homogeneous layer under the hydrostatic approximation, the horizontal pressure gradient is independent of depth. The depth derivatives of velocity obey , . If the motion starts without shear, these derivatives remain zero. Hydrostatic pressure alone does not eliminate a pre-existing freely oscillating shear.
For linear rotating shallow water equations with fixed depth and constant Coriolis parameter , elimination of the horizontal velocity givesAt a depth step this is interpreted distributionally, with continuity of surface elevation and normal volume flux. Because velocity elimination raises the time order, initial height alone is insufficient; the original velocity and continuity equations determine the remaining initial derivatives.
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The Shallow Water Equations (SWE) are a set of hyperbolic partial differential equations that describe the flow of a thin layer of fluid, such as water in rivers, lakes, and coastal areas. These equations are particularly useful in hydraulic and environmental engineering for modeling phenomena like flooding, tsunami propagation, and sediment transport. The SWE are derived under the assumption that the horizontal length scale of the fluid flow is much larger than the vertical scale of the fluid depth.