Shallow-water approximation

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.

Water depth is the local vertical distance from the bed to the free surface. If the bed has elevation $b(x,y)$ and the surface has elevation $\eta (x,y,t)$ in the same vertical coordinate, then $h=\eta -b$. The datum used to define the two elevations cancels from their difference.

Keeping the distinction matters in the shallow water equations: the horizontal hydrostatic pressure gradient depends on surface elevation, while the transported volume depends on depth. A sloping bed with a horizontal free surface can have a varying water depth without a horizontal gravitational driving term. Conversely, the same bed can support a flow driven by a sloping surface.

A seiche is a standing oscillation of water in an enclosed or partly enclosed basin. A disturbance can displace water toward one end; restoring gravity and reflection at boundaries then support repeated oscillation. Wind and atmospheric-pressure changes can excite such motion in natural basins.^[ref]

For an ideal straight basin of length $L$, uniform shallow depth $h$ and two closed ends, no normal flow at the walls gives surface-displacement modes proportional to $\cos (m\pi x/L)$. The shallow-water dispersion relation then gives ${\omega }_m=m\pi \sqrt{gh}/L$ for $m=1,2,\dots$. The fundamental period is $T_1=2L/\sqrt{gh}$.

An illustrative model with $L=2\mathrm{m}$, $h=0.10\mathrm{m}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ gives $T_1\simeq 4.04\mathrm{s}$. A gently disturbed tank can test the period. Variable depth, open connections, friction and finite wave amplitude modify this ideal prediction.

An ideal uniform shallow basin, closed at one end and held at zero displacement at its open end, has its lowest standing wave with a displacement node at the opening and an antinode at the closed end. Its wavelength is four times its length. Real mouths need not satisfy this perfectly reflecting idealization.