Which formula relates wave phase speed C to gravity g and depth d for shallow-water waves?

Which formula relates wave phase speed C to gravity g and depth d for shallow-water waves?

• A. C = sqrt(gd)
• B. C = gd
• C. C = sqrt(g/L)
• D. C = L/T

Answer: (A)

In shallow-water waves the surface motion travels as a single, non-dispersive wave where gravity acting on the water column provides the restoring force and the water’s inertia comes from the entire depth. When you linearize the shallow-water equations for small-amplitude waves, you arrive at c^2 = g d, so the phase speed is C = sqrt(g d). This ties the speed directly to gravity and the depth: deeper water makes waves travel faster, and the speed does not depend on wavelength in this regime. In contrast, deeper or dispersive regimes behave differently, but for shallow water this sqrt(gd) relationship is the key.

In shallow-water waves the surface motion travels as a single, non-dispersive wave where gravity acting on the water column provides the restoring force and the water’s inertia comes from the entire depth. When you linearize the shallow-water equations for small-amplitude waves, you arrive at c^2 = g d, so the phase speed is C = sqrt(g d). This ties the speed directly to gravity and the depth: deeper water makes waves travel faster, and the speed does not depend on wavelength in this regime. In contrast, deeper or dispersive regimes behave differently, but for shallow water this sqrt(gd) relationship is the key.