Why does the shallow water wave speed not depend on wavelength?

In the shallow water limit, the vertical fluid motion is constrained by the bottom. This constraint makes the restoring force of gravity effectively independent of wavelength for long waves, resulting in a non-dispersive system. All long waves propagate at the same phase speed, c = √(gh), similar to sound waves in air. This is why tsunamis, which are shallow-water waves in the deep ocean, travel as a coherent wave train over vast distances.

What does the 'full solution' curve represent, and when do I need to use it?

The full solution, ω² = gk tanh(kh), is the exact dispersion relation from linear wave theory for a constant depth h. You must use it for intermediate depths, where the water depth is comparable to the wavelength (neither very large nor very small compared to it). This is the regime for most ocean waves on continental shelves or in coastal regions. The simulator shows how the full curve smoothly transitions between the deep- and shallow-water asymptotes.

How is the group velocity related to this graph?

The group velocity, which governs the speed of a wave packet's energy, is the slope of the dispersion relation: c_g = dω/dk. On the ω(k) plot, this is the derivative at a point. In deep water, c_g is half the phase velocity. In shallow water, the linear relationship means the slope is constant and equal to the phase velocity, so c_g = c_p = √(gh). The changing slope of the full solution shows how the group velocity varies in intermediate depths.

Does this model apply to ripples from a raindrop or capillary waves?

No, this model neglects surface tension, which is the dominant restoring force for very short wavelengths (typically less than a few centimeters). For such capillary waves, the dispersion relation includes an additional term proportional to k³. This simulator is specifically for gravity-driven waves, which include most ocean swells, tsunamis, and common water waves.

Water Wave Dispersion ω(k)

Water waves exhibit a fundamental property known as dispersion, where their wave speed depends on wavelength. This simulator visualizes the dispersion relation for surface gravity waves, plotting angular frequency (ω) against wavenumber (k). The core physics is governed by the linear wave theory result: ω² = gk * tanh(kh), where g is gravitational acceleration and h is the fluid depth. This single equation unifies two familiar limiting cases. In deep water (where kh is large, meaning depth >> wavelength), tanh(kh) approaches 1, simplifying the relation to ω² = gk. Here, longer waves travel faster. In shallow water (where kh is small, meaning depth << wavelength), tanh(kh) approximates to kh, leading to ω² = gk²h or ω = k√(gh). This yields a non-dispersive regime where all long waves travel at the same speed, √(gh). The simulator plots these three curves—the full tanh solution and its two asymptotic limits—allowing direct comparison. Key simplifications include the assumption of an ideal, inviscid fluid with small-amplitude waves (linear theory), neglecting surface tension (which dominates at very short wavelengths), and assuming a flat, horizontal bottom. By interacting with the graph and adjusting parameters like depth, students learn to distinguish dispersive from non-dispersive wave regimes, understand how depth controls the transition between them, and see how the functional form of ω(k) dictates wave packet spreading and phase versus group velocity.

Who it's for: Undergraduate physics or engineering students studying fluid dynamics, wave mechanics, or geophysical fluid dynamics, as well as educators in these fields.

Key terms

Dispersion relation
Angular frequency (ω)
Wavenumber (k)
Phase velocity
Group velocity
Shallow water approximation
Deep water waves
Hyperbolic tangent (tanh)

How it works

Dispersion links ω and k. Shallow water (λ ≫ h): ω ≈ k√(gh), linear in k (nondispersive phase speed √(gh)). Deep water (λ ≪ h): ω ≈ √(gk) (dispersive). The yellow curve is the full gravity–water relation ω² = g k tanh(k h) — deep and shallow are asymptotes for small/large kh.

Key equations

ω² = g k tanh(kh) → deep: tanh→1 · shallow: tanh→kh

Frequently asked questions

Why does the shallow water wave speed not depend on wavelength?: In the shallow water limit, the vertical fluid motion is constrained by the bottom. This constraint makes the restoring force of gravity effectively independent of wavelength for long waves, resulting in a non-dispersive system. All long waves propagate at the same phase speed, c = √(gh), similar to sound waves in air. This is why tsunamis, which are shallow-water waves in the deep ocean, travel as a coherent wave train over vast distances.
What does the 'full solution' curve represent, and when do I need to use it?: The full solution, ω² = gk tanh(kh), is the exact dispersion relation from linear wave theory for a constant depth h. You must use it for intermediate depths, where the water depth is comparable to the wavelength (neither very large nor very small compared to it). This is the regime for most ocean waves on continental shelves or in coastal regions. The simulator shows how the full curve smoothly transitions between the deep- and shallow-water asymptotes.
How is the group velocity related to this graph?: The group velocity, which governs the speed of a wave packet's energy, is the slope of the dispersion relation: c_g = dω/dk. On the ω(k) plot, this is the derivative at a point. In deep water, c_g is half the phase velocity. In shallow water, the linear relationship means the slope is constant and equal to the phase velocity, so c_g = c_p = √(gh). The changing slope of the full solution shows how the group velocity varies in intermediate depths.
Does this model apply to ripples from a raindrop or capillary waves?: No, this model neglects surface tension, which is the dominant restoring force for very short wavelengths (typically less than a few centimeters). For such capillary waves, the dispersion relation includes an additional term proportional to k³. This simulator is specifically for gravity-driven waves, which include most ocean swells, tsunamis, and common water waves.