Surface Waves (Fluids › Waves in Fluids)

Study of gravity wave propagation on a liquid surface. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Surface waves are disturbances that propagate along the interface between two media, usually water and air. In a gravity wave, a compact mass of water does not travel forward with the crest; instead, a deformation pattern transports energy while fluid particles follow closed or nearly closed paths around their mean positions.

The main restoring force is gravity. If the surface is lifted, the weight of the water tends to return it to equilibrium; if it is depressed, hydrostatic pressure and fluid continuity reorganize the motion. For that reason, depth $h$ and wavelength $\lambda$ are not secondary geometric details: they determine whether the seabed affects the phase speed $c$.

🟢 Essential level

What a surface wave is

A surface wave combines vertical and horizontal displacement. A buoy rises and falls cyclically, but it does not move forward with the same speed as the crest; this shows that the wave is a transfer of energy and oscillatory momentum, not a block of water moving as a whole.

The wavelength $\lambda$ measures the distance between consecutive crests. The depth $h$ measures how much space the water column has to complete its orbital motion. If the bottom is far away compared with the wave scale, the wave behaves as if the ocean were unbounded; if the bottom is close, the base of the wave is slowed, the orbits flatten, and the speed becomes controlled by $h$.

A useful observation is that the water does not travel with the crest over long distances. Small floating objects mainly oscillate while the disturbance carries energy. Depth and wavelength determine whether the bottom affects the motion, so the same surface can support waves with noticeably different speeds and shapes.

🔵 Formal level

Basic formulas

Before applying surface-wave theory, check whether the depth is large or small compared with the wavelength. In deep water the speed depends on wavelength; in shallow water it approaches the square root of g times h, and the model changes.

Dispersion relation and physical limits

The formal reference model is Airy's linear theory for gravity waves. It assumes an incompressible, inviscid, irrotational fluid and an amplitude small compared with $\lambda$ and $h$. Under these hypotheses, the phase speed $c$ is obtained from the general dispersion relation:

General dispersion relation (Airy Theory)

Real student questions

• Why do waves arrive almost parallel to the beach? Because the part of the front in greater depth advances faster than the part in smaller depth. This speed difference rotates the wavefront until it aligns with the shoreline.
• Is a tsunami only a surface wave? Not in the everyday sense. In a tsunami, almost the whole water column moves, from surface to bottom, which is why its energy is enormous even if its open-ocean height is small.
• Why does a wave break as it approaches the coast? Because the bottom slows the base, the crest retains more speed, and the wave shape becomes unstable.

Final synthesis

Surface waves teach how to read a wave by comparing scales. Depth decides whether the bottom is invisible, dominant, or partially relevant; wavelength fixes the spatial scale of the disturbance; and gravity provides the restoring force. Mastering this node means choosing the appropriate model before calculating and then explaining what the resulting speed physically represents.