17.1: Shoaling Waves and Coastal Processes

Wave phase and group velocities are a function of depth when the depth is less than about one-quarter wavelength in deep water. Wave period and frequency are invariant (don’t change as the wave comes ashore); and this is used to compute the properties of shoaling waves. Wave height increases as wave group velocity slows. Wave length decreases. Waves change direction due to refraction. Finally, waves break if the water is sufficiently shallow; and broken waves pour water into the surf zone, creating long-shore and rip currents.

Shoaling Waves

The dispersion relation \((16.1.3)\) is used to calculate wave properties as the waves propagate from deep water offshore to shallow water just outside the surf zone. Because \(\omega\) is constant, \((16.1.3)\) leads to: \[\frac{L}{L_{0}} = \frac{c}{c_{0}} = \frac{\sin \alpha}{\sin \alpha_{0}} = \tanh \frac{2 \pi d}{L} \nonumber \] where \[L_{0} = \frac{gT^{2}}{2 \pi}, \quad c_{0} = \frac{gT}{2 \pi} \nonumber \]

and \(L\) is wave length, \(c\) is phase velocity, \(\alpha\) is the angle of the crest relative to contours of constant depth, and \(d\) is water depth. The subscript \(0\) indicates values in deep water.

The quantity \(d/L\) is calculated from the solution of \[\frac{d}{L} = \frac{d}{L} \tanh \frac{2 \pi d}{L} \nonumber \] using an iterative technique, or from figure \(\PageIndex{1}\), or from Wiegel (1964: A1).

Figure \(\PageIndex{1}\): Properties of waves in intermediate depths between deep and shallow water. \(d\) = depth, \(L\) = wavelength, \(C\) = phase velocity, \(C_{g}\) = group velocity, \(\alpha\) = angle of crests relative to contours of constant depth, \(H\) = wave height. Subscript \(0\) refers to properties in deep water. From values in Wiegel (1964: A1).

Because wave velocity is a function of depth in shallow water, variations in offshore water depth can focus or defocus wave energy reaching the shore. Consider the simple case of waves with deep-water crests almost parallel to a straight beach with two ridges each extending seaward from a headland (figure \(\PageIndex{1}\)). Wave group velocity is faster in the deeper water between the ridges, and the wave crests become progressively deformed as the wave propagates toward the beach. Wave energy, which propagates perpendicular to wave crests, is refracted out of the region between the headland. As a result, wave energy is focused into the headlands, and breakers there are much larger than breakers in the bay. The difference in wave height can be surprisingly large. On a calm day, breakers can be knee high shoreward of a submarine canyon at La Jolla Shores, California, just south of the Scripps Institution of Oceanography. At the same time, waves just north of the canyon can be high enough to attract surfers.

Figure \(\PageIndex{2}\): Sub-sea features, such as submarine canyons and ridges, offshore of coasts can greatly influence the height of breakers inshore of the features. After Thurman (1985: 229).

Breaking Waves

As waves move into shallow water, the group velocity becomes small, wave energy per square meter of sea surface increases, and nonlinear terms in the wave equations become important. These processes cause waves to steepen, with short steep crests and broad shallow troughs. When wave slope at the crest becomes sufficiently steep, the wave breaks (figure \(\PageIndex{3}\), right). The shape of the breaking wave depends on the slope of the bottom, and the steepness of waves offshore (figure \(\PageIndex{3}\), left).

Figure \(\PageIndex{3}\): Left: Classification of breaking waves as a function of beach steepness and wave steepness offshore. Right: Sketch of types of breaking waves.

1. Steep waves tend to lose energy slowly as the waves moves into shallower water through water spilling down the front of the wave. These are spilling breakers.
2. Less steep waves on steep beaches tend to steepen so quickly that the crest of the wave moves much faster than the trough, and the crest, racing ahead of the trough, plunges into the trough (figure \(\PageIndex{4}\)).
3. If the beach is sufficiently steep, the wave can surge up the face of the beach without breaking in the sense that white water is formed. Or if it is formed, it is at the leading edge of the water as it surges up the beach. An extreme example would be a wave incident on a vertical breakwater.

Figure \(\PageIndex{4}\): Steep, plunging breakers are the archetypical breaker. The edge of such breakers are ideal for surfing. From photo by Jeff Devine.

Wave-Driven Currents

Waves break in a narrow band of shallow water along the beach, the surf zone. After breaking, waves continues as a near-vertical wall of turbulent water called a bore which carries water to the beach. At first, the bore surges up the beach, then retreats. The water carried by the bore is left in the shallow waters inside the breaker zone.

Water dumped inside the breaker zone must return offshore. It does this by first moving parallel to the beach as an along-shore current. Then it turns and flows offshore perpendicular to the beach in a narrow, swift rip current. The rips are usually spaced hundreds of meters apart (figure \(\PageIndex{5}\)). Usually there is a band of deeper water between the breaker zone and the beach, and the longshore current runs in this channel. The strength of a rip current depends on the height and frequency of breaking waves and the strength of the onshore wind. Rips are a danger to unwary swimmers, especially poor swimmers bobbing along in the waves inside the breaker zone. They are carried along by the along-shore current until they are suddenly carried out to sea by the rip. Swimming against the rip is futile, but swimmers can escape by swimming parallel to the beach.

Figure \(\PageIndex{5}\): Sketch of rip currents generated by water carried to the beach by breaking waves.

Edge waves are produced by the variability of wave energy reaching shore. Waves tend to come in groups, especially when waves come from distant storms. For several minutes breakers may be smaller than average, then a few very large waves will break. The minute-to-minute variation in the height of breakers produces low-frequency variability in the along-shore current. This, in turn, drives a low-frequency wave attached to the beach, an edge wave. The waves have periods of a few minutes, a long-shore wave length of around a kilometer, and an amplitude that decays exponentially offshore (figure \(\PageIndex{6}\)).

Figure \(\PageIndex{6}\): Computer-assisted sketch of an edge wave. Such waves exist in the breaker zone near the beach and on the continental shelf. After Cutchin and Smith (1973).