As waves approach a shoreline, they enter the transitional depth region in which the wave motions are affected by the seabed.

These effects include reduction of the wave celerity and wavelength, and thus alteration of the direction of the wave crests refraction and wave height shoaling with wave energy dissipated by seabed friction and finally breaking. Wave celerity and wavelength are related through Equations 2, 3a to wave period which is the only parameter which remains constant for an individual wave train :. To find the wave celerity and wavelength at any depth h, these two equations must be solved simultaneously.

However, the wave travelling from C to D traverses a smaller distance, L, in the same time, as it is in the transitional depth region. Hence, the new wave front is now BD, which has rotated with respect to AC.

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In the case of non-parallel contours, individual wave rays i. The wave ray is usually taken to change direction midway between contours. This procedure may be carried out by hand using tables or figures [10] or by computer as described later in this section. Koutitas [11] gives a worked example of a numerical solution to Equations 13 and Consider first a wave front travelling parallel to the seabed contours ie no refraction is taking place.

Making the assumption that wave energy is transmitted shorewards without loss due to bed friction or turbulence, then from Equation 8 ,. The shoaling coefficient can be evaluated from the equation for the group wave celerity, Equation 9 ,. Consider next a wave front travelling obliquely to the seabed contours as shown in Figure 9. In this case, as the wave rays bend, they may converge or diverge as they travel shoreward. Again, assuming that the power transmitted between any two wave rays is constant i.

As the refracted waves enter the shallow water region, they break before reaching the shoreline. The foregoing analysis is not strictly applicable to this region, because the wave fronts steepen and are no longer described by the Airy waveform.

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However, it is common practice to apply refraction analysis up to the so-called breaker line. This is justified on the grounds that the inherent inaccuracies are small compared with the initial predictions for deep-water waves, and are within acceptable engineering tolerances. To find the breaker line, it is necessary to estimate the wave height as the wave progresses inshore and to compare this with the estimated breaking wave height at any particular depth.

As a general guideline, waves will break when.

The subject of wave breaking is of considerable interest both theoretically and practically. In general, the seabed contours are not straight and parallel, but are curved. This results in some significant refraction effects. Within a bay, refraction will generally spread the wave rays over a larger region, resulting in a reduction of the wave heights. Conversely, at headlands the wave rays will converge, resulting in larger wave heights. Over offshore shoals the waves may be focused, resulting in a small region where the wave heights are much larger.

If the focusing is so strong that the wave rays are predicted to cross, then the wave heights become so large as to induce wave breaking. So far, the discussion of shoaling and refraction has been restricted to considering waves of single period, height and direction a monochromatic wave. However, a real sea state is more realistically represented as being composed of a large number of components of differing periods, heights and directions known as the directional spectrum.

Therefore, in determining an inshore sea state due account should be taken of the offshore directional spectrum.

## Coastal engineering books

This can be achieved in a relatively straightforward way, provided the principle of linear superposition can be applied. This implies that non-linear processes such as seabed friction and higher-order wave theories are excluded. The principle of the method is to carry out a refraction and shoaling analysis for every individual component wave frequency and direction and then to sum the resultant inshore energies at the new inshore directions at each frequency and hence assemble an inshore directional spectrum. In the foregoing analysis of refraction and shoaling it was assumed that there was no loss of energy as the waves were transmitted inshore.

In reality, waves in transitional and shallow water depths will be attenuated by wave energy dissipation through seabed friction. Such energy losses can be estimated, using linear wave theory, in an analogous way to pipe and open channel flow frictional relationships. In contrast to the velocity profile in a steady current, the frictional effects under wave action produce an oscillatory wave boundary layer which is very small a few millimetres or centimetres.

In consequence, the velocity gradient is much larger than in an equivalent uniform current that in turn implies that the wave friction factor will be many times larger. Soulsby [14] provides details of several equations which may be used to calculate the wave friction factor. For rough turbulent flow in the wave boundary layer, he derived a new formula which best fitted the available data, given by. The wave height attenuation due to seabed friction is of course a function of the distance travelled by the wave as well as the depth, wavelength and wave height.

BS [15] presents a chart from which a wave height reduction factor may be obtained. Except for large waves in shallow water, seabed friction is of relatively little significance. Hence, for the design of maritime structures in depths of 10 m or more, seabed friction is often ignored. However, in determining the wave climate along the shore, seabed friction is now normally included in numerical models, although an appropriate value for the wave friction factor remains uncertain and is subject to change with wave induced bed forms.

Furthermore, wave energy losses due to other physical processes such as breaking can be more significant.

So far, consideration of wave properties has been limited to the case of waves generated and travelling on quiescent water. In general, however, ocean waves are normally travelling on currents generated by tides and other means. These currents will also, in general, vary in both space and time. Hence two distinct cases need to be considered here. The first is that of waves travelling on a current and the second when waves generated in quiescent water encounter a current or travel over a varying current field. For waves travelling on a current, two frames of reference need to be considered.

The first is a moving or relative frame of reference, travelling at the current speed. In this frame of reference, all the wave equations derived so far still apply. The second frame of reference is the stationary or absolute frame. The concept which provides the key to understanding this situation is that the wavelength is the same in both frames of reference. This is because the wavelength in the relative frame is determined by the dispersion equation and this wave is simply moved at a different speed in the absolute frame.

## Advanced Series on Ocean Engineering

In consequence, the absolute and relative wave periods are different. As the wavelength is the same in both reference frames, the absolute wave period will be less than the relative wave period. The current magnitude must, therefore, also be known in order to determine the wavelength.

From the dispersion Equation 3 it follows that. This equation thus provides an implicit solution for the wavelength in the presence of a current when the absolute wave period has been measured. Conversely, when waves travelling in quiescent water encounter a current, changes in wave height and wavelength will occur. This is because as waves travel from one region to the other requires that the absolute wave period remains constant for waves to be conserved.

Consider the case of an opposing current, the wave speed relative to the seabed is reduced and therefore the wavelength will also decrease. Thus wave height and steepness will increase. In the limit the waves will break when they reach limiting steepness. In addition, as wave energy is transmitted at the group wave speed, waves cannot penetrate a current whose magnitude equals or exceeds the group wave speed and thus wave breaking and diffraction will occur under these circumstances.

Such conditions can occur in the entrance channels to estuaries when strong ebb tides are running, creating a region of high, steep and breaking waves. Another example of wave-current interaction is that of current refraction. This occurs when a wave obliquely crosses from a region of still water to a region in which a current exits or in a changing current field.

The simplest case is illustrated in Figure 13 showing deep-water wave refraction by a current. In an analogous manner to refraction caused by depth changes, Jonsson showed that in the case of current refraction. The wave height is also affected and will decrease if the wave orthogonals diverge as shown or increase if the wave orthogonals converge. For further details of wave-current interactions, the reader is referred to Hedges [16] in the first instance.

Waves normally incident on solid vertical boundaries e.