Apart from the surface elevation h and the velocity potential, there are several other quantities of interest. In the present section we shall look more closely into the velocity of the water due to the waves, the track of the fluid particles and the pressure variations due to the wave motion on the surface.

We recall that the water velocity for the one-dimensional waves we are considering has two components, , and

.

The velocity potential for the regular wave was derived in Chapter 3:

.

Let us for simplicity first consider deep water. For large values of kh we may conveniently write the cosh-factor as follows:

When z is near the surface and , this expression tends to . Note that z gets increasingly negative as we move down into the water, which means that the factor gets smaller and smaller.

For deep water we may thus write

,

from which it follows that

,

and

.

Thus, for a given depth z, both u and w represent running waves with the same amplitude. The waves differ in phase by p/2, however. The amplitude decreases from wa at the surface to times the surface amplitude at the depth z. This decrease is rather fast: for ,

At a depth equal to half the wavelength, the velocity amplitude is only about 4% of its surface value!

For an arbitrary depth the relations are easily seen to be

where we have used the dispersion relation for a slight simplification.

By taking the derivative of the velocities with respect to time, we obtain the fluid accelerations :

Figure 6.1 shows the velocity and acceleration vectors compared to surface elevation. Note that the velocity is directed in the propagation direction of the wave at the wave crest.

Fig. 6.1: Surface elevation along with velocity and acceleration vectors (From Shore Protection Manual, Vol. 1 p. 2-14)

By expanding u and w in a Taylor expansion, e.g.

,

we see by using the expressions for u and w that the first term dominates (assuming that and are of the order of the wave amplitude). Thus to a first approximation we may set

Here we have introduced

If the two equations are integrated with respect to t,

Thus,

We recall that this is the equation of an ellipsis, and the fluid particles thus move in elliptical orbits. In particular, for deep water we have

and the fluid particles move in circles of radius .

This is an approximate result. If we look more closely into the equations for u and w, we see that the u velocity on the top of the orbit is slightly larger than the velocity at the bottom of the orbit. The net result is therefore a slight displacement along the wave direction.

This net motion is called Stokes drift.

Fig. 6.2: Water particle displacement from the mean location for shallow water and deep water waves. ( From the Shore Protection Manual Vol. 1 p. -2-17)

In general the pressure in the water is equal to the atmospheric pressure + the hydrostatic pressure (due to the weight of the water above) and a dynamic part due to the wave motion.

If we return to our form of the Bernoulli Equation, we can recall that

If the wave amplitude is small, we may also here, like we did when we derived the linearized equations, neglect the term and we therefore obtain the following simple expression:

The time varying part is usually called the dynamic pressure and is for the regular small amplitude wave equal to

In this section we have considered a regular wave which we have expressed as

.

The wavenumber k is equal to where l is the wavelength and the angular frequency, w is equal to where T is the period. The dispersion relation connects the wavelength and the period

If the water depth h is larger than about half the wavelength, the water is deep (as the waves are considered) and we may use the simplified relation . If we period and the wavelength, we obtain

that is,

,

where T is measured in seconds. Thus, a 10s wave in deep water has a wavelength of 156m.

On the contrary if , the water is shallow as far as the waves are concerned. Then,

and we obtain

where h is measured in meters and T in seconds. In shallow water the wavelength is thus proportional to the wave period.

In deep water the fluid particles move in circles with constant speed. At the surface, the radius of the circle is equal to the amplitude of the wave. Moreover, the water particle makes one complete turn per wave period. Hence the particle speed at the surface is . The radius of the circle diminishes as as we move downward. When , the radius is only 4% of its surface value, and when only 0.18%! What are the corresponding velocities?

In very shallow water, the fluid moves almost horizontally with an amplitude

.

This table in pdf-format summarizes the Linear Wave Theory. It is similar to a table in the Shore protection manual. Compare both tables and convince yourself that the results are similar. Note that the Shore protection manual uses and that both tables use d instead of h for water depth.