Linear Waves
(3.5.7) This equation can of course be solved directly by the iterative procedure
described earlier. However, for illustration purposes, it is useful to let the
132 Introduction t o nearshore hydrodynamics deep water wave length. Table 3.5.1 below shows the solution to this rela- tion. The columns and in the table represent the corresponding values in the equation. depth (the problem described above), calculate and use the table to get and hence This corresponds to calculating from with known. Then calculate and determine
from the table.
Linear wave propagation over bottom 133
As can be seen, the table also provides values of other functions and parameters encountered in the linear wave theory using any of the columns as input.
Thus, we have determined the wave length at an arbitrary depth from either the wave period T or from the wave length a t reference depth.
Determination of the wave height
The second problem is to deterine the waveheight H at arbitrary points from the reference information. This, however, out to be more com- plicated than determining the wave length, phase speed, etc. While L and c are (assumed t o be) strictly local quantities that depend only on the lo- cal depth, the refraction process will cause the wave height at a point to depend on the entire propagation pattern leading the wave orthogonal to that point.
Here we first present the classical refraction theory with determination of the wave heights. This approach is well suited t o provide insight into the mechanisms and it was for many years used through manual calculations.
Later, we examine the more advanced theories that also include the effect of diffraction, but which can only be solved by computational methods.
The physical mechanism responsible for the change in wave height is the change in local energy density in the wave motion. In the slowly varying waves of refraction theory, energy is not moving across wave orthogonals.
Therefore, when the wave refraction changes the distance between adja- cent wave orthogonals, the energy density changes too and hence the wave heights. To determine the wave height variation we therefore need to de- termine the refraction pattern. This will be discussed later.
However, when the refraction pattern has been determined, we can con- sider the wave motion two adjacent wave orthogonals as shown in Fig. We imagine the reference point is at where the distance be- tween the orthogonals is and wave height and want t o determine the wave height at The energy balance for the control volume between 1 and 2 then states that the energy flowing into the volume at 1, equals the energy flowing out at 2 , plus the energy added or subtracted between 1 and 2 . In mathematical terms
Notice that the energy equation (3.5.9) relates fluxes of energy, not local energy densities.
134 Introduction to nearshore hydrodynamics
The change in energy can be due to bottom friction, or energy dissipation due to wave breaking between 1 and 2, but it could also represent energy added due to wind generation. However, even if the dissipation is only due to the (seemingly small) bottom friction, the term can be substantial if the distance between 1 and 2 is large enough. In the following we will a t first neglect this energy dissipation. However, a warning needs to be issued. The concern is that energy dissipation accumulates. It is therefore not always realistic t o neglect the energy dissipation in a wave propagation problem, even for non-breaking waves.
Fig. 3.5.5 Energy flux through sections between adjacent wave orthogonals
For now we ignore the energy change between the two section. Then the wave height at section 2 can be determined by substituting the expressions (3.3.55) for the energy flux. We get
where G is (again) defined by Using that c =
we can then eliminate the phase velocities, and solving with respect to then gives
- =(-
HI tanh 1+ +
(3.5.11)3.5 Linear wave propagation over uneven bottom 135
It may be noticed that since is given by =
+
G) we can also write (3.5.10) as(3.5.12) or
(3.5.13) To determine we clearly need t o determine the propagation pattern for the wave motion in the x, y domain considered.
(3.5.12) may be modified t o a form which is mathematically more useful.
Consider the closed volume in x, y in Fig. 3.5.2 which has the surface S consisting of sections 1 and 2 and the two orthogonals. Since there is only flux of energy through the two sections 1 and 2 and (3.5.12) has already been integrated over depth we can write (3.5.12) as
n = 0 (3.5.14)
where = We can then apply Gauss’ theorem t o (3.5.14) in which gives that
(3.5.15) And since this must be true a t all points we see that we must have
.
(3.5.16)or
. 0 (3.5.17)
where is the energy flux in the direction of the wave number vector k.
This essentially is the wave version of the conservation of energy equation.
3.5.2.1 Simple shoaling
In the special case of a long straight coast, the bottom contours are straight and parallel. For waves perpendicularly incident on the this represents the simple case also reproduced in a 2DV wave flume. Then,
136 Introduction t o nearshore hydrodynamics
= everywhere. If we continue to neglect the energy dissipation between 1 and 2, (3.5.11) reduces to
As for L and c, it is convenient t o consider deep water as the reference section 1 by which can be written (neglecting index
The ratio is often termed the shoaling coefficient and the case is called simple shoaling.
Fig. 3.5.6 shows the variation of as a function of It illus- trates how the shoaling process causes a significant increase in wave height as the depth is decreasing towards the shore.
I
0.02
Fig. 3.5.6 The variation of and versus The lowest curve represents Green's law (3.5.21)
An interesting feature shown in Fig. 3.5.6 is the minimum for which turns out to occur at = = 0.16. The minimum value of
is 0.913.
3.5 Linear wave propagation over uneven bottom 137
Exercise 3.5-1
Show that the minimum for occurs where the ratio of the group velocity to the deep water phase velocity has a maximum (see Fig. 3.5.6) and verify the minimum value of = 0.913.
Exercise 3.5-2
the functions in (3.5.19). Show that this gives the approximation For
<
1/20, we can use the shallow water approximation for(3.5.20) or, using (3.2.114)
(3.5.21) H h
- = for -
<
0.015Eq. (3.5.21) is called Green's law (Green, and it is one of the oldest results in classical wave theory. It shows that in shallow water, the wave height changes proportional to The curve for (3.5.21) is also shown in Fig. 3.5.6.
Notice also that at the shallow water limit of = 0.05, =
tanhkh is only 0.30 giving = 0.015 (see Table 3.5.1) so that the shallow water limit in terms of is significantly different from the limit in terms of
The wave steepness
Fig. 3.5.6 also shows the variation of the wave steepness relative t o the deep water steepness Writing =
the relative steepness can obviously be written
(3.5.22) As could be expected from the fact that as decreases H increases (beyond the minimum) while L decreases, we find that increases quite rapidly in shoaling water.
This is relevant because one of the basic assumptions leading to the linear wave theory was that was small. Hence, we can expect that as
138 Introduction t o nearshore hydrodynamics
the waves move toward the shore decreasing) the linear theory rapidly becomes a rather poor approximation for the wave motion. This is one of the reasons for the importance of the nonlinear wave theories described in chapters 7 - 9.
It will be shown in those sections that the determining parameter is formed by a combination of wave height H , wave length L and water depth h. It is called the Ursell parameter after F. Ursell who in (1952) defined the parameter as
(3.5.23)