Wave groups

While standing waves were the outcome of adding waves in opposite directions with the same wave period, the addition of two waves propagat- ing in the same direction but with (slightly) different wave frequency will generate another canonical type of wave motion. If the wave compo- nents have the same amplitude, the total wave motion will form what is called a wave group, which is a progressive wave with a (slowly) varying wave amplitude. We first consider this simple case.

Thus, we consider the two wave components propagating toward and given by:

= -

,

- (3.4.17)

To define the problem, we consider

>

which also implies that

<

and

>

since L decreases with T for constant water depth.

3.4 Superposition of linear waves 109

Using a simple trigonometric relation, the sum of and can be written

= -

)

^-

2 2 2

The clearest illustration of this expression is obtained if we consider a case where is only slightly smaller than T2. Then

+

^{- Thus,}

the last factor in (3.4.18) will vary much faster with and t than the first factor. In fact, we will get a wave motion which can be interpreted as a wave motion with a space and time varying amplitude written as

t ) t ) - (3.4.19)

where the frequency w , and the wavenumber are given by

= 2

The wave amplitude is given by

(3.4.20)

=

+

2

t ) = - (3.4.21)

where and are defined by

(3.4.22)

-

2

Fig. 3.4.4 shows this wave motion which is called a wave group because where and when ^{- =} 0 the amplitude t ) ⁼ 0. We see from (3.4.19) that the instantaneous surface (wave) profile propagates with the

I

Fig. 3.4.4 The surface elevation in a wave group. The instantaneous surface profile given by (3.4.19) is shown as a thick line. This motion is oscillating within the amplitude envelop given by (3.4.21) and shown as a thinner line above and a dashed line below in the figure

110

phase velocity

Introduction t o nearshore hydrodynamics

c, = -

and has wave period and wave length given by

(3.4.23)

(3.4.24)

(3.4.25)

However, the amplitude variation, or amplitude modulation, of this wave is also propagating, but with the phase speed

(3.4.26)

While it is evident from (3.4.22) that = and =

>>

L,, it is not a priori whether or is the larger of the two. If we form the difference - we see that this can be written

The assumption

<

will imply

<

because the waves are frequency dispersive and

>

Hence, we see that in general

<

the amplitude envelope in Fig. 5 propagates slower than the instantaneous surface profile.

The exception is for shallow water waves, L ,

>>

h, where we have and are independent of the wave period so that ⁼ c,.

The result of this analysis is that (except in shallow water waves) the individual waves will constantly propagate forward within the amplitude envelope while this is itself propagating. It will look as if the waves emerge at the rear end of the group, propagate forward and disappear at the front of each group.

It is important to notice that the apparent group length between two successive nodes is only This is illustrated in Fig. 3.4.4 by the dis- tinction between the full and the dashed envelope curves.

Superposition of linear 111

Infinitely long wave groups and group velocity

If we assume that so that we see from (3.4.22) that (3.4.28) and we have infinitely long wave groups. From the definition (3.4.27) of we see that this implies

(3.4.29) Since here w and are connected by the dispersion relation =

gktanhkh, we in the limit obtain an expression for by differenti- ation of this relation. We get, keeping h =

2 k h (3.4.30)

Using the relation c = this can after some algebra be reduced to dw 1

+

G) ^(3.4.31)

where G is again defined as G

This expression is usually quoted as the “group velocity” for the waves.

It is worth to notice, however, that this is strictly speaking a misnomer, because the expression (3.4.31) is only valid for infinitely long wave groups where the wave height variation is so slow that there is no wave height variation. In other words the waves look like a train of uniform waves. The correct definition of the velocity of a wave group is given by (3.4.26) and this expression clearly depends on the length of the wave group.

Exercise 3.4-8

Derive the expression (3.4.31). Analyze the deviation

- from this expression for characteristic values of and

Exercise 3.4-9

Show that can also be written

dc (3.4.32)

112 Introduction to nearshore hydrodynamics

and use this to deduce that in shallow water waves c =

The variation of G versus (where = is the deep water wave length of a (linear) wave with period is shown in the Fig. 3.4.5 along with the variation of and = n. We see that n varies from 1 in shallow water to 1/2 in deep water.

1.0

Fig. The variation of the phase velocity c over the deep water value the ratio of group velocity over and G with where is the deep water wave length.

It was shown in Section 3.3 that the energy flux is given by (3.3.55)

1

+

^{G) (3.4.33)}

As mentioned earlier with the expression above for the energy flux can be written as

(3.4.34) where E = is the wave energy density. Thus can actually be

3.4 Superposition linear waves 113 different amplitude and frequency can create a highly complicated signal, with amplitudes of the total signal that changes both in time and space.

This on the other hand leads to the natural question: would it be possible t o represent a real, irregular wave motion in the ocean by simple superposition of a large number of sine wave components?

The answer to this question is: yes. However, it means we need to determine the amplitudes and phases of each of the (very) many wave components, and that turns out t o be a nontrivial task.

This leads to the introduction of wave spectra as a theoretical concept.

This will eventually make it possible t o determine the amplitudes of the wave components that describe the surface variation in a given (mea- sured) time series for a natural wave motion. However, it turns out not to be the simple spetrum of the amplitudes that works but the energy spectrum, which is the spectrum of the energy of each wave component.

It also leads t o development for practical applications of parameterized (standard) wave (energy) spectra that are based on results from numerous practical wave records.

Finally it leads to use of statistical methods for analysing wave records t o determine the gross parameters used to get an overall picture of a wave situation and as parameters in the standard spectra.

There is a rich literature and a large number of spectral and statistical wave models that draw from these, mostly empirical, results. However, it is beyond the scope of the present text to go into the details of this complex area. Therefore we will only look at a simple formulation of the problems involved and point t o some literature for further information.

Fourier representation of a complex wave motion

A registration a t a point of the surface elevation of a wave motion can be thought of as consisting of many wave components, each of which can be written

A, - 6,) ⁼ (3.4.35)

In document Introduction to Nearshore Hydrodynamics (Page 131-136)