Secondary data analysis techniques

3.4.1 Wave propagation direction

In a study of wave transformation, one fundamental property that must be established is the wave propagation direction. This has been achieved using data from the primary analysis, namely, the magnitude and phase of the horizontal velocity vector and the phase behaviour of the surface elevation. Three different techniques are described.

3.4.1.1 Velocity vector techniques

Long-crested waves produce a two-dimensional flow with particle motions contained in a vertical plane. In the horizontal plane this is observed as a straight line at an angle to the coordinate axes that represents the wave propagation direction (figure 3.9(a)). If there are other wave components present, propagating in different directions, the wave is no longer long-crested and the motion in the horizontal plane will take an ellipsoidal form (figure 3.9(b) and (c)). However, the major axis o f this ellipsoid will represent the propagation direction o f the dominant wave component. When the frequencies of the secondary wave components are the same as the dominant wave the motion is that o f a smooth ellipse (figure 3.9(c)). However, when one or more of the secondary wave components have a different frequency, such as if free higher harmonics are present, the ellipsoidal form is much less smooth (figure 3.9(b)).

Two techniques have been used to establish the major axis o f the measured horizontal velocity vector

u = (m,v) and its angle to the coordinate axes. The first relates the gradient m o f the least squares fit v=mu to the ensemble averaged horizontal velocity vector to the wave propagation direction through

Ideal long-crested w ave motion

E nsem ble average velo city ve cto r

1st harm onic v e lo city ve cto r Least S quares Linear Fit M ajor A xis o f Ellipse

Figure 3.9: Recovering the wave propagation direction from the m easured horizontal velocity vector: (a) theoretical velocity vector for long crested waves; (b) m easured velocity vector; (c) first harmonic velocity vector.

/n=tan(a) (figure 3.9(b)). This method is adequate if the vector rem ains essentially linear - that is, the length of the minor axis remains small. As the vector becom es increasingly circular, the length o f the minor axis increases, and the least squares estim ate o f the gradient increasingly underestim ates that of the m ajor axis, tending to zero as the vector becom es circular.

The second method uses the vector formed from the first harm onics o f the ensem ble average horizontal velocity com ponents (figure 3.9(c)). This vector will be an ellipse described param etrically by

u = cos (car - , v = cos (car - <^) [3.12]

The angle between the major axis and the x-coordinate axes is

oc = atan ‘'("'max)

" ("'max) [3.13]

where is the phase at which the velocity vector magnitude, |a | = («^ is m aximum. This is found by substituting equation [3.12] into the expression for the vector m agnitude, differentiating with respect to the phase car and equating to zero. A lgebraic m anipulation o f the resulting equation yields

1 rt,; sin 2(f),^ sin2(|)^

[3.14] rzJcos2(|)^ + rz;cos2(|)^

This can be the phase o f either the maxim um or m inim um magnitude and it is, therefore, essential to evaluate the velocity vector magnitude to confirm which. As the phase difference between the maximum and minimum is n:/2, it is a simple matter to establish the phase o f the m axim um m agnitude

if it is the minimum phase that is recovered from equation [3.14].

3.4.1.2 Wave probe array techniques

An alternative method is to relate the phase difference between surface elevation measurements to the wave propagation direction and phase speed. For a regular wave propagating on a homogeneous medium, the phase difference A ^between two locations with a known separation (Ax, Ay) is a function o f the wave phase speed, c and the wave propagation direction a relative to the measurement axes, namely

A 0 = — (A x c o s a + A y s in a ) [3.15]

Measurements at three locations provide two independent phase differences and thus two equations with which to solve for the two unknowns, a and c, the solutions being

Ax,A(j), - Ax.Acj),

and

c = 0) [Ax,Ay2 - A xjA yJ

11/2 [3.17]

[(Ay^Ac]), - Ayj A4>2)^ + (Ax2A({)^ - Axj A(j)2)^]

With measurements at four or more locations, the system is over-specified and a least squares optimisation is necessary. Defining the following parameters

s1 = E Ax/ ,

Ay/ ,

-^3 = E Ax.Ay. ,

/•I (=i j”i

[3.18]

n n

•^4 = E Ax,.A(J). ,

= E Ay,A((),.

/=i /=i

where n is the number of independent phase differences in the optimisation, the solution for cc and c

can be written, 1S3 1S4 ~ '^1 *^5 ta n a = --- [3.19] ^2 -S4 and c = ---^ [3.20] [ ( i , S5 - + ( j j - s,

The wave probe array allows simultaneous measurements o f the surface elevation at eight known locations (see figure 3.10) providing up to seven independent phase differences. However, in combined wave and current conditions the jet current strength, and hence the wave properties, can vary

wave crest <13 • 4 n(0 Probe 1 Probe 2

Figure 3.10: Recovering the wave propagation direction from surface elevation measurements.

significantly from one end of the array to the other. To improve the situation the array was considered in three sub-groups of four probes. One group contained probes 1, 2, 5 and 6, another contained probes 2, 3 ,5 and 6, and the third contained probes 3, 4, 7 and 8. The wave phase at each location is established from the first harmonic of the ensemble averaged surface elevation.

3.4.1.3 Dispersion relation technique

The linearized Doppler-shifted dispersion relation for a constant over depth current U can be written

{(Ù - k U c o s a Ÿ = gkidir&ikd [3.21]

where f/cosa is the current strength in the wave propagation direction and k is the local wavenumber. Writing the wavenumber k in terms o f the y-component wavenumber, k = m/sin a , this becomes

2