5.2 Wave transformation results
5.2.4 Wave phase celerity
The phase celerity was estimated by first determining the phase relationship of the wave at various positions along the flume relative to that at the position of wave gauge G1. Once the relative phase across the flume is known, the velocity between any two points was determined using the phase difference between these points.
LetG1(t) and G3(t) be the free surface elevations measured by the reference wave gauge G1 and wave gauge G3. The cross correlation function between these two signals is given by :
Rg1g3(τ) = Z T
0
G1(t)G3(t+τ)dt (5.9)
where the time series of G3 is shifted from G1 byτ, andτ = 0,1,2,· · ·. The cross correlation procedure was performed in the Fourier domain using FFTs.
The phase at positionxfor G3, relative to wave gauge G1 is then calculated from, Φ13(x) =Rg1g3(x)
0.02 (5.10)
where 0.02 is the sampling time. Similar expressions were used to get the phase Φ12 using signals from G2 relative to G1.
After getting the relative phase across the flume, the phase difference ∆Φ, between any two points was the calculated from :
∆Φ = Φx+∆x−Φx (5.11)
Figure 5.7 shows the time series of the wave at four positions along the flume together with that measured by gauge G1. The phase relative to gauge G1 is shown in each case. The average phase at a particular position (x) was determined by cross correlating the signal from gauge G1 with the signal from the gauge at positionx. From Figure 5.7 it can be seen that the relative phase between the signal from gauge G1 and that from the other gauge increases away from the generator. The original time series at different positions were sampled at 20 ms. These signals were then interpolated by a factor of 4 using a perfect reconstruction filter which uses a sinc function interpolation routine. This interpolation can easily be implemented using FFTs, followed by zero padding and inverse FFTs (Oppenheim &Schafer [170]). Thus our interpolated signal has a new sampling time of 5 ms and our cross-correlation is accurate to within 5 ms.
Figure 5.8 shows the phase measurement across the flume at positions before and after breaking. At some point, there will be a 2πphase shift between the signals. This has been catered for in Fig. 5.8. As pointed out byKimmoun&Branger [43], the phase shift is due to (i) friction effects on the bottom which slow down velocities near the beach, and (ii) the negative transport near the bottom which acts against the wave. The shear of the current under the crests during the breaking process has been observed by Govender et al.. [42]. Average phase measurements are provided every 0.1 m. It may not be so clear from the figure, but there is a non-linear increase in the measured phase, away from the generator, ranging from 3.0radsto 10.8rads.
The local wave celerity at a particular position,xwas estimated by computing a central difference using phases at x± 0.1 m. This resulted in a local velocity, averaged over a distance of 0.2 m which was calculated from (Thorton &Guza [107]) as :
c=2πf δx
∆Φ (5.12)
where δxis the separation distance between two phases, ∆Φ is the phase difference andf is the funda- mental frequency of the wave.
Figure 5.7: Time series showing phase at that point relative to wave gauge G1 for positions (a) -3.1 m (b) -2.7 m (c) -2.3 and (d) -1.7 m from the still water line mark.
Figure 5.8: Variation of the measured wave phase along the flume for points 0.1 m apart. These were measured relative to the time series of wave gauge G1. Note that the wave propagates from left to right.
As already stated in section 2.2.3, in addition to the shallow water approximation, an empirical alternative arises from observations showing that the phase speed in the surf zone is slightly larger than the linear approximation but still typically proportional toh1/2 (Svendsen et al.[104]). Thus, a simple approach is to model phase speeds with a modified shallow water approximation,
c=δp
gh (5.13)
where δis the constant to be determined. This approach has been used in various wave models owing to its simplicity, with a typical value of δ=1.3 (Schaffer et al. [105]; Madsen et al. [106]). This value is consistent with the surf zone observations of Stive [36], which considered regular laboratory waves.
This value is also consistent with the solitary wave solution Eq. (2.21) using a global value ofH/h=0.78.
Stansby&Feng [71] presented phase speed data from one laboratory condition (regular wave) and their results showed a monotonic cross-shore variation for the proportionality constant in the rangeδ= 1.06 - 1.32.
Figure 5.9: Variation of the measuredaverage wave velocity along the flume and the theoretical estimates calculated using linear and non-linear wave theories. The average wave velocity was computed from the phase difference of pairs of points 0.2 m apart in Fig. 5.8.
Figure 5.9 shows the measured celerity or velocity, together with that predicted by linear theory√ gh and an equation using the roller model concept, 1.3√
gh, developed bySchaffer et al.[105]. The average wave velocity was computed by averaging the phases given in Figure 5.8, over a distance of 0.2 m. As can be seen from Fig. 5.9, wave celerity estimates are bounded by the celerity estimates obtained by linear wave theory and non-linear wave theory, but show a some amount of variability after breaking. Before the break point, (-6.0 m< x < -4.0 m), measured celerities are greater than √
gh by approximately 6
%. It can also be observed that there is a rapid increase in wave celerity just after breaking reaching a peak value of 1.86 m/s and decreasing thereafter. It should be noted that the measured velocities, especially the jump at the break point, correspond to that of the top of the wave rather than the bulk of the wave.
Using a general expression for a phase celerity local in space and time,Stansel &MacFarlane[47] obtained a phase speed of 1.67 m/s at the break point. The maximum phase speed in the vicinity of the break point is greater than that predicted byp
(gh) by approximately 38 % and it is also greater than that predicted byδ√
gh by approximately 5 % where δ = 1.3. After the break point, there is a decrease in the wave phase speed reaching a minimum atx= -2.8 m. Thereafter the measured phase celerity are in good agreement with roller model results as calculated from 1.3√
gh. As pointed out byStive [102], this indicates that non-linear effects are important, as expected for this region. The possible reason for the observed dip in the wave celerity aroundx= - 3.2 m may be due to the undertow reaching its maximum value in that region. The depth averaged undertow measured using video techniques was in the order of 0.15 m/s over that region. However this is still under investigation. Results obtained in this study are similar to those measured byStive [102], who conducted experiments on spilling and plunging waves breaking on a 1:40 plane slope beach. He measured wave phase celerity and obtained deviations from the theoretical wave phase velocity by as much as 28% at the break point for a spilling wave and 19 % for a plunging wave, decreasing close to the shore. Okamoto et al. [171] conducted similar experiments but for waves breaking over a triangular bar consisting of a 1:20 up slope. Their results for positions prior to breaking and immediately after breaking are similar to that in the present investigation. Tissier et al.[50] determined the time lag between two wave height time series recorded by two closely spaced wave gauges, in the field. They used a cross-correlation technique in a study that involved field measurements of wave celerity in the surf zone and obtained an estimate of the local velocity which compares well with δ√
gh, where δ = 1.14. Measurement results by Suhayda & Pettigrew [45] showed that the measured celerity reached 1.2 times the non-linear celerityδ√
gh, at the break point.
Errors associated with the phase measurements are due to errors in estimating the position of the peak in the cross correlation. The position of the cross correlation peaks were estimated to within 5 ms, which represent one source of error. Another source of error is associated with the sampling jitter. This is determined by the speed of the computer clock, which is in the order of nanoseconds. Thus the biggest uncertainty in the position of the cross-correlation peak comes from the 5 ms interpolation. Using an average speed of 1.7 m/s in the surf zone, the average time it takes a wave to traverse a distance of 0.2 m is 118 ms. Thus the 5 ms error translates to a velocity uncertainty of√
2*5 ms/118 ms = 6.0 %. The
√2 factor is due to there being two sources of error from the two wave gauge positions.