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Free surface elevation time series of breaking water waves were measured in a laboratory flume. This was done in order to analyze changes in wave characteristics as the waves propagated from deep water to the shore. A pair of parallel- wire capacitive wave gages was used to simultaneously measure free surface elevations at different positions along the flume. One gage was kept fixed near the wave generator to provide a reference while the other was moved in steps of 0.1 m in the vicinity of the break point. Data from these two wave gages measured at the same time constitute station-to-station free surface elevation time series. Fast Fourier Transform (FFT) based cross-correlation techniques were employed to determine the time lag between each pair of the time series. The time lag was used to compute the phase shift between the reference wave gage and that at various points along the flume. Phase differences between two points spaced 0.1 m apart were used to calculate local mean wave phase velocity for a point that lies in the middle. Results show that moving from deep water to shallow water, the measured mean phase velocity decreases almost linearly from about 1.75 m/s to about 1.50 m/s at the break point. Just after the break point, wave phase velocity abruptly increases to a maximum value of 1.87 m/s observed at a position 30 cm downstream of the break point. Thereafter, the phase velocity decreases, reaching a minimum of about 1.30 m/s.

Water waves that develop on the open sea propagate towards the shore, undergo a series of transformations, the description of which presents both theoretical and experimental challenges [

Wave theory is essential in order to predict and analyze changes in the characteristics of a wave as it propagates from the deep water to the shore. Such theories and empirical formulae have been proposed for the calculation of wave phase velocity and the prediction of breaking as a result of wave shoaling. In an early investigation, Suhayda & Petrigrew [

In this work carefully planned wave phase velocity measurements under known and controlled laboratory conditions are to be conducted in wave flume. Measurements were taken in the vicinity of the break point in laboratory plunging wave flow. The aim is to get some indication of the accuracy of small-amplitude wave theory in predicting the transformation of monochromatic two-dimensional waves as they propagated from intermediate to shallow water depths. Free surface elevation time series measurements were to be made at several positions along the flume using a pair of capacitive wave gages, where one was fixed (reference) and the other mobile. The measurements are to be recorded simultaneously on the computer at each position. Time lags at these positions will be estimated by cross correlating each mobile gage time series with that of the reference wave gage, taken at the same time. This should allow for the computation of relative wave phase along the flume. Local wave phase velocity for points 0.1 m apart will then be computed from relative phases and compared with results from linear shallow water approximation, c = g h . These well-controlled laboratory experiments are necessary as they provide prior information required in model experiments involving turbulent flows and computational fluid dynamics models. As pointed out by Kimmoun & Branger [

A wave that propagates across a surface as a train of crests and troughs is called a progressive wave.

For such a wave propagating in the positive x-direction, η ( x , t ) at a distance x at time t is given by:

η ( x , t ) = A cos ( k x − ω t ) = H 2 cos ( 2 π ( x / L − t / T ) ) (1)

where H 2 is the wave amplitude, ω = 2 π f , f = 1 / T is the frequency of the

wave, and k = 2 π / L is the wave number. Equation (1) is the linear wave equation which is reasonable only for low amplitude waves. For increasing wave amplitude, the surface profile becomes vertically asymmetric with a more peaked wave crest and a flatter wave trough [

Phase velocity of such a wave is the speed at which the phase of a wave propagates and is considered one of the most important parameters for propagating waves. Phase velocity is often predicted using linear shallow water theory given by [

Experiments were conducted in a rectangular, glass-walled flume at the Coastal and Hydraulics Engineering Laboratory located at the Council for Scientific and

Industrial Research (CSIR) in Stellenbosch, South Africa. The flume is approximately 20 m long, 0.75 m wide and has a gentle beach slope of 1:20 which was chosen in order to get a long enough surf zone length over which measurements of wave parameters could later be conducted. A 1:20 (height: length) beach slope has also been used as the standard slope by numerous researchers in similar studies [

The time series of surface elevations were simultaneously recorded at various water depths along the beach. Instantaneous water levels were measured using a pair of parallel-wire capacitance-type wave gages, model WG-50, manufactured by RBR Ltd. The wave gages consisted of a 1 mm wire pair, which were rigidly mounted on stainless steel frames. Each wave gage was connected to an electronic circuit which consisted of two oscillators, one of fixed frequency and the other made variable by means of the changing capacitance of the probe due to changes in water level. The difference in frequency was transmitted to a frequency-to-voltage converter which provided a D.C voltage linearly proportional to the frequency difference. The wave gages were initially calibrated with no wave running in the flume. The wave gage manufacturer specifies a response time of 2 ms for a step change in water level. This more than satisfies our requirement of using 2.5 s period waves. The wave gages have an error of about 0.56% over the wave height range used. This translates to an error of approximately 0.12 cm for a maximum displacement of 21 cm of the water level, which is a typical maximum wave height at the break point in the present experiment.

The waves were allowed to run for approximately 30 minutes to ensure parameters such as the position of the break point and currents, have stabilized. Wave gage G 1 , fixed near the generator at x = − 14.0 m acted as a reference, and the other G 2 , initially placed at x = − 1.5 m , were used to simultaneously measure free surface elevations at several positions along the flume. A computer driven data acquisition system sampled the wave gages at 50 Hz through the WG-50 interfaces and signal conditioning circuits. Then gage G 2 was moved 10 cm towards the generator to a new position and the two sampled again. This process was repeated until a total distance of 5 m spanning the breaking and pre-breaking region was covered.

connected to a piston paddle to generate the specified waves. After the waves in the flume have stabilized, the data acquisition computer simultaneously samples the two wave gages at predetermined times.

Free surface elevation measurements were made for regular 0.4 Hz plunging water waves with a wave height of 12 cm as it propagated and broke on a 1:20 laboratory beach slope. The flow had a Reynolds number of about 30,000 and the break point for this flow was at about 4.0 m from the beach. A computerized measurement system was used to measure discrete time series of surface elevation measurements equally spaced in time as shown in

f is the frequency of the wave, f s is the sampling frequency, t s is the total sampling time, n s is the number of samples captured per period of the wave, N s is the sample size in 120 s and N f is the number of full waves in the time record.

f (Hz ) | f s (Hz) | t s (s) | n s | N s | N f |
---|---|---|---|---|---|

0.4 | 50 | 120 | 125 | 6000 | 48 |

wave motion is affected by the bottom slope and the wave height increases and wavelength decreases to produce a steeper wave, which departs from a sine wave form towards a trochoidal form. It is evident from the figure that as the waves propagate from deep to shallow water, the wave profile changes from being close to sinusoidal to being more peaked at the crest while the troughs become drawn out resulting in crest/trough asymmetry. Amplitudes of wave crests are much higher than amplitudes of wave troughs during wave breaking, leading to the well-known horizontal crest-to-trough asymmetry [

The Fourier transform (FT) of the function f ( x ) is the function F ( ω ) , where:

F ( ω ) = ∫ − ∞ ∞ f ( x ) e − i ω x d x (2)

and the inverse Fourier transform is

f ( t ) = ∫ − ∞ ∞ F ( ω ) e i ω x d ω (3)

The FFT is a fast algorithm for computing the FT providing an accurate method of extracting the dominant frequencies in a signal. FFTs were employed to perform the cross-correlation of free surface elevation time series in order to measure wave phase velocity. The cross-correlation was calculated between two 2-minute time series from two wave gages. The maximum correlation found between the two time series is the average time delay between the surface elevation features at the two positions. We perform the cross-correlation function c r η 1 η 2 ( t ) , a function of time t of free surface elevation η 1 and η 2 recorded by two wave gages located at two positions. By definition, the cross-correlation of two real-valued functions η 1 ( t ) and η 2 ( t ) , is defined as:

c r η 1 η 2 ( τ ) = ∫ 0 T η 1 ( t ) η 2 ( t − τ ) d t (4)

where τ is the time lag, T = 120 sec is the measurement period and time series of η 2 is shifted by τ , and matched with η 1 for τ = 0 , 1 , 2 , ⋯ . Cross-correlation gives an indication of the similarity between two signals for a given value of τ . It has a maximum when the two signals are shifted with respect to each other by some amount. The amount of shift that produces the maximum cross-correlation value indicates the amount by which one signal lags, or leads, the other. Equation (4) can be computed either directly in the spatial/temporal domain or in the frequency domain via FFT algorithms [

The Fourier-based cross-correlation was performed as follows: time series η 1 and η 2 (taken at the same time), were first converted to the frequency space using FFTs as follows:

F η 1 = F F T ( η 1 ) ; F η 2 = F F T ( η 2 ) (5)

where F η 1 and F η 2 are discrete Fourier Transforms of time series η 1 and η 2 , respectively. The time series measured at different positions were sampled at 20 ms. To get a more accurate phase difference from the cross-correlation, the Fourier transformed signals were first interpolated by a factor of 4. This was followed by zero padding and inverse FFTs [

C R η 1 η 2 = F η 1 × F η 2 * (6)

where F η 2 * is the complex conjugate of F η 2 , the Fourier Transform corresponding to η 2 time series. An inverse FFT is then performed on the product to give the cross correlation result given as:

c r η 1 η 2 = F F T − 1 ( C R η 1 η 2 ) (7)

where c r η 1 η 2 is the computed cross correlation function of the two time series. Searching the index corresponding to the correlation maximum gave a time delay of τ at that point. This time delay was used to determine the phase along the flume relative to that at the position of wave gage G 1 , measured at the same time. Once the relative phase across the flume is known, the velocity between any two points was determined using the phase difference between these points.

The period of a 0.4 Hz wave, T = 2.5 s corresponds to 2 π radians, so relative phase at position x is then calculated from,

Φ 12 ( x ) = 2 π τ T (8)

where τ is the time lag between the signals recorded by the two wage gages (

bottom which slow down velocities near the beach, and 2) 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. [

After getting the relative phase across the flume, the phase difference Δ Φ , between any two points Δ x = 0.1 m apart was the calculated from:

Δ Φ = Φ x + Δ x − Φ x (9)

The local wave speed at a particular position, x was estimated by computing a central difference using phases at x and x + 0.1 m. This resulted in a local velocity, averaged over a distance of 0.1 m which was calculated from [

c = 2 π f Δ x Δ Φ (10)

where Δ x is the separation distance between two points and Δ Φ is the phase difference between the two points. Equation (10) reduces to ( [

c = Δ x ( τ x + Δ x − τ x ) (11)

6%. It can also be observed that there is a rapid increase in wave speed just after breaking reaching a peak value of 1.86 m/s and decreasing thereafter.

The maximum phase speed in the vicinity of the break point is greater than that predicted by g h by approximately 38%. Using a general expression for a phase speed local in space and time, Stansell & MacFarlane [

Phase measurements have errors associated with them. The main source comes from 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 used. Using an average speed of 1.5 m/s in the surf zone, the average time it takes a wave to traverse a distance of 0.1 m is 67 ms. Thus the 5 ms error translates to a velocity uncertainty of 2 × 5 ms / 67 ms = 10.6 % . The 2 factor is due to there being two sources of error from the two wave gage positions.

Laboratory experiments were perofrmed in wave flume to determine the accuracy of small-amplitude wave theory in predicting the transformation of monochromatic two-dimensional waves as they propagated from intermediate to shallow water depths. Fourier-based cross correlation techniques were employed to determine the time lag between each pair of the recorded free surface elevation time series. Relative wave phase across the flume was calculated from the time lag. Local wave phase velocity was then calculated for points 0.1 m apart. Results showed that linear shallow water approximation underestimates phase velocity for water levels used. Results also showed that the dispersion equation is relatively satisfactory for predicting the wave phase velocity up to the break point. An important observation from this study is that the wave phase velocity does not depend on depth (linear theory), but increases just after the break point. This is as a result of the energy released by the breaking process. In the vicinity of the break point, the measured phase velocity was 38% higher than linear theory, reaching 1.86 m/s just after the break point, and decreasing thereafter. One of the contributions of this research is a data set of phase velocity measurements for positions prior to and after breaking, which may be valuable in the validation of computational fluid dynamics models.

Experiments reported here were conducted in the Coastal and Hydraulics Engineering Laboratory at CSIR, Stellenbosch in South Africa. The author would like to acknowledge technical support received from CSIR staff during experimental runs. Special mention goes to my mentor Dr. Kessie Govender who introduced the author to this exciting field. Last but not least, financial support received from North West University is sincerely acknowledged.

Mukaro, R. (2018) Cross-Correlation of Station-to-Station Free Surface Elevation Time Series for Breaking Water Waves. Applied Mathematics, 9, 138-152. https://doi.org/10.4236/am.2018.92010