## Abstract

To investigate changes in the instability of Stokes waves prior to wave breaking in shallow water, pressure data were recorded vertically over the entire water depth, except in the near-surface layer (from 0 cm to −3 cm), in a recirculating channel. In addition, we checked the pressure asymmetry under several conditions. The phase-averaged dynamic-pressure values for the wave–current motion appear to increase compared with those for the wave-alone motion; however, they scatter in the experimental range. The measured vertical distributions of the dynamic pressure were plotted over one wave cycle and compared to the corresponding predictions on the basis of third-order Stokes wave theory. The dynamic-pressure pattern was not the same during the acceleration and deceleration periods. Spatially, the dynamic pressure varies according to the faces of the wave, i.e. the pressure on the front face is lower than that on the rear face. The direction of wave propagation with respect to the current directly influences the essential features of the resulting dynamic pressure. The results demonstrate that interactions between travelling waves and a current lead more quickly to asymmetry.

This article is part of the theme issue ‘Nonlinear water waves’.

## 1. Introduction

Mathematical investigations of water waves have been well advanced compared with experimental investigations since the beginning of the nineteenth century. Several mathematicians have attempted to derive satisfactory solutions, e.g. trochoidal waves [1], small-amplitude waves [2], second-order waves [3] and solitary waves and cnoidal waves [4]. In the twentieth century, the Stokes method was extended to the third order by Borgman & Chappelear [5] and to the fifth order by Skjelbreia & Hendrickson [6] and Fenton [7]. Mathematical studies still remain ahead of laboratory studies for many wave problems [8,9]. Accurate wave measurements have been proposed as an acceptable standard for outputs under normal conditions. Researchers are inclined to rely on theory rather than analysing data if a strict interpretation is difficult, and mathematicians often rely on published analysed data when refining their theories.

When the wave height becomes very large in shallow water, a wave becomes unstable and breaks due to energy dissipation and bottom friction. A simple breaking criterion on the limiting waveform was established by Stokes more than 170 years ago [3,10]. Several criteria to start wave breaking have been considered for monochromatic waves in deep water. For example, wave breaking occurs when (i) the particle velocity of the fluid at the wave crest exceeds the phase velocity, (ii) the angle of the wave crest is less than 120°, (iii) the ratio of the wave height to the wavelength becomes approximately 1/7, and (iv) the particle acceleration at the wave crest equals approximately half the gravitational acceleration. To investigate instability leading to wave breaking, various approaches have been applied theoretically [11], experimentally [12] and numerically [13]. Constantin & Strauss [14] proved that the pressure strictly decreases horizontally away from a crest line towards its neighbouring trough lines and increases with depth provided the maximum angle of inclination of the free surface is at most 45°. Recently, using an exact equation relating the time evolution of the maximum and minimum horizontal fluid velocities at the surface, Constantin [15] found that the derivative of the pressure is negative on the forward face of a wave, while it is positive on the rear face. More recently, Constantin [16] and Martin [17] proved mathematically that, for irrotational waves, the maximum and minimum of the dynamic pressure occur at the wave crest and wave trough, respectively. The result is valid without any restrictions on the wave amplitude and agrees with a prediction that the minimum of the pressure is attained on the flat bed or on the free surface [18]. There are also some other noteworthy works related to the dynamic pressure from a theoretical point of view (e.g. [19,20]).

During the past half a century, advances in several optical techniques such as laser Doppler anemometry (LDA) [21], particle image velocimetry (PIV) [22] and particle tracking velocimetry (PTV) [23] have played an important role in fluid flow research. Using two-dimensional LDA, Umeyama [24–26] observed flow structures for three different wave cases: regular, bichromatic and irregular. The experimental results were summarized in flow patterns without, following and opposing a current. Later, Umeyama [27,28] and Umeyama *et al*. [29] investigated the kinematic aspects of surface waves that propagate without or following/opposing a current using PIV and PTV. These techniques have been applied to the prediction of particle trajectories in both Lagrangian and Eulerian schemes, and such optical measurements have been applied to solve for wave breaking inside the breaker zone where no reliable theoretical equations exist.

This paper contrasts sharply with previous studies using optical methods. Our aim is to directly measure the pressure to discover new results for Stokes waves with and without a current. Recently, for solitary waves in irrotational flows, theoretical and experimental studies of the pressure beneath the wave were presented in the papers by Constantin *et al*. [30] and Umeyama [31]. The dynamic-pressure variations over one wave cycle are detected using tiny pressure transducers that move horizontally through the water depth via a carriage. The temporal and spatial images are provided just prior to wave breaking in a shallow water region.

## 2. Theory

### (a) Governing equation

Consider two-dimensional irrotational wave motion in an inviscid and incompressible fluid. The waves are assumed to propagate over a horizontal bottom without changing their form. A rectangular coordinate system (*x*, *z*) is chosen such that the *x*-axis is horizontal and the *z*-axis is directed vertically upwards from the still-water level. The free surface is located at and the bottom at In terms of a velocity potential , the velocity components of the water particles can be expressed as
2.1
and
2.2
where *u*(*x, z, t*) is the horizontal velocity, *w*(*x, z, t*) is the vertical velocity and *t* is time.

Now, consider Stokes waves propagating along the free surface of an incompressible and inviscid fluid. The Laplace equation inside the fluid domain *D* is
2.3
where *x* and *z* indicate the horizontal and vertical coordinates, respectively.

The kinematical and dynamical boundary conditions at the free surface are
2.4
and
2.5
where *η* is the vertical displacement of the water surface measured from *z* = 0 and *g* is the acceleration due to gravity. The pressure at the free surface of the fluid has been normalized to zero. The bottom boundary condition is
2.6

### (b) Perturbation method

The perturbation method is used to solve the governing equations (equations (2.3)–(2.6)). The dependent variables are defined in terms of a power series, with successively smaller terms defined by a perturbation parameter raised to a higher power in each succeeding term. For regular waves following or opposing a uniform current, the velocity potential, the vertical displacement and the angular frequency for the *n*th order can be expressed as
2.7
where is the perturbation parameter and *σ* is the angular frequency. The quantities , and are of the order of .

We replace the free-surface boundary conditions in equations (2.4) and (2.5) with the conditions to satisfy *z* = 0 instead of . The boundary conditions for the first, second and third powers can be obtained by expanding equations (2.4) and (2.5) into a Taylor series in *z* and substituting equations (2.7), (2.8) and (2.9) into equations (2.4) and (2.5) and equation (2.7) into equation (2.6). Umeyama [29] obtained the results of the third-order solution as follows.

The first-order solution is 2.10 2.11 2.12

The second-order solution is 2.13 2.14 2.15 where

The third-order solution is 2.16 2.17 2.18 where

### (c) Dynamic pressure

The pressure field under progressive waves is determined from the unsteady Bernoulli equation for an ideal fluid case [32]. The dynamic pressure, assuming zero atmospheric pressure, can be expressed as
2.19
where *p*_{D} is the dynamic pressure.

## 3. Experiments

The experiments were performed in a wave tank that was 26.0 m long, 0.7 m wide and 0.8 m deep (figure 1*a*). The wave-generating system comprised a servo-controlled device and a vertical steel plate that generated highly accurate nonlinear waves at one end of the tank. A wave absorber consisting of a vinylidene chloride mat with a thickness of 50 mm and a density of 0.075 g cm^{−3} was installed at the other end to reduce wave reflection. The water was recirculated using a variable-speed pump via an 8 cm diameter pipe, and the direction of the flow was determined by manipulating four gate valves (1–4). When the current advanced in the direction of the wave absorber, the water flowed into the channel behind the paddle, turned at an overflow under the wave absorber, and travelled in the return pipe under the channel. When the current advanced in the direction of the wave-maker, the water flowed backwards in the water recirculating system. The test section for the surface and pressure was located 13.4–14.6 m downstream of the paddle. To evaluate the dynamic effect, we obtained the temporal and spatial pressure distributions beneath the surface at four horizontal positions (ch. 1, ch. 2, ch. 3 and ch. 4, which were located 13.4 m, 13.8 m, 14.2 m and 14.6 m downstream of the wave paddle, respectively). A 3 mm diameter pressure transducer (figure 1*c*) was buried in a thin plate that was 400 mm long, 150 mm wide and 30 mm thick and was set on a carriage (figure 1*d*) that could shift vertically in 1 mm intervals. The accuracy of the pressure transducer was Simultaneously, the surface pressure was recorded using a Thermos Recorder (figure 1*e*) that measured the atmospheric pressure, temperature and humidity at intervals of 1 s. Under each test condition, surface data were continuously collected at a frequency of 50 Hz near every position. The accuracy of the gauge was 0.06 cm.

Table 1 lists the experimental conditions. Each run comprised two different tests: a Stokes wave test and a combined wave and current test. To conduct a series of tests, the paddle motion and water discharge were controlled to provide Stokes wave profiles and the mean depth-averaged current velocity. The quiescent water depth varied from 15 cm to 30 cm. In all runs, the surface variations were recorded at four selected sites separated by 40 cm using wave gauges. Run CF is for a current-alone case with a depth-averaged velocity of 8.0 cm s^{−1} at a water depth of 30 cm. Runs W1–W7 are for waves without an artificially generated current. The stroke of the wave paddle was kept constant in each case. Runs WCF1–WCF7 are the waves of runs W1–W7 superimposed over a forward current, and runs WCA4–WCA7 are the waves of runs W4–W7 superimposed over an opposite current with a depth-averaged velocity of 8.0 cm s^{−1} at the corresponding water depth (15 cm or 20 cm).

## 4. Results

Figure 2 shows the water surface and dynamic-pressure displacements in a wave-alone case (W3) at a water depth of 30 cm (indicated by open circles). The solid curves represent the corresponding surface and pressure profiles that were calculated using the third-order Stokes equations. The surface profile is fully nonlinear. The absolute value of the water surface near the crest is larger than that near the trough under the relatively larger wave-height condition. In the mathematical description of the measured pressure *p*(*x*, *z*, *t*) under waves and a current, it is convenient to assume that the instantaneous pressure consists of the atmospheric pressure *p*_{0}, hydrostatic pressure and dynamic pressure *p*_{D}(*x*, *z*, *t*):

Therefore, the dynamic pressure was obtained by subtracting the *in situ* atmospheric pressure and the hydrostatic pressure at any given depth from the experimental data at the sampling point. Prominent features of the dynamic-pressure fluctuations continue from one nonlinear pattern to the next, even though this paper limits the record to a single wave cycle. The general characteristics of the dynamic pressure are easy to understand if the pressure pattern is compared to the surface displacement, i.e. the pressure variation closely resembles the surface variation. There is little pressure distortion in the total layers from those immediately adjacent to the surface to the bed. Getting closer to the surface, the pressure develops in proportion to the increase in its elevation. The profiles mostly retain sinusoidal patterns near the bottom, and finally show nonlinear patterns near the surface. Figure 3 shows a measured pressure plot for a current-alone case (CF) at a water depth of 30 cm. The result demonstrates that the maximum pressure exists near the water surface and that the pressure slowly decreases towards the bottom of the channel, even though most wave theories do not support the existence of a vertical pressure profile when a current flows without waves. These values should not be ignored; however, the effect on waves cannot be adequately proven based on only the presented experimental results.

The phase-averaged vertical distributions of the dynamic pressure in the wave-alone cases, W1, W2 and W3, are shown in figures 4*a*, 5*a* and 6*a*, respectively, while in the wave–current cases, WCF1, WCF2 and WCF3, they are shown in figures 4*b*, 5*b* and 6*b*, respectively. For each of these experimental results, ten profiles are plotted at different phases. Each profile was calculated by averaging the dynamic-pressure data over 50 wave cycles. The consecutive numbers in the legend refer to various phase values within one wave cycle (1.0 s). The solid curves represent the dynamic-pressure profiles calculated using the third-order Stokes equation directly for the wave-alone cases or shifted to track the centre of the average pressure value at the bottom for the wave–current cases. These phase-averaged vertical distributions indicate that the dynamic pressure appears to be symmetrical about the crest or trough line throughout the range of the measured data in the wave-alone cases. Over one wave cycle, the vertical profile of the pressure varies nearly exponentially, increasing its magnitude from the bed to the surface. The change is particularly notable from the mid-depth to the surface in the larger wave case. The pressures increase quickly above the bottom boundary layer when the wave crest arrives. As the wave trough passes, however, the pressure slowly increases from the bottom towards the surface. These measured profiles agree well with the third-order Stokes profiles. The theoretical pressure slightly exceeds the measured one close to the bed when the wave crest arrives. Umeyama [25,26] measured the phase-averaged horizontal- and vertical-velocity distributions in several wave-alone cases using three-beam LDA and indicated that the motion of the water particles is nearly symmetrical throughout the measured range. The particle velocity components are in agreement with linear wave theory over one wave cycle. The phase-averaged dynamic-pressure distributions for waves following a current are depicted in figures 4*b*, 5*b* and 6*b*. As expected, the third-order Stokes equation fails to predict the observed pressure profile in the study. It appears that the dynamic-pressure increase within a specified value with a steady flow is linearly added to the periodic wave. Because there is no theoretically justifiable equation, a comparison was made between the measured profiles and the data predicted by a linear superposition of the steady current and the third-order Stokes waves. Clearly, the comparison of these velocities is most convincing in the total depth, aside from a small amount of scatter in the data. Looking more closely at these pressure distributions in the near-surface layer, one can see that the variation in each dynamic pressure is complex due to the wave–current interaction effect. However, this type of behaviour was already observed by Umeyama [25,26] in his experimental study. He obtained the phase-averaged horizontal-velocity distributions from the depth-averaged velocity for waves following a current. The velocities increase quickly near the bed and slowly above the bottom boundary layer when the wave crest arrives. As the wave trough passes, however, flow reduction occurs at a certain elevation and the particle's speed continues to decrease towards the surface. In general, this change started at *t*/*T* = 2/10 or 3/10 and ended at *t*/*T* = 7/10. This change was particularly noticeable from the mid-depth to the surface in the case of steeper waves. The velocity pattern at a phase during acceleration differs from that of the corresponding phase during deceleration. This deviation increases from the linear addition of the steady and periodic components of the velocity with increasing wave height, decreasing wavelength or increasing water depth.

Figure 7 shows the locations of the pressure and surface measurements in our wave tank. The pressure data were sampled beneath the surface at 13 elevations at four different locations. The pressure transducers were located at intervals of 40 cm. A wave gauge was set at the midpoint between ch. 2 and ch. 3. Two different tests were conducted to collect the dynamic pressures in the middle of the channel. Figure 7 indicates the experimental pressure distributions when the wave crest (*a*) and trough (*b*) arrive at the wave gauge. All pressure transducers are vertically shifted by 1.0 cm intervals. The dynamic pressure observed by the transducer was not fully instantaneous because the measurement was performed 13 times to measure the vertical profiles. Simultaneously, the surface displacement was measured by the gauge for approximately 3 min. Figure 8 shows the observed and calculated dynamic-pressure profiles at four sections in the wave-alone cases. The upper (lower) panels indicate the temporal pressure when the wave crest (trough) arrived at the wave gauge. It was not possible to measure the pressures in the subsurface layer beneath the wave trough; therefore, no data are available for *z* > −3 cm. At each measuring point, the total pressure was observed by a pressure transducer set in a thin plate on a carriage. The output data are the average over 50 wave cycles. Except for either profile at the same surface elevation, the pressure data obtained by the transducer appear to agree reasonably well with the theoretical curve. In each case, there was a maximum difference of approximately 0.01–0.025 kPa between the theoretical and experimental pressures. It was also found that the difference between the measured results whose surface displacements attained identical levels become larger as the wave height increased. In both cases, the dynamic pressure on the front slope was lower than that on the rear slope. Figure 9 shows similar distributions of the dynamic pressure in the case of waves following a current. Because there is no theoretically justifiable equation, a comparison was made between the measured data and the third-order Stokes profiles. The comparison of these pressures is most convincing in the test region, aside from the scatter in the data. The dynamic pressure increased overall when a steady flow was added to the wave field. The maximum difference in the pressures at two locations that were symmetric to the wave gauge was approximately 0.06 kPa; however, the dynamic pressure increased slowly above the bottom boundary layer in these cases, as in the wave-alone cases. In the wave–current cases, the dynamic pressure near the crest on the front slope was notably lower than that on the rear slope.

Figures 10–12 present some comparisons between two contour plots of the dynamic pressure (unit: Pa) in the wave-alone and wave–current cases. The incident waves propagate from right to left. In each figure, the upper panel shows the pressure field for the waves alone, while the lower one shows that for the waves following a current. The wave heights were *H* = 2.02 cm and *H* = 1.86 cm for W1 and WCF1, respectively, increasing gradually for each run: 2.90 cm and 2.75 cm for W2 and WCF2, respectively, and 3.73 cm and 3.42 cm for W3 and WCF3, respectively. Substitution of the velocity potential into the Bernoulli Law yields an equation (equation (2.19)) that gives the dynamic pressure in the wave-alone case. Between the still water and the wave crest, the dynamic pressure has a positive value, and between the still water and the wave trough, it has a negative value. The dynamic pressure is generally largest at the crest and smallest at the trough. The images are nearly symmetrical in the wave-alone cases (W1, W2 and W3). Conversely, the dynamic-pressure field obviously changes when waves propagate with a current (WCF1, WCF2 and WCF3). The steepness of the dynamic pressure near the crest is gentler than that near the trough when compared with that in the corresponding wave-alone case. The contour image is asymmetric, and the asymmetry becomes particularly noticeable when the wave height is relatively large. The area of the dynamic pressure under the positive surface is narrower than that under the negative surface. This means that the dynamic pressure in a wave–current condition is strongly affected by the steepness of the wave.

A comparison between the experimental pressure distributions for cases of waves without, following and opposing a current at a depth of *d* = 15 cm is shown in figure 13 (unit: Pa). The upper panel depicts the dynamic-pressure field for waves (*a*), the middle one for waves following a current (*b*) and the lower one for waves opposing a current (*c*). It is obvious that the wavelength is the longest and shortest when the following and opposing flows are added to the waves, respectively. For W7, the maximum pressure appears in the subsurface layer below the crest and the minimum pressure appears in the near-bottom layer below the trough. It appears that the dynamic-pressure distribution at a distance from the crest on the front slope is relatively lower than that on the rear slope. An identical contour map was obtained for WCF7, even though the area is downsized from that in the wave-alone case. Conversely, for WCA7, waves following a current, there are two minimum pressure points near the nodal points when the surface is lower than the mean water level. However, in this case, symmetry was maintained between the two nodes while the trough passed. In all cases, the dynamic pressure increased quickly while the wave crest passed and slowly while the wave trough passed. Figures 14–16 show similar images for a deeper water depth of *d* = 20 cm; however, the wave height is relatively large (*H* = 5.57–7.27 cm). It is obvious that, for all cases, the dynamic pressure has a symmetrical profile away from the crest lines towards its neighbouring trough lines. Even though the image does not show contour curves under the trough, a similar pressure spreads over a large area.

## 5. Conclusion

Spatial and temporal variations in the dynamic pressure for Stokes waves propagating with or without a steady current before a breaking were experimentally investigated in a wave tank. In this test, the physical characteristics in the combined wave–current motions were examined using simple measuring techniques instead of optical techniques. The *in situ* pressure, atmospheric pressure and surface displacement were measured using pressure transducers, a Thermos Recorder and a wave gauge, respectively. The dynamic pressure was obtained by subtracting the atmospheric pressure and the hydrostatic pressure at a measured depth from the pressure data. The horizontal dynamic-pressure distributions were determined at various phases over one wave period for several wave heights. Wave theory based on the finite-amplitude approximation was adapted to express the surface and pressure profiles for pure waves. There is no theory to predict the dynamic pressure when a steady flow is added to a Stokes wave; however, it appears that the track centre of the experimental profile was just transferred from zero to a certain value. In view of the spatial distribution, the dynamic pressure near the crest on the front slope of the waves was lower than that on the rear slope. Even though the dynamic pressure on the front slope was also lower than that on the rear slope for waves following a current, a large difference was seen between the two pressure values at the same surface elevation. The instantaneous dynamic-pressure fields illustrated a symmetric profile for the wave-alone condition; however, these periodic cycles suddenly lost their shape when a current was added to the wave field. This investigation demonstrates the accuracy of this non-intrusive measuring technique prior to wave breaking. From the presentation of the combined wave–current results, however, it is apparent that a linear superposition of the dynamic pressure due to waves and the current pressure is not likely to predict the horizontal pressure distributions.

## Data accessibility

This article has no additional data.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Footnotes

One contribution of 19 to a theme issue ‘Nonlinear water waves’.

- Accepted September 16, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.