3.1. Surface wave profiles

Three wave frequencies are examined in this study: 12 Hz, 24 Hz and 36 Hz, with and without surfactant. The reconstructed free surfaces using FS-SS for 24 Hz waves with and without surfactant are shown in figure 4. As illustrated in the figure, the FS-SS method achieves a two-dimensional free-surface measurement. Two-dimensional spatial wave profiles are then obtained from the surface reconstruction. The results are obtained by extracting the wave elevations along the central positions (dashed lines in figure 4), which correspond to the locations furthest from the sidewalls and thus minimises sidewall effects. As plane waves are generated in the wave tank, two-dimensional wave profiles provide a suitable basis for evaluating wave dissipation, and the results are shown in figure 5. For accuracy of the method, see Xu & Perlin (Reference Xu and Perlin2021). It can be seen, in all cases, that the addition of surfactants leads to a reduction in wavelength, consistent with the dispersion relation for gravity–capillary waves: $\omega ^2=gk+({Tk^3}/{\rho })$ , where $\omega$ is the angular frequency, $g$ is the gravitational acceleration, $k$ is the wavenumber, $T$ is the surface tension, and $\rho$ is the fluid density. Spatial wave attenuation is evident across all frequencies in figure 5, with the 36 Hz waves exhibiting almost complete dissipation before reaching the end of the tank. As shown in the following section, the 12 Hz waves on clean water decay to approximately 10 % of their initial amplitude when they reach the end of the tank, while cases with surfactants and higher frequencies exhibit even stronger attenuation and barely persist to the tank’s end. Moreover, 12 Hz measurements were performed before the waves reached the far end of the tank, thereby eliminating any potential reflection effects. For example, the group velocity of 12 Hz waves in clean water is 21.5 cm s^–1, thus it takes approximately 1.6 s for the wave energy to reach the end of the tank. To eliminate the initial transient phase, measurements were performed in the interval $t=1\,\rm s$ to $t=1.5\,\rm s$ . For higher frequencies and for cases with surfactants, measurements are conducted after the initial transient phase. In these cases, reflections are not a concern, as the wave energy undergoes strong attenuation and decays substantially before reaching the end of the tank. Figure 5 indicates that the presence of surfactants leads to an enhanced spatial dissipation, with the effect becoming more pronounced at higher wave frequencies. This trend is attributed to the increasing influence of surface tension and Marangoni effects as the wavelength decreases. It should be noted that the 12 Hz waves do not exhibit a sinusoidal waveform. This deviation is attributed to the presence of a superharmonic component, resulting from weak nonlinearity in the system. A small dip can be observed in each wave cycle. The superharmonic components are also evident in the wavenumber spectrum and are aligned with the 24 Hz peaks. The wavenumber spectrum is obtained by applying a spatial Fourier transform to the wave profile. In the FS-SS measurements in the present study, the spatial resolution is 0.33 mm, which, according to the Nyquist sampling theorem, limits the maximum resolvable wavenumber to approximately 10 000 rad m⁻¹.

3.2. Enhanced dissipation by surfactants

Energy dissipation is typically characterised in the temporal domain, with wave attenuation primarily attributed to the intrinsic viscosity of water. In this study, we establish a connection between temporal dissipation and the spatial attenuation observed in the experiments. Lamb (Reference Lamb1924) describes the temporal decay of wave amplitude due to viscosity using the following:

(3.1) \begin{equation} \eta (x,t) = \eta _0 \mathrm{e}^{\mathrm{i}(kx - \omega t) - 2\nu k^2 t}, \end{equation}

where $\eta _0$ is the initial wave amplitude, and $\nu$ is the kinematic viscosity. By relating the spatial coordinate to time through the relation $x = c_g t$ , where $c_g$ is the group velocity, and substituting $c_g$ for gravity–capillary waves into (3.1), the expression yields

(3.2) \begin{equation} \eta (x,t) =\eta _0 \mathrm {e}^{\mathrm{i}(kx - \omega t) - \xi _tx}, \end{equation}

with the theoretical exponential decay rate (Crapper Reference Crapper1984)

(3.3) \begin{equation} \xi _t=\dfrac {4\nu k^3}{\omega } \dfrac {1 + \frac {T k^2}{\rho g}}{1 + \frac {3Tk^2}{\rho g}}. \end{equation}

To compute the theoretical spatial decay rate, the wavenumber $k$ is extracted from the measured wave profiles, and the surface-tension values $T$ are solved from the dispersion relation. For clean water, the surface tension is determined to be 0.074 N m⁻¹, whereas for water with Triton X-100 added, the surface tension decreases to 0.043 N m⁻¹. (Note that the slightly larger than usual value of surface tension for clean water is calculated using the FS-SS measured wavelengths, and it is then averaged over the 12 experiments.) The measured value of 0.043 N m⁻¹ is reported here as a reference effective surface tension of the surfactant solution. It should be noted that under wave propagation, the surfactant distribution along the interface becomes variable, leading to local and time-dependent differences in surface tension that drive the Marangoni flow. Direct measurement of the dynamic surface tension field is beyond the scope of the present experimental set-up; nevertheless, the reported value provides a useful reference for characterising the surfactant-laden interface. These values are assumed to remain consistent across all clean-water cases and across all surfactant cases.

Peak values extracted from the wave elevations (see figure 5) are used to estimate the measured spatial decay rate, $\xi _m$ , using the expression $m\lambda \xi _m=\ln ({\eta _n}/{\eta _{n+m}})$ , where $\eta _n$ is the peak value of the $n$ th wave crest, $\eta _{n+m}$ is the peak amplitude of the crest $m$ wavelengths downstream, and $\lambda$ is the wavelength. The comparison of theoretical and experimental spatial decay rates is shown in table 1. As shown in the table, the measured decay rates are significantly higher than the theoretical ones. This discrepancy may be attributed to sidewall contact line friction and the nonlinear nature of the generated waves. As the theoretical decay rate is derived from linear wave theory, nonlinear effects inherently lead to enhanced dissipation, causing the theoretical prediction to systematically underestimate the actual dissipation rate. Table 1 also shows that the presence of surfactants increases the decay rate. Interestingly, while theoretical values suggest that smaller-scale waves experience a more pronounced enhancement in wave decay due to surfactants, this trend is less evident in the experimental measurements. This is because the theoretical model accounts only for the change in surface tension, and smaller waves are more sensitive to it. It should be noted that merely altering the surface tension does not directly enhance energy dissipation, as surface tension itself is not a dissipative mechanism. The observed increase in the spatial decay rate, denoted by $\xi$ , reflects spatial rather than temporal attenuation. In other words, within the theoretical model, surfactants do not increase dissipation; instead, they modify the wavelength and group velocity, resulting in faster spatial decay of the wavefield. However, real-world scenarios involve additional mechanisms caused by surfactants – such as alterations to subsurface flow patterns – that can substantially influence wave dissipation, even in larger waves (e.g. at 12 Hz) where changes in surface tension are less influential. These aspects will be examined in more detail in the following section. It should be noted that, in the 12 Hz cases, energy is leaked to superharmonic components, as evidenced by the wave profile and wavenumber spectra (figure 4). This energy leakage at 12 Hz causes the primary wave crest to decay more rapidly. As the decay estimation method is based on local peak amplitudes, the additional energy carried by superharmonic components is not explicitly resolved. Superharmonics are more prominent in the 12 Hz waves than in the higher-frequency cases examined in this study; the resulting $\xi _m$ is larger, causing a larger discrepancy between the theoretical and measured values compared with the higher-frequency cases. Subharmonics and other low-frequency components from nonlinear wave–wave interactions (possibly resonant triads with spatial spread) are also present in the 36 Hz spectra. The wave profiles exhibit slight peak modulation. While these features are less pronounced than the superharmonics observed at 12 Hz, they can still affect the accuracy of $\xi _m$ . Thus, the $\xi _m$ values in table 1 should be viewed as indicators of enhanced dissipation in the surfactant-laden cases rather than precise estimates of the true dissipation rate.

Table 1. Spatial decay rate comparison.

3.3. Surface particle trajectories

Velocity fields beneath the free surface were obtained using PIV. Figure 6 shows the results for 12 Hz waves, resolved to a depth of approximately 3 mm. With interrogation windows of 32 $\times$ 32 pixels containing on average approximately five tracer particles, the PIV resolution is sufficient to reliably capture velocity field and velocity gradients. A 75 % interrogation window overlap is applied, resulting in a vector spacing of eight pixels; this increases the sampling density for numerical differentiation, while the true spatial resolution remains set at the 32 $\times$ 32 pixel window size. It is evident that in the presence of surfactants, the vertical velocity components become more pronounced, particularly on the rear face of the wave. Moreover, the velocity gradient of the vertical component is noticeably larger with surfactant addition. This is reflected in the velocity field plot, where the amplitude of the vertical velocity decays more rapidly with depth in figure 6(b) compared with the surfactant-free case in figure 6(a). These observations suggest a pronounced alteration in surface-particle motion due to the presence of surfactants. For particles located further below the surface, the differences between them are less pronounced.

Figure 6. Velocity fields beneath 12 Hz waves: (a) clean water, (b) water with Triton X-100.

To characterise the surface flow in more detail, a particle tracking approach is employed. These particles, visible as high-intensity regions in the recorded images, serve as Lagrangian markers of the surface fluid motion. A series of sequential local images of near-surface particles (as shown in figure 7) are acquired to capture their motion. Each image is first converted to greyscale, and a Gaussian smoothing filter is applied to reduce noise. A fixed intensity threshold, set at 70 % of the maximum intensity, is then used to segment the bright particle regions. Centroid positions of the detected particles are extracted from each frame using connected-component labelling. The particles are then linked across consecutive frames using a nearest-neighbour matching approach: for each particle in frame $n$ , the distances to all particles in frame $n + 1$ are computed, and the closest match is assigned, provided the displacement falls below a predefined threshold. This produces continuous two-dimensional trajectories for each particle over time. The trajectories of surface particles beneath 12 Hz waves, along with the recorded images, are shown in figure 7. It should be noted that at the high particle concentrations required for PIV, clustering and surface reflections can make it difficult to accurately distinguish individual particles. To address this, particle tracking is performed only in near-surface regions where the concentration is sufficiently low and particles are well isolated. Accordingly, the overall particle concentration in the surface tracking experiments is reduced compared with that used in the PIV measurements.

To evaluate the accuracy of the particle tracking method, the particle trajectories are compared with the wave profiles measured using the FS-SS method. At specific locations, particle displacement is compared with the corresponding wave height obtained from FS-SS measurements. The experiments are conducted at a fixed wavemaker stroke and frequency, with two runs performed for each case – one using FS-SS and the other using particle tracking. As illustrated in figure 8, results for 12 Hz waves in both clean water and water with added Triton X-100 show that the wave heights measured by particle tracking closely match those obtained using FS-SS at the same locations. The particle used in this comparison is located upstream of those shown in figures 7 and 9, and therefore exhibits a higher wave height. These findings confirm that both techniques provide comparable accuracy in representing the wave motion.

Figure 7. Near-surface particle trajectories of 12 Hz waves: (a) clean water, (b) with Triton X-100. Reference lengths are indicated in the panels.

Figure 8. Wave profiles and particle trajectories of 12 Hz waves: (a) clean water, (b) water with Triton X-100. Solid lines, wave profiles; dashed lines, near-surface particle trajectories. Note that the ordinate scales differ.

Figure 9. Surface particle trajectories over one wave period. Left column, clean water; right column, water with Triton X-100. (a) 12 Hz, (b) 24 Hz, (c) 36 Hz. The red star indicates the initial particle location. (Note that the scales differ).

Figure 10. Normalised surface particle trajectories over one wave period. Blue circles, clean water; red circles, Triton X-100 solution. (a) 12 Hz, (b) 24 Hz, (c) 36 Hz.

Figure 11. Normalised surface particle trajectories over one wave period, under two Triton X-100 concentrations. Red circles, 0.037 g l ⁻¹; blue square, 0.018 g l⁻¹. (a) 12 Hz, (b) 24 Hz, (c) 36 Hz.

The surface particle trajectories of 12 Hz, 24 Hz and 36 Hz waves are shown in figure 9. Note that the scales in the figure differ because higher-frequency waves exhibit much smaller spatial dimensions. To facilitate comparison across different scales, a normalised trajectory is presented in figure 10. The length scale is normalised by dividing the position by the corresponding wavelength, and the trajectory is shifted such that its centre aligns with the origin. As can be seen, on the clean surface, the particle trajectories follow a circular motion, slightly distorted due to wave nonlinearity and measurement noise. The dip observed in the wave profile in § 3, attributed to the presence of superharmonics, is also evident in the surface particle trajectories. This observation is consistent with classic wave theory, which predicts that particles on the wave surface move in circular orbits. However, in the case with surfactant, the surface particles exhibit significantly different motion patterns. For 12 Hz waves, the presence of the surfactant prevents the particles from following a circular trajectory. Rather, particle motion becomes a combination of vertical oscillation and horizontal drift. In the 24 Hz cases, the particle trajectories form elliptical-like shapes. The particle trajectories at two different Triton X-100 concentrations are shown in figure 11. The change in concentration from 0.018 g l⁻¹ to 0.037 g l⁻¹ does not alter the particle trajectory significantly, indicating that other surface properties, such as surface viscosity, play only a minor role in determining the surface flow pattern. Rather, the Marangoni effect, which depends not on the absolute surfactant concentration but on the surface-tension gradient, dominates the alteration of the surface flow. The dissipation rate caused by viscosity on a unit volume of fluid in a two-dimensional flow can be written as

(3.4) \begin{equation} \phi =\mu \left[2\left(\frac {\partial u}{\partial x} \right)^2+2\left(\frac {\partial w}{\partial z} \right)^2+ \left(\frac {\partial w}{\partial x}+\frac {\partial u}{\partial z} \right)^2 \right]\!, \end{equation}

where $\mu$ is the dynamic viscosity of water, and $u$ , $w$ are the $x$ and $z$ component velocities, respectively. As shown in (3.4), the dissipation is proportional to the square of the velocity gradient. Consequently, altering the flow pattern from circular to elliptical or vertical increases velocity gradients such as $\partial u/\partial x$ and $\partial w/\partial z$ , thereby enhancing the overall dissipation rate. Using the velocity field obtained from the PIV data, the dissipation rate $\phi$ is calculated throughout the resolved flow field for both the clean water and Triton X-100 cases, and the spatially averaged values $\bar \phi$ are shown in figure 6. The results demonstrate that the addition of surfactant leads to a significant increase in the dissipation rate.

The results in this study are consistent with very recent numerical predictions of Panda et al. (Reference Panda, Kahouadji, Tuckerman, Shin, Chergui, Juric and Matar2025), who showed that Marangoni stresses arising from surfactant gradients can oppose inertial flows, effectively ‘rigidifying’ the interface; the numerical simulations in Panda et al. (Reference Panda, Kahouadji, Tuckerman, Shin, Chergui, Juric and Matar2025) assumes the surfactant is insoluble. An experimental study by Strickland et al. (Reference Strickland, Shearer and Daniels2015) demonstrated the determination of insoluble surfactant concentrations at the air–water interface using fluorescence of 7-nitrobenz-2-oxa-1,3-diazol-4-yl-tagged phosphatidylcholine, under Faraday waves. For soluble surfactants such as Triton X-100 used in the present work, direct tracking of the surface concentration is more challenging. A preliminary attempt is made in this study to visualise the spatial distribution of Triton X-100 at the air–water interface under 12 Hz waves. Nile Red is employed as a fluorescent probe due to its preferential partitioning into hydrophobic environments, enabling co-localisation with Triton X-100 films at the interface. In non-polar media, it exhibits strong red fluorescence with an emission maximum near 600 nm when excited by blue light (488–520 nm). In the experiments, methanol-based Nile Red is first dispersed in the bulk water, followed by the addition of a Triton X-100 stock solution. During the initial mixing stage, Triton X-100 is distributed transiently throughout the bulk, and its distribution is clearly visualised under blue laser illumination, confirming the efficacy of Nile Red as a tracer. Within approximately 30–60 s, Triton X-100 accumulated at the air–water interface and formed a coherent surface layer. The resulting fluorescence is captured using the high-speed imager, a long-pass filter (LP590-52, MidOpt) installed in front of the camera to reject scattered excitation light and isolate the red emission light. In the recorded image (figure 12 b), regions of higher fluorescence intensity correspond to higher Triton X-100 concentrations. The results reveal a spatially inhomogeneous distribution of surfactants, with higher concentrations in the regions between the crest and trough, where the free surface experiences compression. Although further refinement of the experimental technique is needed, particularly to address the insufficient fluorescence intensity for high-speed imaging and the lack of time-resolved distributions, our findings demonstrate that surfactants distribute non-uniformly, leading to surface-tension gradients on the surface, and show that Nile Red is a promising probe for future quantitative investigations. Due to the complexity involved in preparing an insoluble surfactant monolayer, their investigation is reserved for future work.

Figure 12. (a) Conceptual diagram of surface flow with surfactants; red highlights regions of high surfactant concentration, black dashes with an arrow show distorted particle paths, and red dashed arrows indicate Marangoni flows. (b) Nile Red fluorescence image, where brighter regions correspond to higher concentration.

It should be noted that, for the 36 Hz waves, the motion is similar to that observed in the 12 Hz cases, but rather than exhibiting a forward drift, the particles appear to drift backward. The theoretical Stokes drift velocity of surface particles can be estimated by $\bar {u}_t=(ka)^2C$ , where $a$ is the wave amplitude, and $C$ is the phase velocity. The drift velocities are measured using the recorded image frames. As the wave propagates and its steepness decreases, the drift velocity is expected to decrease. In this study, the measured drift velocities $\bar {u}_m$ are quantified as the mean horizontal particle velocities in five full wave periods. For comparison, the theoretical prediction accounts for the spatial decay of $ ka$ , by using the experimentally measured decay rate. For instance, in 12 Hz clean water, $ ka$ decreases by approximately $ 11\,\%$ after one wavelength and by $ 45\,\%$ after five periods. The theoretical drift velocity $ \bar {u}_t$ is then calculated by averaging the Stokes drift over this five-period span. It is worth noting that, within five periods, fluid particles do not drift a significant distance. For example, in the case of 12 Hz waves, the particle drift over five periods was approximately 1.7 mm, which corresponds to less than one-tenth of a wavelength. Thus, the spatial amplitude decay has only a minor influence on the mean drift velocity. The results are summarised in table 2. Five measurements are performed for each case, and the mean value together with its uncertainty is reported in the last column of table 2. The uncertainty, $\sigma _u$ , is calculated using $\sigma _u = \sigma / \sqrt {N}$ , where $\sigma$ is the standard deviation and $N$ is the number of measurements ( $N=5$ in this study). As shown in table 2, for the clean-water cases, the measured drift velocities closely match the theoretical predictions. In contrast, for surfactant-laden cases, the measured velocities are consistently lower than the predicted values. This discrepancy becomes more pronounced at higher frequencies, indicating that the presence of surfactants increasingly suppresses horizontal motion. This is attributed to Marangoni flows induced by surface-tension gradients. As the wave frequency increases, the effect becomes more pronounced. At sufficiently high frequencies, the Marangoni effect becomes dominant, eventually overcoming the wave-induced Stokes drift and reversing the net surface particle motion (negative drift velocity at 36 Hz). This is discussed in detail in the following section.

Table 2. Surface particle drift comparison.

3.4. Marangoni effects on surface flow with waves

The deformation of the free surface (waves) leads to non-uniform surfactant concentrations, with the concentration reduced at both wave crests and troughs. Regions of higher surfactant concentration correspond to lower surface tension, whereas regions of lower surfactant concentration correspond to higher surface tension. The resulting surface-tension gradient drives flow from the lower-surface-tension region toward the higher-surface-tension region, which is the called Marangoni effect. This induced surface flow opposes the wave-driven horizontal particle motion, transforming roughly circular trajectories into elliptical ones. This phenomenon was theoretically predicted for water waves with insoluble surfactants (Manikantan & Squires Reference Manikantan and Squires2020), and is shown qualitatively in figure 12. This Marangoni stress is more pronounced in shorter waves, as their shorter crest-to-trough distance produces a larger surface-tension gradient.

For soluble surfactants (such as Triton X-100 in this study), surface-tension gradients induced by surface deformation can be compensated by adsorption and desorption processes. If the surfactant concentration at the interface remains in rapid equilibrium with the bulk solution, the resulting surface-tension gradient is typically insufficient to generate noticeable Marangoni flows. However, if adsorption and desorption are slow, such that the number of surfactant molecules at the interface remains nearly constant during deformation, the surface behaves similarly to that of an insoluble surfactant. In this case, Marangoni flows suppress the horizontal motion of surface fluid particles. As the wave scales decrease, the Marangoni effect becomes increasingly significant, eventually causing the surface motion to become predominantly vertical. In this limit, the free surface behaves like an incompressible sheet (Manikantan & Squires Reference Manikantan and Squires2020).

A surface-tension gradient along the air–water interface generates a tangential Marangoni stress. The interfacial stress boundary condition (assuming a one-dimensional surface) is

(3.5) \begin{equation} \mu \frac {\partial u}{\partial z}\bigg |_{z=0} = \frac {\partial\! T}{\partial x}, \end{equation}

where $\mu$ is the dynamic viscosity, $u$ is the surface–parallel velocity (for small wave steepness, assumed horizontal), and $T$ is surface tension. Scaling the gradients as $\partial u / \partial z \sim U_M / h$ gives

(3.6) \begin{equation} U_M \sim \frac {h}{\mu } \frac {\partial\! T}{\partial x}, \end{equation}

where $h$ is the viscous penetration depth within the subphase, i.e. the depth over which the Marangoni stress drives shear. For oscillatory flow this depth is

(3.7) \begin{equation} h = \sqrt {\frac {2\nu }{\omega }}, \end{equation}

where $\nu$ is the kinematic viscosity and $\omega$ is the wave frequency.

Wave-induced convergence or divergence along the interface perturbs the interfacial concentration $\varGamma$ , producing a dynamic surface-tension gradient:

(3.8) \begin{equation} \frac {\partial\! T}{\partial x}\approx E\frac {\partial (\delta \varGamma /\varGamma )}{\partial x}, \end{equation}

where $E$ is the dilational (Marangoni) elasticity, which changes with oscillating frequency, and $\delta \varGamma$ denotes the change in surfactant concentration. To relate wave-driven surface motion to surfactant redistribution, the surfactant conservation equation for a (locally) insoluble interface is used (neglecting surface diffusion):

(3.9) \begin{equation} \frac {\partial (\delta \varGamma )}{\partial t}\;+\;\varGamma _{0}\,\frac {\partial u}{\partial x}\;\approx \;0, \end{equation}

where $\varGamma _{0}$ is the initial surfactant concentration and $u$ is the wave-induced surface velocity with the order $u \sim a\omega$ . Noting that the concentration oscillates at frequency $\omega$ and $u$ varies spatially with $k$ , we obtain

(3.10) \begin{equation} \frac {\delta \varGamma }{\varGamma _0} \sim \frac {1}{\omega } \left | \frac {\partial u}{\partial x} \right | \sim \frac {1}{\omega } \left ( k a\omega \right )= ka. \end{equation}

Hence, the relative concentration variation induced by the surface wave is

(3.11) \begin{equation} \frac {\delta \varGamma }{\varGamma _0} = O(ka). \end{equation}

As this variation occurs over a distance of order $1/k$ (one wavelength), the surface gradient of the concentration field scales as

(3.12) \begin{equation} \left | \frac {\partial }{\partial x} \left ( \frac {\delta \varGamma }{\varGamma _0} \right ) \right |\sim k(ka)=k^2 a, \end{equation}

hence,

(3.13) \begin{equation} U_M \sim \frac {h}{\mu } E\, k^2 a. \end{equation}

For a two-dimensional regular wave, the characteristic scale of the velocity of surface particles is $U_W \sim a\omega$ , and the ratio $U_M/U_W$ therefore quantifies the relative strength of the Marangoni-driven flow compared with the wave-induced motion. Determining the dilational (Marangoni) elasticity $E$ requires measurements that are beyond the scope of the present experimental set-up. Rather, values of the dynamic Marangoni elasticity of Triton X-100 in the literature (Langevin Reference Langevin2014; Ma et al. Reference Ma, Gong, Xu, Jin, Zhang, Ma and Zhang2025) are used as a rough estimation; the value range is $10$ – $50\,\mathrm{mN\,m^{- 1}}$ . For water ( $\nu = 10^{-6}\,\rm m^2\,s^{- 1}$ ) driven at 12 Hz ( $\omega =75.36\,\rm rad\,s^{- 1}$ ), we obtain $h \approx 1.6 \,\rm mm$ . Substituting these values into the scaling estimate yields the ratio $U_M/U_W \sim O(1)$ . Therefore, Marangoni flow is sufficiently strong to compete with the wave-induced particle motion and, over multiple wave periods, may alter or even reverse the net drift direction. It should be noted that the calculation presented here is only a coarse estimate. Detailed measurements are required to quantitatively compare the Marangoni flow velocity with the wave-induced velocities.

As Triton X-100 is a soluble surfactant, the adsorption and desorption processes may re-balance the surface-tension gradient. Yet, the experimental results in the present study match the predicted elliptical-like particle motion for insoluble surfactant (especially in the 24 Hz case). This is because dynamic surface-tension data indicate diffusion-controlled adsorption of Triton X-100 with characteristic times of $\gt$ 0.05–200 s at the air–water interface (Fainerman et al. Reference Fainerman, Lylyk, Aksenenko, Liggieri, Makievski, Petkov, Yorke and Miller2009a ,Reference Fainerman, Lylyk, Aksenenko, Makievski, Petkov, Yorke and Miller b ; Gassin et al. Reference Gassin, Martin-Gassin, Meyer, Dufrêche and Diat2012). Our wave frequencies/periods (12–36 Hz/0.083–0.028 s) are comparable to or shorter than these times, so adsorption cannot equilibrate within each cycle and thus is slow relative to the wave motion. Moreover, the amphiphilic nature of Triton X-100 promotes its adsorption at the air–water interface. Under these conditions, the interface behaves similarly to that of an insoluble monolayer. The results in this study provide clear experimental evidence that Marangoni effects significantly alter surface flow patterns beneath small-scale waves, transforming the typical circular motion into elliptical or even near-linear trajectories. This modification increases the shear stress between the surface layer and the fluid immediately below, resulting in enhanced dissipation rates.