2.1 Insect collection

This study was conducted on the water strider Gerris paludum. We collected adult and larval stages regularly from April to August 2014, from a freshwater pond in Tours, France ( $47^{\circ }22^{\prime }01.3^{\prime \prime }\text{N}{-}0^{\circ }41^{\prime }29.5^{\prime \prime }\text{E}$ ). The insects were kept in aquariums ( $37\times 17~\text{cm}$ ) at a constant temperature of $23\pm 2\,^{\circ }\text{C}$ , under a natural light cycle, until their use in experiments. We provided frozen fruit flies for feeding every other day. We validated and calibrated the profilometry technique on 70 individuals selected for static measurements. We established five groups for immature stages (first to fifth instar, the word instar designating an insect in any one of its periods of postembryonic growth between moults) and one group of adults, based on body length. Large adults generated large waves, with large surface gradients, during their leg strokes. For dynamic measurements, the analysis was restricted to two groups of instars (8 individuals in total from the two groups: six second instars and two third instars), because the synthetic schlieren method (Moisy, Rabaud & Salsac Reference Moisy, Rabaud and Salsac2009) (detailed below) is accurate only for small surface-profile slopes. The waves generated by first instar insects were undetectable. We believe, but cannot definitively demonstrate, that the absence of visible wave generation during first-instar propulsion is the consequence of the limited resolution of our measurement technique (see below). Note that early instar striders move their legs at a speed below the critical velocity of $23~\text{cm}~\text{s}^{-1}$ (see Raphaël & de Gennes Reference Raphaël and de Gennes1996), falling into the framework of Dennys paradox on the impossibility of wave generation. Note, however, that this situation applies only to straight and steady motion. Here, we show experimentally that strider leg motion is neither regular nor rectilinear, and that there should, therefore, always be waves, even if their amplitude is too small for detection. In addition, Hu & Bush (Reference Hu and Bush2010) argued that the smallest water walkers, generating negligible deformations of the surface, make use principally of microscale contact forces generated by the brushing of the legs against the air–water interface. The ratio between leg curvature and the brushing forces is dependent on the morphological and functional characteristics of the tarsi and their integuments (Sun et al. Reference Sun, Zhao, Jiang and Zheng2018).

2.2 Synthetic schlieren method

Several profilometry techniques have been developed for measuring the local topography of the air–water interface. We chose to use the synthetic schlieren method initially developed by Moisy et al. (Reference Moisy, Rabaud and Salsac2009). This innovative method is based on the capture, from above, of high-speed video camera images of a random dot pattern at the bottom of the basin transmitted through the liquid interface. Changes in the topography of the liquid–air interface result in changes in the refracted dot pattern images. The insect is also captured by the camera, albeit in blurred images, because the camera has a limited depth of field and is focused on the dot pattern at the bottom of the basin. Below, $\unicode[STIX]{x1D701}(x,y)$ denotes the free surface displacement with respect to the equilibrium position, and $\unicode[STIX]{x1D6FF}\boldsymbol{r}(x,y)$ denotes the resulting change in refracted dot pattern. Moisy et al. (Reference Moisy, Rabaud and Salsac2009) demonstrated that, for small surface gradients, namely $\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}\ll 1$ , or equivalently $\unicode[STIX]{x1D701}\ll \unicode[STIX]{x1D706}$ , where $\unicode[STIX]{x1D706}$ denotes the smallest wavelength, the distortion field $\unicode[STIX]{x1D6FF}\boldsymbol{r}$ is simply proportional to the free surface gradient. As in particle image velocimetry (PIV) techniques, digital image correlation algorithms can then be used to determine the apparent displacement field between the refracted image and the reference image obtained when the surface is flat. Finally, numerical integration of this displacement field, based on a least-squares inversion of the gradient operator, can be used to reconstruct the instantaneous surface height. This technique and the mathematical simplification used during reconstruction of the surface height are valid only for the first-order paraxial angle approximation (see Moisy et al. Reference Moisy, Rabaud and Salsac2009), implying that the camera must be placed at a large distance from the target.

2.3 Static measurements

As described by Moisy et al. (Reference Moisy, Rabaud and Salsac2009), we first validated the full measurement procedure, using a solid Plexiglas model of a wavy surface of wavelength 20 mm and amplitude 0.5 mm, which we borrowed from F. Moisy. We carried out the static experiments in a $185\times 125~\text{mm}$ plastic tank, filled with tap water ( $9~\text{mm}$ deep for adults and $3~\text{mm}$ deep for instars). Images of the water surface were taken with a $1632\times 1200$ pixels high-speed camera (Vision Research Phantom V9.0) located above the tank. The distance $H$ between the water surface and the high-speed camera, and the resulting width $L$ of the field of view were adapted according to insect body length. We used $H_{ad}=71~\text{cm}$ and $L_{ad}=27.5~\text{mm}$ for adults, $H_{la.5,4}=45~\text{cm}$ and $L_{la.5,4}=15~\text{mm}$ for fifth and fourth instars, $H_{la.3}=39~\text{cm}$ and $L_{la.3}=12~\text{mm}$ for third instars and $H_{la1,2}=36~\text{cm}$ and $L_{la1,2}=7~\text{mm}$ for first and second instars. For each experiment, an insect was placed in the tank and recorded when it was within the field of view of the camera and resting at the surface with no water movement. We estimated the displacement field $\unicode[STIX]{x1D6FF}\boldsymbol{r}$ of the dot pattern, using a digital image correlation algorithm, with interrogation windows of $16\times 16$ pixels for the calculation of correlation functions. With such small interrogation windows, and the use of classical subpixel interpolation, it was possible to achieve a resolution of 0.1 pixels for the estimation of dot pattern displacement (Feldmann & Mayinger Reference Feldmann and Mayinger2001). We thus obtained a precision for the estimation of $\unicode[STIX]{x1D701}(x,y)$ of $\pm 6.2~\unicode[STIX]{x03BC}\text{m}$ for adults and $\pm 1.2~\unicode[STIX]{x03BC}\text{m}$ for young instars.

Figure 1.

(a) Static water surface topography deformation around three different instars of non-moving Gerris paludum. The colour scale indicates the elevation of the water surface. The scale is indicated by the black line (5 mm). (b) Mass of displaced fluid in the menisci as a function of insect mass, from first instar to adult. Linear regression results: $t=1.0401$ , $df=68$ , $p\_\text{value}=0.30$ .

According to the vertical force balance of a water strider in a static situation (see Keller Reference Keller1998 and Hu & Bush Reference Hu and Bush2010), the mass of the volume displaced by the insect at the free surface is equal to the mass of the insect. Indeed, given the free surface displacement $\unicode[STIX]{x1D701}(x,y)$ resulting from an external pressure field $p_{ext}(x,y)$ , with $\hat{\unicode[STIX]{x1D701}}(k)$ and $\hat{p}(k)$ as the corresponding Fourier transforms, the volume of the displaced fluid $V_{d}$ can be linked directly to $\unicode[STIX]{x1D701}(x,y)$ as follows:

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle V_{d}=\iint \,\text{d}x\,\text{d}y\,\unicode[STIX]{x1D701}(x,y)=-\underset{k\rightarrow 0}{\text{lim}}\hat{\unicode[STIX]{x1D701}}(k). & \displaystyle\end{eqnarray}$$

In addition, as $\hat{\unicode[STIX]{x1D701}}(k)$ is proportional to $\hat{p}(k)$ (see e.g. Raphaël & de Gennes Reference Raphaël and de Gennes1996), and $\hat{p}(0)=Mg$ , it can be shown that:

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle V_{d}={\displaystyle \frac{M}{\unicode[STIX]{x1D70C}}}, & \displaystyle\end{eqnarray}$$

where $M$ is the mass of the insect and $\unicode[STIX]{x1D70C}$ denotes the water density. Interestingly, Archimedes law is not modified by capillarity (see (2.2)). Individual insects were weighed separately, as follows. The insects were frozen, defrosted and weighed, immediately after defrosting, with a microbalance (Sartorius Supermicro S4). They were placed on absorbent paper during defrosting, to remove any excess water. The weight of 70 individuals (from first instars to adults), as measured with the microbalance, was compared with that predicted by the schlieren technique. Excellent agreement was achieved, for all instars (see figure 1). The shade thrown over the measurement field by the body of the insect might have been expected to be problematic for the evaluation of surface elevation. Fortunately, the deformation of the pattern does not seem to be significantly blurred by the presence of the body. The schlieren technique can, therefore, be considered an accurate and non-intrusive technique appropriate for this task (for an alternative optical method based on shadows, see Zheng et al. Reference Zheng, Lu, Yin, Tao, Shi and Tian2016).

2.4 Dynamic measurements

We filmed individuals propelling themselves in the middle of a shallow tank ( $7\times 5~\text{cm}$ ). We acquired images with a $1632\times 1200$ camera, and divided the full image into $101\times 74$ interrogation areas of $16\times 16$ pixels. The distance between interrogation areas determines the spatial resolution of the elevation surface, i.e. the distance between estimates of elevation $\unicode[STIX]{x1D6FF}d_{s}$ ( $\unicode[STIX]{x03BC}\text{m}$ ). The spatial resolution of the surface elevation $\unicode[STIX]{x1D6FF}d_{s}$ can be adjusted modifying the distance between the camera and the water, and also depends on (i) the focal distance of the optical system, (ii) the distance between the lens and the camera sensor and the camera and (iii) the size and resolution of the camera sensor. Table 2 (see appendix B) reports the mass of the individuals, their velocities and their accelerations. In this table, we also provide the distance between the camera and the water surface, $h_{camera}$ (m), the resting water surface elevation $h_{water}$ , the spatial resolution $\unicode[STIX]{x1D6FF}d_{s}$ of the images ( $\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$ ), the spatial resolution of the elevation $\unicode[STIX]{x1D6FF}d_{s}$ ( $\unicode[STIX]{x03BC}\text{m}$ ) and the resulting resolution of the elevation $\unicode[STIX]{x1D702}_{min}=\unicode[STIX]{x1D6FF}d_{s}\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}_{min}$ , where $\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}_{min}$ denotes the minimal surface gradient that can be measured. In turn, $\unicode[STIX]{x1D702}_{min}$ is dependent on the minimum measurable random dot pattern displacement $\unicode[STIX]{x1D6FF}x_{min}$ and the distance $h^{\ast }$ , where:

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle h^{\ast }=\left({\displaystyle \frac{1}{\unicode[STIX]{x1D6FC}h_{water}}}+{\displaystyle \frac{1}{h_{camera}}}\right)^{-1}, & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D6FC}$ is the refractive index of water. Using a two-dimensional FFT (fast Fourier transform) correlation algorithm providing subpixel interpolation, we measured surface displacements with a resolution of 0.1 pixels (Feldmann & Mayinger Reference Feldmann and Mayinger2001), the real size of a pixel being given by the spatial resolution $\unicode[STIX]{x1D6FF}x$ , such that $\unicode[STIX]{x1D6FF}x_{min}=0.1\unicode[STIX]{x1D6FF}x$ , and:

(2.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\unicode[STIX]{x1D701}_{min}={\displaystyle \frac{0.1\unicode[STIX]{x1D6FF}x}{h^{\ast }}}. & & \displaystyle\end{eqnarray}$$

By combining (2.3) and (2.4), we can see that increasing the water depth at rest $h_{water}$ results in a decrease in the minimum measurable slope $\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}_{min}$ and, thus, a decrease in elevation resolution $\unicode[STIX]{x1D702}_{min}$ . Furthermore, as $h_{water}$ increases, the refracted image of the pattern above the wave crests includes increasingly elongated patches, particularly for large waves (Moisy et al. Reference Moisy, Rabaud and Salsac2009). These patches are very difficult to analyse with a cross-correlation algorithm. We expected to see relatively large waves for large third instar individuals, resulting in large surface gradients, and we therefore had to keep $h_{water}$ relatively low (1 to 5 mm) while increasing the distance between the camera and the surface $h_{camera}$ , to increase $\unicode[STIX]{x1D6FF}d_{s}$ . The balance between the spatial resolution $\unicode[STIX]{x1D6FF}d_{s}$ and the minimal measurable slope $\unicode[STIX]{x1D735}\unicode[STIX]{x1D701}_{min}$ resulted in an almost constant resolution of elevation $\unicode[STIX]{x1D702}_{min}$ for all measurements.

We recorded 10 single strokes made by eight different individuals with masses of 0.127–4.85 mg, at 1000 frames per second, and fixed the shutter time to $500~\unicode[STIX]{x03BC}\text{s}$ to ensure that the position of the fast-moving middle legs was accurately estimated. We determined position at various time points of the body and the two middle legs, the joint between the body and the coxa, the joint between the femur and tibia and the end point of the tarsus. Figure 2(c) shows the dimensionless velocity of the body during the 10 strokes, the velocity being normalised by the final constant velocity of the body. The averaged dimensionless velocity (red curve) shows that the typical duration of a stroke $\unicode[STIX]{x1D70F}_{imp}$ ranged between 10 and 15 ms. By performing several kinetic measurements, we were able to evaluate data dispersion, which we found to be limited. For the sake of clarity, we then chose the most representative measurement for comparison with the theory (see § 3).

Figure 2.

Body and leg kinetics during a stroke. (a) Definition of the morphology of an insect leg. (b) The water strider body is oriented along the $x$ axis. At rest, the 3.75 mm-long median leg makes a $\unicode[STIX]{x03C0}/4$ angle with the $x$ axis. During the 10 ms stroke, the leg rotates by approximately $\unicode[STIX]{x03C0}/4$ rad. (c) Dimensionless velocity of the body for 10 strokes. The small superimposed sketch illustrates the movement of an individual, over a period of 10 ms.

3.1 Infinite depth

Consider an incompressible, inviscid, infinitely deep liquid with a free surface at rest located at $z=0$ . As stated above, Bühler’s original model is based on the assumption that the stroke can be considered impulsive and localised. We shall also assume that the impulsive force is horizontal, aligned with the $x$ axis. The impulsive force field applied to the fluid is taken to be of the form:

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{F}(\boldsymbol{r},t)=\boldsymbol{f}(\boldsymbol{r})\unicode[STIX]{x1D6FF}(t), & \displaystyle\end{eqnarray}$$

where the spatial part $\boldsymbol{f}$ can be expressed as follows:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{f}(\boldsymbol{r})=f_{0}\unicode[STIX]{x1D6FF}(x)\unicode[STIX]{x1D6FF}(y)\unicode[STIX]{x1D6FF}(z+h)\boldsymbol{e}_{x}, & \displaystyle\end{eqnarray}$$

and where $h$ typically corresponds to the depth of the meniscus created by the water strider at rest. It should be noted, however, that, in this model, the free surface is considered to be flat at $t=0$ , ignoring the static deformation of the interface. (This simplifying assumption is discussed in § 5.1.) The resulting pressure field $p$ can be calculated from the Poisson equation:

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}p=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{f}, & & \displaystyle\end{eqnarray}$$

subject to the boundary conditions $p(x,y,z=0)=0$ and $\unicode[STIX]{x1D735}p|_{r\rightarrow \infty }\rightarrow \mathbf{0}$ , where $r=|\boldsymbol{r}|$ . Equation (3.3) takes into account the divergence of the linearised Euler equation and the condition of incompressibility. Green’s function of (3.3) can be calculated by the method of images to ensure a vanishing solution at the interface (see Bühler Reference Bühler2007). By convolving the resulting Green’s function with the right-hand side of (3.3), we obtain the impulsive pressure field $p$ , which, at first order in $h$ can be expressed as follows:

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle p(\boldsymbol{r})=-h{\displaystyle \frac{3f_{0}}{2\unicode[STIX]{x03C0}}}{\displaystyle \frac{xz}{r^{5}}}+O(h^{2}). & \displaystyle\end{eqnarray}$$

A knowledge of $p$ makes it possible to calculate the resulting free surface displacement. Below, we establish the free surface displacement in response to a given surface pressure field. The velocity potential $\unicode[STIX]{x1D719}(\boldsymbol{r},t)$ is determined by solving the Laplace equation $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=0$ together with the boundary conditions:

(3.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\left[\unicode[STIX]{x2202}_{t}^{2}\unicode[STIX]{x1D719}+g\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D719}-\unicode[STIX]{x1D6FE}\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D6E5}_{x,y}\unicode[STIX]{x1D719}+{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x2202}_{t}P\right]_{z=0}=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(3.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D719}\big|_{z\rightarrow -\infty }=0, & \displaystyle\end{eqnarray}$$

$g$

$\unicode[STIX]{x1D6FE}$

$P(\boldsymbol{r},t)$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(\boldsymbol{r},t)=\int {\displaystyle \frac{\text{d}^{2}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{2}}}\,A(\boldsymbol{k},t)\,\text{e}^{\text{i}(ux+vy)}\text{e}^{kz}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{k}=(u,v)$ . Projecting (3.5a ) along the vertical axis $\boldsymbol{e}_{z}$ and incorporating (3.6) leads to:

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle k\left[\unicode[STIX]{x2202}_{t}^{2}A+\unicode[STIX]{x1D714}(k)^{2}A\right]=-{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x2202}_{t}{\mathcal{F}}\big[\unicode[STIX]{x2202}_{z}P\big|_{z=0}\big], & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}^{2}=\unicode[STIX]{x1D6FE}k^{3}/\unicode[STIX]{x1D70C}+gk$ denotes the dispersion relation for capillary–gravity waves, and ${\mathcal{F}}$ denotes the spatial two-dimensional Fourier transform operator (see appendix A). The kinematic boundary condition $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D701}=\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D719}|_{z=0}$ can be written in Fourier space as:

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\hat{\unicode[STIX]{x1D701}}=kA, & & \displaystyle\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},t)={\mathcal{F}}[\unicode[STIX]{x1D701}]$ . Incorporating  (3.8) into (3.7) and integrating over time yields:

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}^{2}\hat{\unicode[STIX]{x1D701}}+\unicode[STIX]{x1D714}(k)^{2}\hat{\unicode[STIX]{x1D701}}=-{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}{\mathcal{F}}\big[\unicode[STIX]{x2202}_{z}P\big|_{z=0}\big]. & \displaystyle\end{eqnarray}$$

Equation (3.9) is of prime importance in that it can be used to calculate the surface displacement $\unicode[STIX]{x1D701}(\boldsymbol{x},t)={\mathcal{F}}^{-1}[\hat{\unicode[STIX]{x1D701}}]$ for any given pressure distribution $P(\boldsymbol{r},t)$ and for any given set of initial conditions. The general solution can be written as the sum of the solution to the homogeneous equation and a particular solution to the non-homogeneous equation, namely $\hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},t)=\hat{\unicode[STIX]{x1D701}_{h}}(\boldsymbol{k},t)+\hat{\unicode[STIX]{x1D701}_{p}}(\boldsymbol{k},t)$ where:

(3.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D701}}_{\text{h}}(\boldsymbol{k},t)=\hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},0)\cos (\unicode[STIX]{x1D714}(k)t)+\unicode[STIX]{x2202}_{t}\hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},0){\displaystyle \frac{\sin (\unicode[STIX]{x1D714}(k)t)}{\unicode[STIX]{x1D714}(k)}}, & \displaystyle\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D701}}_{\text{p}}(\boldsymbol{k},t)=-{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}\int _{0}^{t}\text{d}\unicode[STIX]{x1D70F}\,{\displaystyle \frac{\sin (\unicode[STIX]{x1D714}(k)(t-\unicode[STIX]{x1D70F}))}{\unicode[STIX]{x1D714}(k)}}\,{\mathcal{F}}\big[\unicode[STIX]{x2202}_{z}P\big|_{z=0}\big], & \displaystyle\end{eqnarray}$$

$\hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},0)=0$

$\unicode[STIX]{x2202}_{t}\hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},0)=0$

$\hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},t)=\hat{\unicode[STIX]{x1D701}_{p}}(\boldsymbol{k},t)$

$P(\boldsymbol{r},t)=p(\boldsymbol{r})\unicode[STIX]{x1D6FF}(t)$

$p(\boldsymbol{r})$

$P(\boldsymbol{r},t)$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D701}}(\boldsymbol{k},t)=-{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}{\displaystyle \frac{\sin (\unicode[STIX]{x1D714}(k)t)}{\unicode[STIX]{x1D714}(k)}}{\mathcal{F}}\big[\unicode[STIX]{x2202}_{z}p\big|_{z=0}\big], & \displaystyle\end{eqnarray}$$

where, at first order, in $h$ :

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{F}}\left[\unicode[STIX]{x2202}_{z}p\big|_{z=0}\right]=-{\mathcal{F}}\left[{\displaystyle \frac{3f_{0}}{2\unicode[STIX]{x03C0}}}{\displaystyle \frac{hx}{\left(x^{2}+y^{2}+h^{2}\right)^{5/2}}}\right]. & \displaystyle\end{eqnarray}$$

Fourier transform in polar coordinates (see appendix A) yields ${\mathcal{F}}[\unicode[STIX]{x2202}_{z}p|_{z=0}]=\text{i}f_{0}k_{x}\,\text{e}^{-hk}$ , such that taking the inverse Fourier transform of  (3.11) and changing to a polar set of integration variables finally yields:

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}_{imp}(\boldsymbol{x},t)={\displaystyle \frac{f_{0}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}}}{\displaystyle \frac{x}{\sqrt{x^{2}+y^{2}}}}\int _{0}^{\infty }\,\text{d}k\,k^{2}\text{e}^{-hk}J_{1}\left(k\sqrt{x^{2}+y^{2}}\right){\displaystyle \frac{\sin (\unicode[STIX]{x1D714}(k)t)}{\unicode[STIX]{x1D714}(k)}}. & \displaystyle\end{eqnarray}$$

Equation (3.13) provides the change in the free surface after an impulsion.

3.2 Finite depth

As indicated above, the measurements must be performed in shallow water due to the constraints of optical resolution. We therefore extend the theory to the case of shallow water. With the same assumptions as above, we consider an impulsive force field located at $x=y=0$ and depth $z=-h$ , where $h>0$ in shallow water of depth $H>h$ . The force field is given by (3.1) and (3.2), and the resulting pressure field is found by solving the Poisson equation (see (3.3)) but with a different set of boundary conditions: $p(z=0)=0$ , $\unicode[STIX]{x2202}_{z}p(z=H)=0$ and $\unicode[STIX]{x1D735}p|_{r\rightarrow \infty }\rightarrow \mathbf{0}$ . For the sake of readability, the details of the calculations are reported in appendix A. The solution for the pressure field is as follows (see appendix A):

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle p_{h}=-{\displaystyle \frac{f_{0}}{\unicode[STIX]{x03C0}H}}\cos (\unicode[STIX]{x1D703})\mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D6EC}_{n}\sin \left(\unicode[STIX]{x1D6EC}_{n}z\right)\sin \left(\unicode[STIX]{x1D6EC}_{n}h\right)K_{1}\left(r\unicode[STIX]{x1D6EC}_{n}\right), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6EC}_{n}=(2n+1)\unicode[STIX]{x03C0}/(2H)$ and where $r=\sqrt{x^{2}+y^{2}}$ . The change in the free surface, calculated as described above, is given by (see appendix A):

(3.15) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}_{imp}^{H}(\boldsymbol{x},t)={\displaystyle \frac{f_{0}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}}}{\displaystyle \frac{x}{r}}\int _{0}^{+\infty }\,\text{d}k\,k^{2}\left[\cosh (kh)-\sinh (kh)\tanh (Hk)\right]J_{1}(kr){\displaystyle \frac{\sin (\unicode[STIX]{x1D714}(k)t)}{\unicode[STIX]{x1D714}(k)}},\quad & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}^{2}=(\unicode[STIX]{x1D6FE}k^{3}/\unicode[STIX]{x1D70C}+gk)\tanh (kh)$ is the dispersion relation for finite-depth capillary gravity waves Lamb (Reference Lamb1993). In deep water (i.e. $H\rightarrow \infty$ ), we recover (3.13) (see appendix A).

Figure 3.

Comparison of the experimentally measured surface elevation with theoretical predictions. Panels (a,d,g,j,m) represent the experimental surface elevation generated during the propulsion of a Gerris paludum individual ( $m_{gerris}=0.348~\text{mg}$ , $V_{gerris}=0.38~\text{m}~\text{s}^{-1}$ ); (b,e,h,k,n) represent the theoretical surface elevation produced by a single impulse in the direction of motion, with $\unicode[STIX]{x1D701}_{imp}$ as expressed in (3.13) (Bühler Reference Bühler2007); (c,f,i,l,o) show the results for continuous forcing in shallow water along the trajectory of the water strider legs (see (4.1) and (4.2)).

4.1 Qualitative comparison, from impulsive to continuous forcing

Figure 3(a,d,g,j,m) displays the time-resolved surface topography around an individual of Gerris paludum, during the 10 ms of a leg stroke. Figure 3(b,e,h,k,n) shows a theoretical estimate of topography according to Bühler’s method for impulsive forcing. It is clear at a glance that the lack of agreement between theory and experimental results leads to poor qualitative prediction of the wave pattern. Indeed, the experiments showed the wave to be of an elongated circular shape, reflecting the circular displacement of the leg during the stroke, whereas theory predicted a symmetric shape. (Furthermore, over a period of 1 to 10 ms, the wave produced after the impulse in experimental conditions increased, reaching a maximal value at 8 ms. By contrast, the theoretical prediction of surface elevation given by (3.13) yields a maximal wave crest at 4 ms (Bühler Reference Bühler2007).) In addition, theory predicted many more oscillations than were observed in the experiment. Interestingly, this particular feature can be resolved by accounting properly for finite depth, consistent with the shallow-water conditions in which the experiments were performed. Another small improvement can be obtained by taking into account the length of the strider’s leg in contact with the water surface (leg length is $l_{leg}=3.75~\text{mm}$ ).

However, a real improvement emerges from the observation that impulsive forcing is a very rough approximation of the real continuous forcing of the legs. During a leg stroke, the leg tip imparting momentum to the interface is not localised in space but instead follows a circular trajectory during the duration of the stroke. Accounting for the temporal extension of the rowing period, consistent with the linearity of the governing equation, we superimposed the impulsive solutions along the trajectory of the leg. Denoting $\boldsymbol{x}_{leg}(t)=(x_{leg}(t),y_{leg}(t))$ for $t\in [0,\unicode[STIX]{x1D70F}_{imp}]$ the trajectory of a leg during the stroke phase, we obtain:

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}_{cont}(\boldsymbol{x},t)=\int _{0}^{t}{\displaystyle \frac{\text{d}s}{\unicode[STIX]{x1D70F}_{imp}}}\,\unicode[STIX]{x1D6E9}(s<\unicode[STIX]{x1D70F}_{imp})\,\unicode[STIX]{x1D701}_{imp}^{H}(\boldsymbol{x}-\boldsymbol{x}_{leg}(s),t-s), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6E9}$ is the Heaviside step function, and $\unicode[STIX]{x1D701}_{imp}^{H}$ is given by (3.15). The time-dependent positions of the tips of the left and right legs were obtained by fitting two parametric equations $\boldsymbol{x}_{leg}(t)=(x_{leg}(t),y_{leg}(t))$ . We calculated the total surface deformation numerically, by discretising (4.1) as follows:

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}_{cont}(\boldsymbol{x},\unicode[STIX]{x1D70F})=\mathop{\sum }_{n=0}^{N}\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70F}-n\unicode[STIX]{x0394}t)\unicode[STIX]{x1D701}_{imp}(\boldsymbol{x}-\boldsymbol{x}_{leg}(n\unicode[STIX]{x0394}t),\unicode[STIX]{x1D70F}-n\unicode[STIX]{x0394}t){\displaystyle \frac{\unicode[STIX]{x0394}t}{\unicode[STIX]{x1D70F}_{imp}}}, & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x0394}t$ the discretisation time step, $N=\unicode[STIX]{x1D70F}_{imp}/\unicode[STIX]{x0394}t$ the number of time iteration steps and $\unicode[STIX]{x1D6E9}$ the Heaviside step function. The results are displayed in figure 3(c,f,i,l,o). As we can see, the typical elongated circular wave produced by the legs is very well reproduced by the simulation of continuous forcing in shallow water.

4.2 Quantitative fitting

As described above, fitting the theoretical surface to the experimental data by minimising the mean square difference makes it possible to obtain the total exerted force $f_{0}$ as a fitting parameter. We obtain $f_{0}=1.3\times 10^{-10}~\text{kg}~\text{m}^{4}~\text{s}^{-1}$ . (Note that the units of this force are quite uncommon. Indeed, an impulsive forcing involves delta function. In the definitions provided in (3.1) and (3.2), the delta functions have the inverse dimension of their arguments. As such, $f_{0}$ is a force multiplied by a volume multiplied by a time.)