3.1. Mathematical framework

We aim at reformulating the surface-elevation field $\zeta (\boldsymbol{x},t)$ from (2.1) as a superposition of monochromatic capillary-gravity waves emitted along the trajectory $\boldsymbol{X}(t)$ . Using the delayed Green’s function from an initial time $t_0$ (Appel Reference Appel2007), the wave field results from the continuous emission of waves along the trajectory and can be written as the following convolution integral with respect to $\tau \in [t_0, t]$ (Gierczak et al. Reference Gierczak, Fadle, Arutkin, Raphaël and Benzaquen2020):

(3.1) \begin{equation} \zeta (\boldsymbol x,t)=\frac {1}{(2\pi )^2} \int _{t_0} ^t \text{d} \tau \iint \text{d} \boldsymbol k \,\frac {1}{2i \omega (k)} \left ( -\frac {k}{\rho } \widehat p_e(\boldsymbol k,\tau ) \right )e^{i \phi \left ( \tau , \boldsymbol k, t;\boldsymbol x\right )}+ \text{c.c.}, \end{equation}

where c.c. denotes the complex conjugate and the phase is defined as

(3.2) \begin{equation} \phi = \boldsymbol {k} \boldsymbol{\cdot }\left [\boldsymbol {x} - \boldsymbol {X}(\tau )\right ] - \omega (k)(t - \tau ). \end{equation}

In catastrophe optics (Nye Reference Nye1999; Poston & Stewart Reference Poston and Stewart2014), the phase is expressed in terms of the state variables $(\tau , k_x, k_{\!y})$ , with respect to which the conditions of stationarity and degeneracy are defined. The spatio-temporal coordinates $(x, y, t)$ act as control parameters, whose variation may induce abrupt changes in the degeneracy of the associated critical points. In this framework, the codimension is defined as the number of independent control parameters required to describe, or equivalently to unfold, the singularity.

Since this integral involves a double integration over $\boldsymbol{k}$ and $\tau$ , the method of stationary phase yields two types of extremum conditions. The first condition, $\boldsymbol{\nabla} _k \phi =\boldsymbol{0}$ , for large $\Delta t$ , determines the ray trajectory $\boldsymbol {r}$ along which the energy of a wave packet with central wavenumber $k$ , emitted at $\Delta t$ , propagates. Using the polar basis $(\boldsymbol {e}_k, \boldsymbol {e}_\theta )$ in Fourier space where $\boldsymbol e_k= \boldsymbol k/k$ , this condition leads to the ray equation (Lighthill Reference Lighthill2001; Whitham Reference Whitham2011):

(3.3) \begin{equation} \boldsymbol r(\tau , t,k,\theta )=\boldsymbol X(\tau )+{c}_g(k) \boldsymbol e_k(\theta ) \Delta t. \end{equation}

In the limit of large $k$ , the wakefront also emerges from the coherent interaction of waves with a stationary phase satisfying $\partial _\tau \phi =0$ giving the second condition: $\omega (k,\tau )=\boldsymbol k \boldsymbol{\cdot }\dot {\boldsymbol X}(\tau )$ (Brown Reference Brown2001; Whitham Reference Whitham2011). Using the polar decomposition of the velocity, with $\dot X=U \cos (\psi )$ and $\dot Y=U \sin (\psi )$ , this stationary condition for the wavefront is rewritten into

(3.4) \begin{equation} U(\tau )\cos [\theta - \psi (\tau )] ={c_\phi (k)}. \end{equation}

Here, the velocity amplitude $U(\tau )$ and the steering angle $\psi (\tau )$ can vary along the trajectory such that resonant condition can be extended to unsteady motion. When the supercritical motion condition $U(\tau )\gt c_\phi$ is satisfied for a set of $k$ including $\kappa$ , a local angle of emission $\theta ^e_\pm (\tau )$ along the trajectory $\boldsymbol X(\tau )$ can be defined as

(3.5) \begin{equation} \theta _\pm ^e(\tau ,k)=\psi (\tau )\pm \arccos \left ( \frac {c_\phi (k)}{ U (\tau )}\right )\!. \end{equation}

This angle specifies the emission direction along which two concentric waves emitted at times $\tau$ and $\tau + \mathrm{d}\tau$ interfere constructively, as illustrated by the wavefront model (figure 2 d). The symbol $\pm$ denotes the angles of emission from the two sides of the trajectory.

By combining the two extremum conditions (3.3) and (3.5), we obtain the set of curves $\boldsymbol {x}_\phi (\tau ,t,k)$ defining wavefronts as

(3.6) \begin{equation} \boldsymbol {x}_\phi (\tau ,t,k)=\boldsymbol X(\tau )+{c}_g(k) \boldsymbol e_k\left (\theta _e(\tau ) \right ) \Delta t. \end{equation}

For uniform rectilinear motion ( $U = \text{const}, \psi = \text{const}$ ), the wavefront $\boldsymbol{x}_\phi$ is given by the set of Fourier modes $(k_x,k_{\!y})$ satisfying both the dispersion relation and the stationary-phase conditions, $\partial _\tau \phi =0$ . In this steady case, these modes also coincide with the steady resonant modes $\boldsymbol{k}$ , i.e. the poles of the Cauchy integral in (2.3) satisfying $\omega (k)=\boldsymbol U \boldsymbol{\cdot }\boldsymbol k$ for $n=0$ , where $\boldsymbol U$ is a constant velocity. By parametrising the set of stationary modes as $\boldsymbol{k}(s)$ , where $s$ is a scalar parameter, caustics and cusps occur when the dispersion curve $\boldsymbol{k}(s)$ exhibits inflection points, i.e. when $\vert \boldsymbol{k}''(s)\vert =0$ (Appel Reference Appel2007). These cusps induced by dispersion have been extensively studied in steady pure gravity-wave (Lamb Reference Lamb1924; Lighthill Reference Lighthill1967, Reference Lighthill2001) and capillary-gravity wakes (Wehausen & Laitone Reference Wehausen and Laitone1960), where a swallowtail catastrophe can occur (Binnie Reference Binnie1965; Moisy & Rabaud Reference Moisy and Rabaud2014; Liang & Chen Reference Liang and Chen2018), although it was not identified as such.

For the slalom, the Fourier modes $\boldsymbol{k}(\tau )$ satisfying the stationary-phase condition (3.4) do not coincide with the resonant modes or poles $\boldsymbol{k}_n(s)$ , which satisfy the Doppler-shifted resonant condition in (2.3), with $\omega (k_n)=\boldsymbol{k}_n \boldsymbol{\cdot }\boldsymbol{U}+n\omega _s$ , where $n\in \mathbb{Z}$ . Applying the inflection-point criterion to each family of resonant modes $\boldsymbol{k}_n(s)$ amounts to extending the wavefront bifurcation reported by Binnie (Reference Binnie1965), Moisy & Rabaud (Reference Moisy and Rabaud2014) and Liang & Chen (Reference Liang and Chen2018) for a pulsating source in uniform translation (Lighthill Reference Lighthill1967; Roccuzzo & Herault Reference Roccuzzo and Herault2026). However, this bifurcation occurs for a critical speed $U$ that is nearly twice $c_{\textit{min}}$ , whereas the singularity described in § 2 develops in the wake for $U \gt c_{\textit{min}}$ and after a finite time.

3.2. Wavefront folding

In contrast, we consider here local folding of the wavefront caused by the unsteadiness of the trajectory $\boldsymbol{X}(\tau )$ , where such caustic persists in the non-dispersive limit (d’Hardemare et al. Reference d’Hardemare, Wildeman, Eddi and Fort2019). This elementary folding occurs when $\partial _\tau \boldsymbol x_\phi =\partial _\tau (\boldsymbol{\nabla} _{\boldsymbol k} \phi ) = \boldsymbol{0}$ in addition to the stationary-phase conditions. At this point, the norm of the local tangent vector to the wavefront $\boldsymbol x_\phi$ (3.6) vanishes, corresponding to the coherent interference of two wave packets along the unsteady wavefront. In catastrophe theory (Poston & Stewart Reference Poston and Stewart2014), a cusp can be interpreted as the locus, in the parameter space $(x,y)$ of co-dimension 2, of a degenerate stationary point of the phase $\phi (\boldsymbol{k},\tau ; \boldsymbol x,t)$ defined in the variable space $(k_x,k_{\!y},\tau )$ .

To simplify this analysis, we consider the wavefront $\boldsymbol{x}_{0} = \boldsymbol{x}_{\phi }(\tau , t,k= \kappa )$ associated with the steady wavenumber $\kappa$ and connecting the source term at $\boldsymbol{x}_{0}(\tau =t,t,\kappa )=\boldsymbol X(t)$ . Its dispersion relation is greatly simplified, as it corresponds to the iso-phase $\phi =0$ , for which $\phi ( \kappa ,\theta _e; \boldsymbol x_0,\tau ) = (c_\phi (\kappa ) - c_g(\kappa ) )\Delta t = 0$ for all $\Delta t \in \mathbb{R}$ , where $c_\phi (\kappa ) =c_g(\kappa )=c_{\textit{min}}$ . The wavefront $\boldsymbol{x}_{0}$ is represented in figure 2 by dashed curves for both capillary-gravity waves (panel a) and non-dispersive waves (panels b–d). In these figures, the wavefront follows an isophase, except in the region below the folding (see for instance figure 2 c). This discrepancy is likely due to the sharp phase variations at the folding, which are smoothed by the dissipative term and the limit of spatial resolution. Using the specific branch $\boldsymbol x_0$ allows us to illustrate the strong connections between non-dispersive waves and dispersive wave with local minimum of phase velocity where $c_\phi =c_g$

The tangent vector of the caustic $\boldsymbol{x}_{0}$ given in (3.6) reads

(3.7) \begin{equation} \frac {{\rm d} \boldsymbol x_0 }{{\rm d} \tau }= \left ( \dot {\boldsymbol X} \boldsymbol{\cdot }\boldsymbol e_\theta + {c}_{\textit{min}} \Delta t\frac {{\rm d} \theta ^e}{{\rm d} \tau }(\tau ,\kappa )\right ) \boldsymbol e_\theta \left (\theta ^e\right )\!. \end{equation}

Using (3.4) for $k=\kappa$ , we show that the norm of this tangent vector cancels out when

(3.8) \begin{equation} \Delta t_c \left ( \pm \frac {\mathcal{U}^{-1}}{\sqrt { \mathcal{U}^2-1}} \frac {\hbox{d} \mathcal{U}}{\hbox{d}\tau } +\frac {\hbox{d} \psi }{\hbox{d}\tau }\right )=\pm \sqrt {\mathcal{U}^2-1} , \end{equation}

such that cusps emerge at $\tau _c = t - \Delta t_c$ . We introduced the dimensionless velocity $\mathcal{U}(\tau ) = U(\tau )/c_{\textit{min}}$ , which is analogous to the instantaneous Mach number for non-dispersive waves. The sign depends on which edge of the wake is considered: $+$ for $\sin (\theta _e - \psi )\gt 0$ , corresponding to the superior edge, and $-$ for the inferior edge.

For instance, (3.8) predicts the elementary wavefront folding produced by uniform rotational motion with $\psi (\tau ) = \omega _s \tau$ and $U = \omega _s R$ , where $\omega _s$ is the rotation rate and $R$ the gyration radius. It leads to the well-known cusp time $\omega _s \tau _c =\sqrt {(\omega _s R /c_{\textit{min}})^2-1 }$ on the inner branch ( $\sin (\theta _e - \psi )\lt 0$ ) for $t=0$ (d’Hardemare et al. Reference d’Hardemare, Wildeman, Eddi and Fort2019).

3.3. Swallowtail catastrophe

The higher-order catastrophe induced by a non-uniform motion is then described by the interaction of three wave packets, for which the additional condition $\partial _\tau ^2 \boldsymbol{\nabla} _{\boldsymbol k} \phi =\boldsymbol{0}$ must be satisfied. This point is identified as a swallowtail catastrophe and is described by a parameter space of co-dimension three, where the locus of this catastrophe occurs in the spatiotemporal space $(\boldsymbol x,t)$ .

To characterise the emergence of the swallowtail catastrophe associated with slalom motion, we consider a simplified kinematic model with constant swimming speed $U_s$ and a harmonic steering angle $\psi (t) = \psi _s \cos (\omega _s t)$ . This representation is equivalent, to first order in $\psi _s$ (Eloy Reference Eloy2013), to the formulation introduced in § 2 using the Jacobi–Angers identities. By identification, we obtain $U_X = U_s J_0(\psi _s)$ and $\omega _s Y_s = 2 U_s J_1(\psi _s)$ , which leads to the following parameters: $U_s \simeq 45.5$ cm s⁻¹ and $\psi _s\simeq \pi /9$ .

Expressed in terms of the parameters $(\psi _s, U_s, \omega _s)$ , (3.8) then reduces to a single transcendental equation:

(3.9) \begin{equation} \omega _s( \tau _c- t ) \sin (\omega _s \tau _c)=\pm \frac {\sqrt {\mathcal{U}^2-1}}{ \psi _s}. \end{equation}

For $\mathcal{U} \gt 1$ and at a given time $t$ , the set of solutions $\{\tau _c\}$ (blue circles in figure 3 a) is given by the intersections with the levels $\pm \sqrt {\mathcal{U}^2 - 1}/\psi _s$ (dashed red lines in figure 3 a) of the function $f_t(\tau _c) = \omega _s(\tau _c - t)\sin (\omega _s \tau _c)$ , shown as the blue curve, which is the product of the affine function $\omega _s(\tau _c-t)$ (black dotted lines) and a sinusoidal function. As $t$ increases, the conic envelopes (dashed black lines) shift towards larger values of $\tau$ , thereby increasing the amplitude of the sinusoidal envelope at fixed $\tau$ .

The time $t = t_{{sw}}$ selected in figure 3(a) corresponds to the formation of a degenerate cusp on the upper branch ( $y\gt 0$ ) of the wavefront $\boldsymbol{x}_0$ , shown in red in figure 3(a), appearing at $x = U_x \tau _c \simeq -32.5\,\mathrm{cm}$ . At this instant, the level $\sqrt {\mathcal{U}^2 - 1}/\psi _s$ (dashed red line in figure 3 a) is tangent to the maximum of the function $f_{t_{{sw}}}(\tau _c)$ (blue curve). This degeneracy, associated with an extremum of $f_{t_{{sw}}}$ , corresponds to the condition $\partial _\tau ^2 \boldsymbol{x}_0 = \partial _\tau ^2 \boldsymbol{\nabla} _{\boldsymbol{k}} \phi = 0$ , at which three wavepackets converge to form the degenerate cusp. For $t \gt t_{{sw}}$ , two cusps emerge from the degenerate cusp as the red line will intersects the blue curve at two points on each side of the local extremum, producing a double fold of the wavefront $\boldsymbol{x}_0$ (purple curves above the red curve in figure 3 b), characteristic of the swallowtail catastrophe.

Figure 3.

(a) Numerical representation of the emergence criterion obtained from (3.9). The intersections between the blue curve and the two level lines $\pm \sqrt {\mathcal{U}^2-1}/\psi _s$ define the delay times $\tau _c$ associated with the appearance of the cusps. The horizontal axis represents the distance from the body position using the mapping $\tau \mapsto U_x \tau$ . The time was chosen as $t=t_{sw}$ , corresponding to the appearance of a degenerate cusp at $U_x \tau _c \simeq -32.5$ cm. (b) Illustration of the wavefront folding process $\boldsymbol x_0(t,\tau )$ for different times $t$ , with $t_{sw}$ shown in red. Each filled circle along the trajectory (black curve) corresponds to the most upstream position of the body $\boldsymbol X(t)$ . (c) Unfolding of the swallowtail manifold by representing all the wavefronts from panel (b) in the Mach-cone frame using the change of frame given by (3.10).

Figure 3. Long description

A: Numerical representation of the emergence criterion. B: llustration of the wavefront folding process. C: Unfolding of the swallowtail manifold by representing all the wavefront in the Mach-cone frame

Since $(\tau - t) \lt 0$ , the maxima of $f_t(\tau )$ are located near the minima of the lateral oscillations $Y(t)$ of the trajectory, and conversely for the minima of $f_t$ . This property is consistent with the mechanism identified in § 2 and in figure 1(d), whereby wavefront folding occurs because rays emitted near a minimum of $Y(\tau )$ does not reach the edge of the upper boundary ( $y\gt 0$ ).

Finally, an elegant representation of the canonical swallowtail structure for $y\gt 0$ is obtained by mapping the wavefronts $\boldsymbol{x}_0(\tau ,t)$ into the Mach-cone reference frame $(\tilde {\boldsymbol{x}},t)$ via the transformation

(3.10) \begin{equation} \left (\boldsymbol{x} , t\right )\mapsto \left (\tilde {\boldsymbol{x}} = \boldsymbol{R}(\varTheta _{\mathcal{U}})\boldsymbol{x} - c_{\textit{min}} t \,\tilde {\boldsymbol{e}}_{\!y} , t\right )\!. \end{equation}

Here, $\boldsymbol{R}$ is the canonical rotation matrix in SO(2) defined by the Mach angle $\varTheta _{\mathcal{U}} = \arcsin (\mathcal{U}^{-1})$ . We also introduce the unit vector $\tilde {\boldsymbol{e}}_{\!y} = (0,1)^T$ . Hence, the wavefronts shown in figure 3(b) exhibit a clear swallowtail catastrophe manifold in the $(\tilde {\boldsymbol{x}},t)$ represented in figure 3(c).

The interpretation of this surface within the framework of catastrophe theory applied to the wake is as follows. The swallowtail surface in parameter space represented in figure 3(c) corresponds to the locus of fold bifurcations, forming the wavefronts where the two stationarity conditions of the phase are satisfied. The singular lines of this surface correspond to cusp catastrophes, which, in the wake context, are associated with caustics where two wave packets locally coalesce on the wavefront. These singularities propagate in space–time and meet at the instant $t=t_{{sw}}$ , corresponding to a higher-order degeneracy where three wave packets locally coincide in physical space.