2. Formulation and stationary response

We consider a vertical pressure source $\mathcal{P}(x,t)$ applied at the interface of a deep, inviscid, irrotational fluid of density $\rho$ , surface tension $\gamma$ and gravity $g$ , and translating along it at constant speed $V$ . The unperturbed interface lies on the $z=0$ plane, and the source moves along the $x$ -axis, its centre at $x=-Vt$ (with $V\ge 0$ ). Its amplitude is modulated harmonically at frequency $\varOmega$ ,

(2.1) \begin{align} \mathcal{P}(x,t)={p_e}(x+Vt)\,\cos (\varOmega t), \end{align}

where $p_e(x)$ is the spatial profile of the pressure distribution. The source is invariant along the transverse $y$ -direction, so the wave field lies in the $(x,z)$ plane and has straight crests: the surface elevation is $\zeta (x,t)$ , and the wavevector has the single component $k$ . Expanding the $x$ -dependence in Fourier modes $\zeta (x,t)=\int _{-\infty }^{\infty } ({\mathrm{d}k}/{2\pi })\,\mathrm{e}^{\mathrm{i}kx}\,\hat \zeta (k,t)$ , the wavenumber $k$ ranges over all real values; a free wave of wavenumber $k$ oscillates at the frequency $\omega (k)=\sqrt {g|k|+(\gamma /\rho )|k|^3}$ (which depends only on $|k|$ ) (Lighthill Reference Lighthill1978). Since $\zeta$ is real, $\hat \zeta (-k,t)=\hat \zeta ^*(k,t)$ , and it suffices to take $k\gt 0$ , for which $\omega (k)=\sqrt {gk(1+k^2/k_c^2)}$ is the capillary–gravity dispersion relation, where $k_c=\sqrt {\rho g/\gamma }$ . The phase speed $c(k)\equiv \omega (k)/k$ is minimal at the capillary wavenumber $k_c$ , where it equals $c_{\textit{min}}$ , and $\omega _c=c_{\textit{min}}k_c=\sqrt {2gk_c}$ ; the group velocity is $c_g(k)\equiv \mathrm{d}\omega /\mathrm{d}k$ . We write $\tilde {k}=k/k_c$ , $\tilde {V}=V/c_{\textit{min}}$ and $\tilde {\varOmega }=\varOmega /\omega _c$ ; the dispersion relation then reads $\omega (\tilde {k})/\omega _c=\sqrt {(\tilde {k}+\tilde {k}^3)/2}$ , and the phase speed $c/c_{\textit{min}}=\sqrt {(1+\tilde {k}^2)/(2\tilde {k})}$ is minimal at $\tilde {k}=1$ . The forcing of each surface mode is set by the spatial Fourier transform of the pressure distribution $p_e$ , $\hat {p}_e(k)=\int _{-\infty }^{\infty }{p_e}(x)\,\mathrm{e}^{-\mathrm{i}kx}\,\mathrm{d}x$ . For definiteness we take this distribution concentrated along a line, ${p_e}(x)=p\,\delta (x)$ . Its Fourier transform is then constant, $\hat {p}_e(k)=p$ .

Linearising the equation of motion and the boundary condition at the free surface, and eliminating the velocity potential (Lighthill Reference Lighthill1978), $\zeta (x,t)$ obeys, in Fourier space, the forced-oscillator equation

(2.2) \begin{align} \partial _{tt}\hat \zeta (k,t)+\omega ^2(k)\,\hat \zeta (k,t) =-\frac {k\,p}{2\rho }\bigl (\mathrm{e}^{\mathrm{i}\omega _{f1}t} +\mathrm{e}^{\mathrm{i}\omega _{f2}t}\bigr ), \end{align}

where the cosine modulation has split the forcing into two sidebands of the Doppler frequency $kV$ ,

(2.3) \begin{align} \omega _{f1}=kV+\varOmega ,\qquad \omega _{f2}=kV-\varOmega . \end{align}

Note that, for a general pressure profile ${p_e}(x)$ of finite extent, the constant amplitude $p$ on the right-hand side of (2.2) is replaced by the Fourier transform $\hat {p}_e(k)$ .

The long-time solution of (2.2) is, for $k\gt 0$ ,

(2.4) \begin{align} \hat \zeta (k,t)=\frac {k\,p}{2\rho } \left [\frac {\mathrm{e}^{\mathrm{i}\omega _{f1}t}}{D_1-2\mathrm{i}\omega _{f1}\varepsilon } +\frac {\mathrm{e}^{\mathrm{i}\omega _{f2}t}}{D_2-2\mathrm{i}\omega _{f2}\varepsilon }\right ], \qquad D_{\kern-1pt j}=\omega _{\textit{fj}}^2-\omega ^2 , \end{align}

where the Sommerfeld radiation condition is enforced by adding an infinitesimal friction, $2\varepsilon \,\partial _t\hat \zeta$ , to the left-hand side of (2.2) (Lighthill Reference Lighthill1978). Each radiating mode ( $k\gt 0$ ) appears as $\mathrm{e}^{\mathrm{i}(kx+\omega _{\textit{fj}}t)}$ , a wave whose crests travel at the phase velocity $-\omega _{\textit{fj}}/k$ . As the source advances in the $-x$ direction ( $x=-Vt$ ), the sign of $\omega _{\kern-1pt f}$ sets which way these crests travel: $\omega _{\kern-1pt f}\gt 0$ moves them in the same direction as the source ( $-x$ ; prograde), $\omega _{\kern-1pt f}\lt 0$ in the opposite direction ( $+x$ ; retrograde).

The radiating modes are the real positive roots of $D_{\kern-1pt j}$ ; in dimensionless form $D_{\kern-1pt j}\propto -P_{\kern-1pt j}(\tilde {k})$ , introducing the two cubic polynomials that organise the whole analysis

(2.5) \begin{align} P_1(\tilde {k})&=\tilde {k}^3-2\tilde {V}^2\tilde {k}^2+(1-4\tilde {V}\tilde {\varOmega })\,\tilde {k}-2\tilde {\varOmega }^2, \\[-12pt] \nonumber \end{align}

(2.6) \begin{align} P_2(\tilde {k})&=\tilde {k}^3-2\tilde {V}^2\tilde {k}^2+(1+4\tilde {V}\tilde {\varOmega })\,\tilde {k}-2\tilde {\varOmega }^2. \\[10pt] \nonumber \end{align}

Three of their roots recur below: $\tilde {k}_{u}$ and $\tilde {k}_\ast$ , the smallest positive root of $P_1$ and the retrograde root of $P_2$ (both long-wavelength modes, $\sim \tilde {\varOmega }^2$ , at small $\tilde{\varOmega}$ ), and $\tilde {k}_{\textit {d}}$ , a double root, one at each radiation transition. The damping term in (2.4), $-2\mathrm{i}\omega _{\textit{fj}}\varepsilon$ , makes the radiation condition enter with the sign of $\omega _{\textit{fj}}$ : negative for the upper sideband, reversed for the lower one once $\varOmega \gt kV$ . The interface profile was first studied by Pramanik (Reference Pramanik1980); a representative one is shown in figure 1. How many of these roots are real and positive – and hence which waves radiate – depends on $(\tilde {V},\tilde {\varOmega })$ ; the resulting partition of the plane into four radiation regimes is the phase diagram of figure 2 (§ 4).

Figure 1.

Interface profile $\zeta (x')$ in the source frame ( $x'=x+Vt$ ) at $t/T=0$ , $T=2\pi /\varOmega$ , from (2.4), for $\tilde {V}=1.3$ , $\tilde {\varOmega }=0.15$ : two short capillary ripples ( $\tilde {k}\approx 2.7$ and $3.3$ ) on one side; a longer wave ( $\tilde {k}\approx 0.62$ ) with a slowly varying long-wave mode ( $\tilde {k}\approx 0.03$ ) on the other. The source is located at $x'=0$ (dotted line).

Figure 2.

Phase diagram in the $(\tilde {V},\tilde {\varOmega })$ plane. Each region is labelled by the numbers $(n_1,n_2)$ of real positive roots of $(P_1,P_2)$ and, beneath, by the split of the radiating modes into upstream ( $\uparrow$ , energy ahead of the source) and downstream ( $\downarrow$ , behind); the two counts are equal in every region (the narrow $(3,1)$ region is labelled via an arrow). The green and red curves are the radiation transitions of $P_1$ and $P_2$ ; both emanate from the bifurcation point $(\tilde {V},\tilde {\varOmega })=(1,0)$ , and the cusp of the $P_1$ curve lies at $(\tilde {V},\tilde {\varOmega })\approx (0.768,0.174)$ . The expression for $\langle R\rangle$ in each region is given by (3.3).

3. Mean wave resistance

The wave resistance is the streamwise force exerted by the fluid on the source (Havelock Reference Havelock1918), reckoned positive when it opposes the motion

(3.1) \begin{align} R(t)=-\int _{-\infty }^{+\infty }\mathcal{P}(x,t)\,\partial _x\zeta (x,t)\,\mathrm{d}x , \end{align}

where $\mathcal{P}(x,t)$ is given by (2.1). Using Parseval’s theorem, we substitute (2.4) and average over a modulation period; only the $\cos ^2\varOmega t$ contributions survive, giving

(3.2) \begin{align} \langle R\rangle =\int _0^\infty \frac {k^2p^2}{\rho } \left [\frac {\omega _{f1}\varepsilon }{D_1^2+4\omega _{f1}^2\varepsilon ^2} +\frac {\omega _{f2}\varepsilon }{D_2^2+4\omega _{f2}^2\varepsilon ^2}\right ]\frac {\mathrm{d}k}{2\pi }. \end{align}

In the radiation limit $\varepsilon \to 0^+$ each Lorentzian collapses onto a signed delta, $\omega _{\kern-1pt f}\varepsilon /(D^2+4\omega _{\kern-1pt f}^2\varepsilon ^2)\to (\pi /2)\,\mathrm{sgn}(\omega _{\kern-1pt f})\,\delta (D)$ , and the integral reduces to a sum of residues over the roots of $P_1$ and $P_2$ . In terms of the dimensionless wavenumber $\tilde {k}=k/k_c$ , this gives

(3.3) \begin{align} \langle \tilde {R}\rangle \equiv \frac {\langle R\rangle }{p^2/\gamma } =\frac {1}{4}\bigl (\tilde S_1+\tilde S_2^{\gt }-\tilde S_2^{\lt }\bigr ), \end{align}

where

(3.4) \begin{align} \tilde S_1=\!\!\sum _{\substack {P_1(\tilde {k}_n)=0\\ \tilde {k}_n\gt 0}}\, \frac {\tilde {k}_n^2}{|P_1'(\tilde {k}_n)|}, \qquad \tilde S_2^{\gtrless }=\!\!\sum _{\substack {P_2(\tilde {k}_n)=0\\ \tilde {k}_n\tilde {V}\gtrless \tilde {\varOmega }}}\, \frac {\tilde {k}_n^2}{|P_2'(\tilde {k}_n)|}. \end{align}

The three sums in (3.4) are non-negative and enter (3.3) with the sign of $\omega _{\textit{fj}}$ : since $\omega _{f1}=kV+\varOmega \gt 0$ at every $k\gt 0$ , $\tilde S_1$ enters with a plus, as do the prograde lower-sideband waves, the retrograde ones with a minus. Writing $\tilde \omega (\tilde {k})=\omega /\omega _c$ , one has $P_{\kern-1pt j}=2[\tilde \omega ^2-(\tilde {k}\tilde {V}\pm \tilde {\varOmega })^2]$ , so the positive roots of $P_1$ solve $\tilde \omega -\tilde {k}\tilde {V}=\tilde {\varOmega }$ , and those of $P_2$ solve $\tilde \omega -\tilde {k}\tilde {V}=-\tilde {\varOmega }$ (prograde) or $\tilde \omega +\tilde {k}\tilde {V}=\tilde {\varOmega }$ (retrograde). The last equation has exactly one root $\tilde {k}_\ast$ , its left-hand side rising monotonically from $0$ to $\infty$ : there is always exactly one retrograde wave.

The three sums combine in (3.3) with $\tilde S_2^{\lt }$ subtracted, so $\langle R\rangle$ is not manifestly positive. It is nevertheless strictly positive for every $\tilde {V}\gt 0$ and $\varOmega \neq 0$ . To prove this, let $\tilde {k}_{u}$ be the smallest positive root of $P_1$ , which exists since $P_1(0)=-2\tilde {\varOmega }^2\lt 0$ . Because $\tilde \omega +\tilde {k}\tilde {V}\gt \tilde \omega -\tilde {k}\tilde {V}$ for $\tilde {k}\gt 0$ , the retrograde root $\tilde {k}_\ast$ lies below $\tilde {k}_{u}$ ; and the identity $\tilde \omega \,c_g/c_{\textit{min}}= (1+3\tilde {k}^2)/4$ then gives $\tilde {k}_{u}^2/P_1'(\tilde {k}_{u})\gt \tilde {k}_\ast ^2/P_2'(\tilde {k}_\ast )$ . (This inequality follows from the derivatives: with $\tilde \omega _{u}=\tilde {k}_{u}\tilde {V}+\tilde {\varOmega }$ and $\tilde \omega _\ast =\tilde {\varOmega }-\tilde {k}_\ast \tilde {V}$ the dispersion at the two roots, both positive, $P_1'(\tilde {k}_{u})=1+3\tilde {k}_{u}^2-4\tilde {V}\tilde \omega _{u}$ and $P_2'(\tilde {k}_\ast )=1+3\tilde {k}_\ast ^2+4\tilde {V}\tilde \omega _\ast$ . The second is positive term by term; the first is non-negative, $\tilde {k}_{u}$ being the first root of $P_1$ , and vanishes only at a radiation transition, where $\langle R\rangle$ diverges. Except at a transition, both denominators are positive, and the inequality is equivalent to $\tilde {k}_{u}^2P_2'(\tilde {k}_\ast )-\tilde {k}_\ast ^2P_1'(\tilde {k}_{u})\gt 0$ . In this difference the $3\tilde {k}_{u}^2\tilde {k}_\ast ^2$ terms cancel, leaving $\tilde {k}_{u}^2-\tilde {k}_\ast ^2+4\tilde {V}(\tilde {k}_{u}^2\tilde \omega _\ast +\tilde {k}_\ast ^2\tilde \omega _{u})$ , positive since $\tilde {k}_\ast \lt \tilde {k}_{u}$ .) The left-hand side is one term of $\tilde S_1$ and the right-hand side is all of $\tilde S_2^{\lt }$ , so $\tilde S_1\gt \tilde S_2^{\lt }$ : the single upper-sideband wave $\tilde {k}_{u}$ already outweighs the retrograde one $\tilde {k}_\ast$ , which therefore cannot reverse the balance. We conclude that the mean wave resistance $\langle R\rangle$ is always positive, and vanishes only on the steady below-threshold axis $\varOmega =0$ , $\tilde {V}\lt 1$ , where nothing is radiated, and on the axis $\tilde {V}=0$ , where a source that does not advance radiates symmetrically. Note that the sign of the instantaneous wave resistance $R(t)=\langle R\rangle (1+\cos 2\varOmega t)-C\sin 2\varOmega t$ (with $C$ the reactive, non-radiating amplitude) varies in time, dipping to $\langle R\rangle -\sqrt {\langle R\rangle ^2+C^2}\lt 0$ within each period.

The identity used above also recasts (3.3) in far-field form: it gives $|P_{\kern-1pt j}^{\prime}(\tilde {k}_n)|=4\tilde \omega _n|\tilde c_{g,n}-\epsilon _n\tilde {V}|$ with $\epsilon _n=\mathrm{sgn}(\omega _{f,n})$ and $\tilde{c}_{g,n} = c_{g}(\tilde{k}_n)/c_{min}$ , so that each term is an energy flux and $\langle R\rangle =\sum _n(k_n/\omega _{f,n})\varPhi _n$ , with $\varPhi _n$ the power radiated by mode $n$ (Grue & Palm Reference Grue and Palm1985). This does not supersede the residue calculation. A pulsating source works through its oscillation as well as its advance, so the energy budget reads $\varPhi _{\textit{tot}}=V\langle R\rangle +\varOmega W$ , with $W$ the pulsation contribution: conservation fixes this sum but not $\langle R\rangle$ separately, and only the residues disentangle them.

4. Below-threshold radiation and the phase diagram

The central consequence of (3.3) is immediate. For any $\varOmega \neq 0$ the small root $\tilde {k}_\ast$ of $P_2$ exists at any speed $V\gt 0$ : a genuinely propagating mode, radiating power to infinity. It is retrograde ( $\omega _{f2}\lt 0$ ) and thus enters the residue sum (3.3) through the subtracted term $\tilde S_2^{\lt }$ . The upper sideband is never silent either: $P_1(0)=-2\tilde {\varOmega }^{2}\lt 0$ guarantees a positive root $\tilde {k}_{u}$ of $P_1$ for every $\varOmega \neq 0$ , and by the inequality of § 3 that single wave already outweighs the retrograde one $\tilde {k}_\ast$ , so that $\langle R\rangle \gt 0$ . The steady threshold $V=c_{\textit{min}}$ therefore does not carry over to the modulated problem – as soon as the source oscillates, radiation occurs at all speeds, and the below-threshold silent regime of the steady source disappears. Whirligig beetles radiate below $c_{\textit{min}}$ through their leg strokes (Voise & Casas Reference Voise and Casas2010), an effect ascribed to their unsteadiness (Closa et al. Reference Closa, Chepelianskii and Raphael2010); their periodicity alone suffices.

The number of radiating modes changes across radiation transitions, at which two roots of $P_{\kern-1pt j}$ coalesce into a double root $\tilde {k}_{d}$ . This is the condition $c_g(\tilde {k}_{d})=V$ on the group velocity (i.e. $c_g/c_{\textit{min}}=\tilde {V}$ ), which traces one curve for each polynomial in the $(\tilde {V},\tilde {\varOmega })$ plane and partitions it into the four regions of the phase diagram, figure 2. The criterion, and the partition it induces – including the cusp below – were obtained by Pramanik (Reference Pramanik1980), within the general framework of Lighthill (Reference Lighthill1967). In the variables $(n\tau ,\tilde {V})$ with $\tau =\varOmega V/g=2\tilde {\varOmega }\tilde {V}$ , where $n$ is the harmonic index, the same partition is the phase diagram of Roccuzzo & Herault (Reference Roccuzzo and Herault2026, figure 13a), reached there independently; each point of that diagram refers to a single sideband, whereas figure 2 records the joint state $(n_1,n_2)$ of both. Their critical point is the cusp located below, whose coordinates we give in closed form. Both transition curves emanate from the steady bifurcation point $(\tilde {V},\tilde {\varOmega })=(1,0)$ ; the $P_1$ curve possesses a cusp at $\tilde {k}_{\textrm {f}}=\sqrt {(2\sqrt 3-3)/3}\approx 0.393$ , with coordinates $\tilde {V}_{\textrm {f}}=(3-\sqrt 3)/[2(2\sqrt 3-3)^{1/4}]\approx 0.768$ and $\tilde {\varOmega }_{\textrm {f}}=(2\sqrt 3-3)^{1/4}(3-\sqrt 3)/6\approx 0.174$ .

The same criterion fixes where each radiated wave is found relative to the source: a mode sits upstream (ahead) when its energy outruns the source, downstream (behind) otherwise. A prograde mode ( $\omega _{\textit{fj}}\gt 0$ ) carries its energy forwards, so it sits upstream when $c_g\gt V$ , i.e. when $P_{\kern-1pt j}'(\tilde {k}_n)\gt 0$ . A retrograde mode ( $\omega _{f2}\lt 0$ ) carries its energy backwards and is therefore always downstream, as in the steady picture. Each sideband favours one side. The upper sideband ( $\omega _{f1}$ ) supplies one net extra upstream mode, its long prograde wave $\tilde {k}_{u}\sim \tilde {\varOmega }^2$ at small $\tilde {\varOmega }$ . The lower sideband ( $\omega _{f2}$ ) supplies one net extra downstream mode, the retrograde long wave $\tilde {k}_\ast$ . The radiating modes therefore split into equal numbers upstream and downstream in every region, as annotated in figure 2. This balance has no counterpart in the pure-gravity problem, where one wave travels upstream and three downstream below the resonance $\varOmega V/g= 1/4$ , only two downstream waves remaining above it (Kaplan Reference Kaplan1957; Debnath & Rosenblat Reference Debnath and Rosenblat1969; Dagan & Miloh Reference Dagan and Miloh1982); the capillary–gravity case of a submerged source is treated by Debnath & Vann (Reference Debnath and Vann1974). Surface tension adds two capillary waves, both upstream (Pramanik Reference Pramanik1980).

The transitions leave a clear signature in $\langle R\rangle$ : at a double root $P_{\kern-1pt j}'(\tilde {k}_{d})=0$ , the term $\tilde {k}_{d}^2/|P_{\kern-1pt j}'(\tilde {k}_{d})|$ diverges and $\langle R\rangle$ develops an integrable singularity.

Close to a transition, reached at $\tilde {\varOmega }=\tilde {\varOmega }_{d}$ for a given $\tilde {V}$ , the two coalescing roots form a saddle node, $\tilde {k}_\pm =\tilde {k}_{d}\pm \sqrt {\alpha \,(\tilde {\varOmega }-\tilde {\varOmega }_{d})}$ , with the saddle-node coefficient $\alpha =-2\,\partial _{\tilde {\varOmega }}P_{\kern-1pt j}/P_{\kern-1pt j}''$ evaluated at $(\tilde {k}_{d},\tilde {\varOmega }_{d})$ , so that $|P_{\kern-1pt j}'(\tilde {k}_\pm )|\propto |\tilde {\varOmega }-\tilde {\varOmega }_{d}|^{1/2}$ and the resistance diverges as an integrable, one-sided square root, $\langle \tilde {R}\rangle \simeq A(\tilde {V})\,|\tilde {\varOmega }-\tilde {\varOmega }_{d}|^{-1/2}$ on the side where the pair is real (for instance $A\approx 0.11$ at $\tilde {V}=0.95$ ). This square-root law degenerates at the bifurcation point $(\tilde {V},\tilde {\varOmega })=(1,0)$ , where the coalescence is pinned at the dispersion minimum $\tilde {k}=1$ and the two transition curves originate. There the steady limit of $\langle \tilde {R}\rangle$ already diverges as $\langle \tilde {R}\rangle \simeq (1/4)(\tilde {V}-1)^{-1/2}$ along $\tilde {\varOmega }=0$ , while along the threshold $\tilde {V}=1$ the modulated wave resistance diverges as $\langle \tilde {R}\rangle \simeq (1/8)\,\tilde {\varOmega }^{-1/2}$ as $\tilde {\varOmega }\to 0$ ; the bifurcation point is thus the common origin of the steady and modulated square-root singularities. At the cusp of the $P_1$ curve the two saddle-node branches meet, the radiating root becoming triple, and the one-sided square root is replaced by a stronger, two-sided divergence $\langle \tilde {R}\rangle \propto |\tilde {\varOmega }-\tilde {\varOmega }_{\textrm {f}}|^{-2/3}$ (figure 3 b). Conversely, for $\tilde {V}$ below the cusp speed a vertical cut meets no $P_1$ transition and $\langle R\rangle$ rises smoothly as $\tilde {\varOmega }^{5}$ at small $\tilde {\varOmega }$ to a broad maximum without resonance, after which it decays as $\tilde {\varOmega }^{-1/3}$ (figure 3 a).

Figure 3.

Dimensionless mean wave resistance $\langle \tilde {R}\rangle =\langle R\rangle /(p^2/\gamma )$ (logarithmic scale) versus $\tilde {\varOmega }=\varOmega /\omega _c$ at four fixed speeds, sweeping $V$ through the $P_1$ cusp and the steady threshold. Throughout, $\langle \tilde {R}\rangle$ diverges integrably at the radiation transitions (dotted asymptotes), where a radiating root of $P_1$ or $P_2$ becomes double, and stays finite elsewhere; it decays as $\tilde {\varOmega }^{-1/3}$ at high frequency. (a) At $V=0.5\,c_{\textit{min}}$ , below the cusp speed, the vertical cut meets no $P_1$ transition: $\langle \tilde {R}\rangle$ rises as $\tilde {\varOmega }^{5}$ at small $\tilde {\varOmega }$ and passes a broad maximum, without any resonance. (b) At the cusp speed $V=0.768\,c_{\textit{min}}$ the two $P_1$ saddle-node branches meet and the radiating root becomes triple at $\tilde {\varOmega }_{\textrm {f}}=0.174$ , where $\langle \tilde {R}\rangle$ diverges on both sides as $|\tilde {\varOmega }-\tilde {\varOmega }_{\textrm {f}}|^{-2/3}$ . (c) At $V=0.95\,c_{\textit{min}}$ , although $V\lt c_{\textit{min}}$ , the source radiates at every $\varOmega \neq 0$ ( $\langle \tilde {R}\rangle$ vanishes only at $\varOmega =0$ and is too small to be visible before the first transition), crossing two $P_1$ transitions ( $\tilde {\varOmega }\approx 0.048,\,0.134$ ). (d) Above threshold, $V=1.3\,c_{\textit{min}}$ , the resistance starts from its finite steady value, crosses a low-amplitude $P_1$ resonance ( $\tilde {\varOmega }\approx 0.097$ ) and a high-amplitude $P_2$ resonance ( $\tilde {\varOmega }\approx 0.393$ ) – both integrable divergences, of very different prefactor – beyond which a radiating mode disappears and the resistance drops.

Figure 3. Long description

Panel A: A line graph shows the dimensionless mean wave resistance on a logarithmic scale versus the dimensionless modulation frequency at a fixed speed below the cusp speed. The vertical axis represents the wave resistance, and the horizontal axis represents the variable. The graph rises as a power law at small values and passes a broad maximum without any resonance. Panel B: A line graph shows the dimensionless mean wave resistance on a logarithmic scale versus the same variable at the cusp speed. The graph features a triple root where the resistance diverges on both sides. Panel C: A line graph shows the dimensionless mean wave resistance on a logarithmic scale versus the same variable at a speed where the source radiates at every value. The graph crosses two transitions. Panel D: A line graph shows the dimensionless mean wave resistance on a logarithmic scale versus the same variable at a speed above the threshold. The resistance starts from a finite steady value, crosses two resonances, and drops beyond a certain point.

Just below the steady threshold, by contrast, a vertical cut does cross both $P_1$ transitions and $\langle R\rangle$ exhibits sharp resonant peaks (figure 3 c): a slow modulation of a marginally below-threshold source suffices to trigger a pronounced radiative response.

5. Limiting and asymptotic behaviour

As $\varOmega \to 0^+$ the two long sideband roots $\tilde {k}_{u}$ and $\tilde {k}_\ast$ tend to zero and their contributions vanish, while the remaining roots of $P_1$ and $P_2$ merge onto those of $P_0(\tilde {k})=\tilde {k}^3-2\tilde {V}^2\tilde {k}^2+\tilde {k}$ . The formula (3.3) then reduces to $\langle \tilde {R}\rangle = (1/2)\!\sum _{P_0(\tilde {k}_n)=0,\,\tilde {k}_n\gt 0}\, \tilde {k}_n^2/|P_0'(\tilde {k}_n)|$ , which is zero for $\tilde {V}\lt 1$ (no real positive root of $P_0$ ) and positive for $\tilde {V}\gt 1$ . This is exactly one half of the steady RdG resistance (1.1), the factor $(1/2)$ being the time-average $\langle \cos ^2\varOmega t\rangle$ of the modulation.

In the drag-free corner of the steady problem ( $V\lt c_{\textit{min}}$ , small $\varOmega$ ), the long roots $\tilde {k}_{u}$ and $\tilde {k}_\ast$ are nearly degenerate and their residues almost cancel, leaving

(5.1) \begin{align} \langle \tilde {R}\rangle \;\xrightarrow [\tilde {\varOmega }\to 0]{}\;24\,\tilde {V}\,\tilde {\varOmega }^{5}. \end{align}

At the opposite end the radiating wavenumbers grow without bound, $\tilde {k}_n\sim \tilde {\varOmega }^{2/3}$ ; at such large $\tilde {k}$ the capillary term $\tilde {k}^3$ dominates the dispersion relation over the gravity term $\tilde {k}$ , so the radiated waves are short, pure-capillary ripples. The order- $\tilde {\varOmega }^0$ terms cancel, and

(5.2) \begin{align} \langle \tilde {R}\rangle \;\xrightarrow [\tilde {\varOmega }\to \infty ]{}\; \frac {2^{1/3}}{9}\,\tilde {V}\,\tilde {\varOmega }^{-1/3}, \end{align}

with the universal coefficient $2^{1/3}/9\approx 0.140$ . Since $c_{\textit{min}}$ enters only through the scales, below- and above-threshold sources alike approach the same capillary decay, differing only through the prefactor $\propto \tilde {V}$ . Below threshold this raises $\langle R\rangle$ from zero; above threshold it drives $\langle R\rangle$ well below the steady RdG value. Finally, for $V\to \infty$ at fixed $\varOmega$ the radiation is carried by short capillary ripples ( $\tilde {k}_n\sim 2\tilde {V}^2$ ) and the resistance flattens onto a plateau, $\langle \tilde {R}\rangle \to (1/2)+\tilde {\varOmega }/(4\tilde {V}^{3})+3/(16\tilde {V}^{4})$ . The leading $(1/2)$ is again $\langle \cos ^2\varOmega t\rangle$ , but that factor fails at higher order – the $\tilde {V}^{-4}$ coefficient is $3/16$ , not the $1/4$ that a uniform halving of the steady result (1.1) would give – so the expansion is non-analytic at $\varOmega =0$ and the limits $V\to \infty$ and $\varOmega \to 0$ do not commute.

Because each radiating mode contributes to $\langle R\rangle$ in proportion to $\tilde {k}_n^2$ (3.4), the resistance is dominated by the short, capillary-scale waves throughout most of the $(\tilde {V},\tilde {\varOmega })$ plane – everywhere, that is, except the below-threshold corner where the radiated waves are long. As a result the two single-physics reductions – approximating the dispersion relation by capillarity alone or by gravity alone – are far from equally accurate. In both we retain the full physical system, so that $c_{\textit{min}}$ and $\omega _c$ remain well defined, and simplify only the dispersion relation used to compute the radiated field. Retaining only the capillary term ( $\omega ^2=\gamma k^3/\rho$ ) reproduces $\langle R\rangle$ to within a few per cent wherever the radiating waves are short, at high speed ( $\tilde {V}\gtrsim 2$ ) or high frequency; but it has no threshold, and below $c_{\textit{min}}$ at small $\tilde {\varOmega }$ it misses the near cancellation of residues, so that by the analysis of (5.1) it overestimates $\langle R\rangle$ by orders of magnitude. Discarding surface tension instead, that is using the pure-gravity dispersion $\omega ^2=gk$ , misrepresents $\langle R\rangle$ by more than $10\,\%$ over almost the entire $(\tilde {V},\tilde {\varOmega })$ plane (figure 4): although $\gamma$ is dynamically negligible for the long gravity waves, it controls the short, dominant capillary-scale waves and therefore $\langle R\rangle$ itself almost everywhere.

Figure 4.

Failure of the pure-gravity approximation for the mean wave resistance $\langle R\rangle$ of a line source, obtained by discarding surface tension (dispersion $\omega ^2=gk$ ). Contours mark the relative error reaching $10\,\%$ (solid), $20\,\%$ (dashed) and $50\,\%$ (dotted): the approximation is accurate to within $10\,\%$ only along the thin band where its single gravity branch happens to mimic the exact response, while the error exceeds $50\,\%$ over most of the $(\tilde {V},\tilde {\varOmega })$ plane. Neglecting $\gamma$ thus misrepresents $\langle R\rangle$ almost everywhere.

For a source of finite width $b$ , $p$ is replaced by $\hat {p}_e(k)$ , as noted after (2.3), reweighting each radiating mode by $|\hat {p}_e(k_n)/p|^{2}$ . The phase diagram, the divergence exponents and the $\tilde {\varOmega }^{5}$ law are unchanged; the $\tilde {\varOmega }^{-1/3}$ tail, carried by the shortest ripples, is cut off once $k_n$ $b\gtrsim 1$ .