2.1 General solution

For fully three-dimensional flow in the presence of vorticity, potential theory is not an option, so to solve the flow problem we must turn to the Euler equations, describing inviscid flow. The flow is assumed to be incompressible. We write the full velocity and pressure field on the form

Here $U(z)$ is the basic shear current of constant vorticity $S\geqslant 0$ , and the hatted quantities are perturbations due to the waves. We shall assume all perturbations to be small, and work to linear order in these quantities. We have assumed the surface of the water to be at rest with respect to the coordinate system in order that results can be immediately compared to previous work: this is easily generalised by an overall Galilean coordinate transformation.

The wave source (‘ship’) is modelled as a superimposed localised pressure $\hat{p}_{ext}$ of constant shape and strength, travelling with velocity $\boldsymbol{V}$ which makes an angle ${\it\beta}$ with the $x$ axis, and hence the shear flow. The situation is sketched in figure 1. We consider only stationary solutions as seen from the boat, hence all physical quantities will depend on surface position $\boldsymbol{x}=(x,y)$ and time $t$ only through the combination ${\bf\xi}=\boldsymbol{x}-\boldsymbol{V}t$ . Such a stationary wake may be interpreted as a continuous series of ring waves emitted by the travelling source, and all our results might instead be derived based on such a formalism (Li & Ellingsen Reference Li and Ellingsen2016).

The flow is a solution to the Euler equation

which we linearize with respect to perturbations. We use a Fourier decomposition of perturbation quantities into plane waves according to

Following Ellingsen (Reference Ellingsen2014b ), we can eliminate $u,v$ and $p$ to find a simple Rayleigh equation (the inviscid form of the Orr–Sommerfeldt equation) for $w$ alone, $w^{\prime \prime }=k^{2}w.$ Solving this subject to the boundary condition that $w(\boldsymbol{k},-h)=0$ (no vertical velocity at the bottom), we find the solution to the full flow field as follows

Here, $A(\boldsymbol{k})$ is an unknown coefficient. These solutions are the ship wave equivalents of the general solutions reported by Ellingsen (Reference Ellingsen2014a ). Note that the motion introduced by this wave solution is itself rotational since it shifts and twists the vortex lines of the background flow, unlike any wave motion described by potential theory (Ellingsen Reference Ellingsen2016).

Let the surface elevation (relative to its equilibrium state) be ${\it\zeta}({\bf\xi})$ and the external pressure be $\hat{p}_{ext}({\bf\xi})$ , and let their Fourier transforms in the manner of (2.3), be $B(\boldsymbol{k})$ and $p_{ext}(\boldsymbol{k})$ , respectively. We can now write down the linearized kinematic boundary condition (stating that a particle on the surface stays on the surface)

(note that $U(0)=0$ by choice), and dynamic boundary condition (stating that normal stress, as given by the pressure through Euler equation, is continuous at the surface),

Eliminating $A(\boldsymbol{k})$ , we find $B(\boldsymbol{k})$ which we integrate over the $\boldsymbol{k}$ plane to find ${\it\zeta}$ . One now encounters the same difficulty always encountered when considering waves in quasi-stationary or quasi-periodic wave systems, namely that the integral is indeterminate due to poles on the axes of integration. The criterion that our system, while being a stationary description, still knows the difference between past and future, must be imposed through a radiation condition. We use the procedure employed, e.g. by Lighthill (Reference Lighthill1978, § 3.9), presuming that the external pressure has been turned on very slowly since $t=-\infty$

where ${\it\epsilon}$ is defined to be dimensionless for convenience. The same slow temporal behaviour will affect all perturbation quantities, hence throughout our analysis this transition amounts to the replacement rule $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}\rightarrow \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}+\text{i}kV{\it\epsilon}$ . The tiny addition to the ship’s ‘frequency’, $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}$ , can be neglected except where it moves the pole slightly off the integration path, to complex values of $\boldsymbol{k}$ .

The resulting expression for the surface elevation is now well defined and reads

This result accords perfectly with that of Havelock (Reference Havelock1922) when $S=0$ . It is quite possible to use this expression directly for numerical purposes, keeping ${\it\epsilon}$ small but finite, as was done by Moisy & Rabaud (Reference Moisy and Rabaud2014). The effect of ${\it\epsilon}$ is to attenuate the waves slightly away from the boat. In § 2.4 we shall apply the Cauchy integral theorem as well as path of steepest descent techniques to obtain an expression for the far field only.

2.2 Wave resistance

The theory for wave resistance on a travelling pressure disturbance was laid out long ago in a series of papers by Havelock (Reference Havelock1917, Reference Havelock1919, Reference Havelock1922). This and other analytical models, as well as experimental results available at the time, were famously reviewed by Wehausen (Reference Wehausen1973). The effect of elongation of an ellipsoidal Gaussian pressure distribution was recently investigated by Benzaquen et al. (Reference Benzaquen, Darmon and Raphaël2014). The effect of shear upon wave resistance has never been considered before to our knowledge.

Havelock’s idea for calculation of wave resistance was to identify it as the horizontal component of the applied pressure force acting on the surface, in the direction of ship motion whereby the wave resistance may be found by an integral over the whole water surface:

where $\boldsymbol{{\rm\nabla}}_{{\it\xi}}=(\partial /\partial {\it\xi}_{x},\partial /\partial {\it\xi}_{y})$ . Inserting (2.8), we recognise the complex conjugate of $p_{ext}$ and find

The limiting expression in deep water is again given by letting $\coth kh\rightarrow 1$ .

As shown by Ellingsen (Reference Ellingsen2014b ) the presence of a shear current beneath the boat will result in an asymmetric wake when the angle ${\it\beta}$ is not $0$ or ${\rm\pi}$ . This, in turn, will give a lateral force on the wave source, normal to the direction of motion. In a slight misuse of terminology we term it the ‘lateral wave resistance’ $R_{\bot }$ , and it is found in analogous fashion

A positive value of $R_{\bot }$ means a lateral force directed towards the right (starboard) with respect to the ship’s direction of motion. As for $R$ we find

2.3 Dispersion relation

Much of the physics of any wave problem may be discerned from analysis of the dispersion relation. As discussed in Ellingsen (Reference Ellingsen2014b ) (and numerous previous expositions without shear current present), the pole in the integrand of (2.8) – that is, the zero of the denominator – corresponds to values of $\boldsymbol{k}$ which simultaneously satisfy the dispersion relation and the condition of stationariness,

ensuring wave crests which look stationary as seen from the moving source. $c(\boldsymbol{k})$ is the phase velocity of a plane wave with wave vector $\boldsymbol{k}$ .

Letting the denominator of (2.8) (say) equal zero and solving with respect to $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}$ immediately gives two solutions for the phase velocity,

where $\boldsymbol{k}=(k\cos {\it\theta},k\sin {\it\theta})$ . As discussed by Ellingsen (Reference Ellingsen2014a ), to each wave vector $\boldsymbol{k}$ there are two associated phase velocities. One of these phase velocities is positive, the other negative, and given the sign of $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}$ , the appropriate solution is chosen.

For later reference let us denote the angle between $\boldsymbol{k}$ and $\boldsymbol{V}$ as ${\it\gamma}={\it\theta}-{\it\beta}$ , so that

The constituents of the Fourier integral are plane waves with ‘frequency’ $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}$ which is negative when $|{\it\gamma}|>{\rm\pi}/2$ , and a wave of negative frequency and wave vector $\boldsymbol{k}$ has phase velocity in the direction of $-\boldsymbol{k}$ . Thus the integral in (2.8) (say) obtains two identical pole contributions, one from a wave of phase velocity $c_{+}$ whose wave vector has a forward component ( $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}>0$ ) and one of phase velocity $c_{-}$ whose wave vector has a rearward component ( $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}<0$ ). Both of these plane waves appear stationary as seen by the ship and give identical contributions.

Figure 2.

Definitions of the different angles used in the analysis.

While the coordinate system defined in figure 1 is easiest for the sake of the preceding derivations, for further analysis we shall want to use a polar system relative to the ship’s motion, in the manner of Ellingsen (Reference Ellingsen2014b ). We define

Using ${\it\gamma}$ and ${\it\phi}_{{\it\beta}}$ instead of ${\it\theta}$ and ${\it\phi}$ corresponds to rotating the coordinate system so that the boat moves along a new $x$ axis while the current, in general, does not. Definitions of the various angles used in the analysis are summarised in figure 2.

2.4 Near-field and far-field contributions

We wish now to extract the far-field contribution to the ship waves. The analysis is carried out in a detailed and careful manner, thereby exposing weaknesses in a more cavalier method used in the recent literature, including Darmon et al. (Reference Darmon, Benzaquen and Raphaël2014) and Ellingsen (Reference Ellingsen2014b ). In order not to clutter the reading overly with mathematics, some calculations are found in appendices.

We will re-write the expression for the surface elevation (2.8) with a Gaussian disturbance in a different form which is suitable for further analysis. First, let us write the integral over $\boldsymbol{k}$ in polar form with dimensionless quantities,

where we have defined

and the functions (for frequent reference below)

Our system is described by four non-dimensional parameters: $Fr,Fr_{\mathit{s}},Fr_{h}$ and ${\it\beta}$ , where the three Froude numbers are

Here $Fr$ is based on the size of the source (the ship), $Fr_{\mathit{s}}$ is based on a ‘shear depth’ $g/S^{2}$ which is half the depth at which the dynamic and hydrostatic pressure of the shear flow are equal, and $Fr_{h}$ is based on the finite water depth.

We will consider the integral $I({\it\gamma})$ by forming a closed contour in the complex $K$ plane. The contour is formed of the positive real $K$ axis (the original integration path) and closed either in the upper or lower plane, depending on the exponent function $E(K,{\it\gamma})$ so that the resulting path gives a finite and well defined integral. To wit we shall choose the path of steepest descent (cf. e.g. Bender & Orszag Reference Bender and Orszag1991, § 6.6),

This path is parallel to the real $K$ axis and lies either above or below the latter depending on the sign of $\cos ({\it\gamma}-{\it\phi}_{{\it\beta}})$ . We connect it to the original path of integration while noticing that ${\it\Gamma}(K,{\it\gamma})$ has a series of zeros along the imaginary axis which we wish to avoid, and therefore choose the connection path (arbitrarily) at $45^{\circ }$ to the real axis. The path is closed at infinity where the integrand is exponentially zero. The closed path of integration, ${\it\Lambda}$ , is shown in figure 3.

Figure 3.

Contours of integration: the original integration path along the positive real $K$ axis is closed with a steepest descent path and a connection path at an angle with both axes. The position of the pole is indicated, its position relative to the real $K$ axis depending on the sign of the function $F(K_{0},{\it\gamma})$ . (a) $\cos ({\it\gamma}-{\it\phi}_{{\it\beta}})>0$ , (b)  $\cos ({\it\gamma}-{\it\phi}_{{\it\beta}})<0$ .

Now we define $K_{0}({\it\gamma})$ as the (real) value of $K$ for which ${\it\Gamma}(K,{\it\gamma})$ has a zero, i.e. it is implicitly defined by

Only in the deep water limit $H\rightarrow \infty$ is the expression for $K_{0}$ explicit. For a given set of parameters, it is not certain that a positive solution of (2.25) exists. Note that the trivial solution $K_{0}=0$ is not a pole of the integrand since it is cancelled by a factor $K$ in the numerator. The existence or non-existence of a positive solution $K_{0}({\it\gamma})$ is related to the question of a critical velocity, and is discussed in § 3. There, an approximate, explicit solution for $K_{0}({\it\gamma})$ is also given.

Importantly, when $K_{0}$ is inserted for $K$ , the exponent function $E(K_{0},{\it\gamma})$ does not depend on the source size $b$ , i.e. on the Froude number $Fr$ . This is important because the pole at $K=K_{0}$ gives far-field waves (to be shown below), and the exponent function $E(K_{0},{\it\gamma})$ is what determines the width of the Kelvin wake. Hence, just as was for deep water (Ellingsen Reference Ellingsen2014b ) (and is well-known to be the case without shear), the source Froude number has no influence on the Kelvin angle, although it strongly affects the apparent wake angle at which the waves have the largest amplitude (see, e.g. Darmon et al. Reference Darmon, Benzaquen and Raphaël2014; Noblesse et al. Reference Noblesse, He, Zhu, Hong, Zhang, Zhu and Yang2014).

We now proceed to solve the contour integral $I({\it\gamma})$ . We show in § A.2 that the integrals $I_{s.d.}$ and $I_{conn.}$ make up the near field of the wake, i.e. a surface deformation following the source which falls off quickly as $X$ increases. The all-important far field of the wave pattern is given by the contribution from the pole, provided it is found inside the contour. This should be no surprise, since exactly this decomposition has been reported numerous times in the literature (e.g. Lighthill Reference Lighthill1978) for ship waves as well as other wave systems.

Evaluating the contribution from the pole (details may be found in § A.1) we obtain the following expression for the far field of the wave pattern

Here ${\it\Theta}$ is the Heaviside step function.

Crucially, once the far field is identified as the contribution to $I({\it\gamma})$ from the pole, it implies that a wave of propagation direction ${\it\gamma}$ contributes to the far field in the real space direction ${\it\phi}$ if and only if the pole where ${\it\Gamma}+\text{i}{\it\epsilon}{\it\Psi}=0$ lies inside the contour ${\it\Lambda}$ . Much of the below analysis rests upon this insight, and we shall see that the two different ways in which the pole can fall outside the contour, corresponding to the two Heaviside functions in (2.26), each have their different physical interpretations.

Figure 4.

Illustration of wake waves calculated using the expressions for (a) the full expression in (2.8), (b) the far-field expression in (2.26), and (c) only the near field. In (a–c) $Fr=Fr_{\mathit{s}}=0.8$ , $Fr_{h}=0$ and ${\it\beta}={\rm\pi}/2$ . Arrows indicate direction of shear flow as defined in figure 1.

The separation into near field and far field are shown in figure 4. One may note that the far field as calculated with the Cauchy integral theorem gives a butterfly like surface deformation near the source, which is particularly visible with strong side-on shear such as shown in figure 4. The artefact is not a worry since the far-field expression is only accurate far from the source.

The requirement that the pole most lie inside the closed contour in order to contribute led to the Heaviside factor ${\it\Theta}[-\!\cos ({\it\gamma}-{\it\phi}_{{\it\beta}})\cos {\it\gamma}]$ in (2.26). A careful inspection of the angles involved reveals that this factor restricts the contribution to the far field to including only the partial waves whose direction of propagation has a positive component towards the ship. In other words, the waves in the far field always follow the ship and are not allowed to run ahead of it. This accords well with what one must expect, but we note that arguments based on the group velocity, say, would be complicated since in the presence of shear the phase and group velocities do not in general have the same direction.

The radiation condition used here automatically allows for the perhaps surprising result that with side-on shear near the critical velocity, waves can in fact be seen in front of the moving ship on one side (see § 4 and Ellingsen (Reference Ellingsen2014b )). This is not in violation of the radiation condition because these waves have been sufficiently refracted by the shear to still be able to follow the moving source according to the above definition.

Note finally that, beyond being a generalization, the far-field expression in (2.26) in fact differs slightly from those reported in Ellingsen (Reference Ellingsen2014b ), Darmon et al. (Reference Darmon, Benzaquen and Raphaël2014), Benzaquen et al. (Reference Benzaquen, Darmon and Raphaël2014). In these references the Sokhotsky–Plemelj formula was used to extract the far-field contribution from an expression similar, but not identical, to (2.8). We explain in appendix B why the use of this theorem is treacherous; it yields a far-field expression which but for a factor $2$ is identical to ours in the limit $X\rightarrow \infty$ behind the ship, but which contains a spurious wake also in front of the ship which is equal but of opposite sign. Clearly such a far field does not satisfy the radiation condition at positions within the spurious Kelvin wedge in front of the ship, the reason for which appendix B elucidates. When Sokhotsky–Plemelj is used, the spurious wake must be manually removed from the far-field expression, for instance by simply not plotting it. This is easy and clear cut in deep waters with no shear current, when the wave train is famously contained within a Kelvin wedge with half-angle $19^{\circ }28^{\prime }$ , but not so straightforward when the wake angle grows large or even extends beyond  $90^{\circ }$ . Moreover, Ellingsen (Reference Ellingsen2014b ) and Darmon et al. (Reference Darmon, Benzaquen and Raphaël2014) perform integration over ${\it\gamma}$ only from $-{\rm\pi}/2$ to ${\rm\pi}/2$ , resulting in a further factor $2$ difference and a far-field expression which is an overall factor $4$ smaller than our (2.26). This does not affect any of the main conclusions in these references.

3.1 Condition of criticality

We will now derive and discuss the critical velocity in the present case in which both finite depth (parameterised though $Fr_{h}$ ) and shear current (parameterised through $Fr_{\mathit{s}}$ ) are present. Either of the two on its own will give rise to a finite critical velocity, and combining the two naturally yields a critical velocity which depends on both $Fr_{h}$ and $Fr_{\mathit{s}}$ , as well as the angle ${\it\beta}$ between ship motion and shear current.

We shall describe a situation given by parameter triplet $Fr_{h},Fr_{\mathit{s}},{\it\beta}$ as supercritical if a finite sector of ${\it\gamma}$ values exists wherein the criterion (3.4) is false. It is slightly easier mathematically to work with the rearranged condition, equivalent to (3.4),

(we presume $\cos {\it\gamma}\neq 0$ ). The situation is supercritical if

where the notation means the minimum of the left-hand side with respect to ${\it\gamma}$ is taken. The minimum is found at $\tan {\it\gamma}=-(Fr_{\mathit{s}}\sin {\it\beta})/2$ , which we reinsert and conclude that the situation is supercritical if

The sub and supercritical regions of the $Fr_{\mathit{s}}$ – $Fr_{h}$ parameter plane are shown in figure 5 for some values of ${\it\beta}$ .

Figure 5.

Critical lines as function of $Fr_{\mathit{s}}$ and $Fr_{h}$ for different values of ${\it\beta}$ . The shaded regions below the critical lines are subcritical.

Figure 6.

Kelvin angles for constant $Fr_{\mathit{s}}=0.5$ (a), constant $Fr_{h}^{2}=0.75$ (b) and constant $Fr_{h}^{2}=1.25$ (c). The case $Fr_{\mathit{s}}=0$ is also shown in (a) for comparison. See also figure 8.

Inserting the definitions of $Fr_{\mathit{s}}$ and $Fr_{h}$ , (3.8) can be solved with respect to $V$ to obtain the critical velocity given in (3.1). The known limits when $S=0$ (no shear current) when $V_{\mathit{crit}}=\sqrt{gh}$ (Havelock Reference Havelock1908), and as $h\rightarrow \infty$ when $V_{\mathit{crit}}=S/g\cos ^{2}({\it\beta}/2)$ (Ellingsen Reference Ellingsen2014b ), are regained. The latter limit may be seen easily if one notes that (3.8) can instead be written

We plot $Fr_{crit}$ in figure 7.

Figure 7.

$Fr_{crit}$ as a function of $h$ and $Fr_{Sb}$ for different values of ${\it\beta}$ . (a) ${\it\beta}=0$ ;(b) ${\it\beta}={\rm\pi}/4$ ; (c) ${\it\beta}={\rm\pi}/2$ ; (d) ${\it\beta}=3{\rm\pi}/4$ ; (e) ${\it\beta}={\rm\pi}$ .

5.1 Lateral wave ‘resistance’

Now consider the lateral ‘wave resistance’ $R_{\bot }$ which is calculated exactly like $R$ was above, using (2.14) while noting that $(\mathbf{e}_{z}\times \boldsymbol{V})\boldsymbol{\cdot }\boldsymbol{k}=kV\sin {\it\gamma}$ ,

The lateral force is plotted in figure 13 for the same values of $H$ as used in figure 12. Just like the standard wave resistance, the lateral component also tends to pull the source in the direction where the wave field is strongest, i.e. downstream (as seen from the system where the undisturbed water surface is at rest). The lateral force behaves more or less like the wave resistance as a function of parameters, but is smaller in magnitude. Its maximum value is when the current is approximately side on ( ${\it\beta}\sim {\rm\pi}/2$ ), in which case its value can amount to about 10–20 % of the value of the corresponding no-shear wave resistance when the shear is strong ( $Fr_{Sb}\sim 1$ ).