3.1 Intrinsic group and phase velocities

In the following we will discuss in detail solutions of the dispersion relation in different directions of plane-wave propagation. The discussion follows the same lines as that in § 3.7.1 of Mei, Stiassnie & Yue (Reference Mei, Stiassnie and Yue2005).

The intrinsic group velocity of a plane wave of wave vector $\boldsymbol{k}$ is given by the vectorial quantity

(3.3) $$\begin{eqnarray}\boldsymbol{c}_{g}(\boldsymbol{k})=\unicode[STIX]{x1D735}_{k}\unicode[STIX]{x1D70E}_{\pm }(\boldsymbol{k}),\end{eqnarray}$$

where $\unicode[STIX]{x1D735}_{k}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}k_{x},\unicode[STIX]{x2202}/\unicode[STIX]{x2202}k_{y})$ is the gradient operator in the $\boldsymbol{k}$ plane and $\unicode[STIX]{x1D70E}_{\pm }$ is defined in (2.4b ). In particular, $\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{\pm }(\boldsymbol{k})/\unicode[STIX]{x2202}k$ is the component of $\boldsymbol{c}_{g}$ along the direction $\boldsymbol{k}$ , which we term the radial component. The intrinsic frequencies and velocities are independent of the motion of the source and are velocities measured in the ‘lab’ frame of reference, i.e. where the fluid surface is at rest. Relative velocities are measured relative to the moving source. The relative group velocity is

(3.4) $$\begin{eqnarray}\boldsymbol{c}_{g}^{R}(\boldsymbol{k})=\boldsymbol{c}_{g}(\boldsymbol{k})-\boldsymbol{V}=\unicode[STIX]{x1D735}_{k}(\unicode[STIX]{x1D70E}_{\pm }-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}).\end{eqnarray}$$

Likewise, the intrinsic phase velocities are $\unicode[STIX]{x1D70E}_{\pm }(\boldsymbol{k})/k$ , one of which positive, the other negative. A negative intrinsic phase velocity for a wave component $\boldsymbol{k}$ means that wave phase is transported in the direction $-\boldsymbol{k}$ . On the other hand, the relative phase velocity of a wave $\boldsymbol{k}$ is $\unicode[STIX]{x1D70E}_{\pm }(\boldsymbol{k})/k-V\cos \unicode[STIX]{x1D6FE}$ , which, by virtue of the dispersion relation (3.1), equals $\unicode[STIX]{x1D714}_{0}/k$ and is necessarily positive; hence, the relative phase velocity is always directed along $\boldsymbol{k}$ . The relative group velocity, however, can have a negative component along the direction $\boldsymbol{k}$ .

In non-dimensional terms, the intrinsic and relative group velocities are

(3.5a,b ) $$\begin{eqnarray}\boldsymbol{C}_{g}(K,\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D735}_{K}\unicode[STIX]{x1D6F4}_{\pm }(\boldsymbol{K}),\quad \boldsymbol{C}_{g}^{R}(K,\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D735}_{K}(\unicode[STIX]{x1D6F4}_{\pm }(\boldsymbol{K})-K\mathit{Fr}\cos \unicode[STIX]{x1D6FE}).\end{eqnarray}$$

Especially the radial component, $\unicode[STIX]{x2202}_{K}\unicode[STIX]{x1D6F4}_{\pm }$ , will be important in the following discussions.

3.2 Dispersion relation in finite water depth

In the following, we consider graphical solutions of the dispersion relation $\unicode[STIX]{x1D6FA}_{\pm }(\boldsymbol{K})=0$ . In our theory, physical quantities are given by Fourier integrals over the whole $\boldsymbol{k}$ plane, such as (2.2), and far from the source only the waves satisfying the dispersion relation will be present (cf. e.g. Lighthill Reference Lighthill1978). A shear current introduces a richness of solutions, which we find it necessary to discuss carefully and in some detail.

We shall first consider the case of finite depth, which, despite being formally a little more cumbersome, is more straightforward in principle than the deep-water case.

In each case we proceed by solving the dispersion relation (3.2) graphically by considering intersections between the straight lines $\unicode[STIX]{x1D6FA}_{0}+K\mathit{Fr}\cos \unicode[STIX]{x1D6FE}$ and the curves $\unicode[STIX]{x1D6F4}_{\pm }(K,\unicode[STIX]{x1D6FE})$ in different circumstances. Each intersection corresponds to a possible far-field wave, and the graphs in figures 3–5 afford some immediate physical insights: considering an intersection $P$ (say) for one particular propagation direction $\unicode[STIX]{x1D6FE}$ , the (non-dimensional) intrinsic phase velocity of the corresponding far-field wave is the slope of the straight line connecting the point $P$ to the origin, the radial component of the intrinsic group velocity is the slope of the tangent of the curve $\unicode[STIX]{x1D6F4}_{\pm }$ at $P$ , and the velocity of source motion projected onto the direction $\boldsymbol{k}$ is the slope of the straight line connecting $P$ to the point ( $0,\unicode[STIX]{x1D6FA}_{0}$ ).

Figure 3.

Graphical solutions of (3.2) when $V=0$ .

3.2.2 Moving source, $V>0$

A far richer situation ensues once the source is in motion relative to the water surface, $\mathit{Fr}>0$ . The situation is sketched in figure 4(a). Which far-field waves now exist for different propagation angles can vary strongly as a function of propagation direction $\unicode[STIX]{x1D6FE}$ . The figure is to be understood in a qualitative sense since the curves $\unicode[STIX]{x1D6F4}_{\pm }(K,\unicode[STIX]{x1D703})$ also depend on $\unicode[STIX]{x1D6FE}$ when the direction of motion, $\unicode[STIX]{x1D6FD}$ , is fixed, but since the sign of the curvature of these graphs remains the same for all $\unicode[STIX]{x1D703}$ (see figure 3), the sketch is sufficient to visualize in a qualitative way the possible cases that occur.

Figure 4.

(a) Graphical solutions of (3.2) in different situations. See text for discussion and details. (b) Behaviour of $\unicode[STIX]{x1D6F4}_{+}(K)$ as $H$ tends to infinity, when $\cos \unicode[STIX]{x1D703}<0$ is assumed ( $\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}=-2$ in the figure). Values of $H$ shown as numbers on respective graphs.

Figure 5.

Graphical solutions of dispersion relation (3.2) in infinite water depth. Intrinsic frequencies $\unicode[STIX]{x1D6F4}_{\pm }(K,\unicode[STIX]{x1D6FE})$ are the curved lines, $\unicode[STIX]{x1D6FA}_{0}+K\mathit{Fr}\cos \unicode[STIX]{x1D6FE}$ are the straight lines. Intersections of the two are wave solutions, marked with circles. The point $(0,-\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703})$ is marked with a triangle. The panels show situations $0<-\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}<\unicode[STIX]{x1D6FA}_{0}$  (a), $-\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}<0$ (b), $0<\unicode[STIX]{x1D6FA}_{0}<-\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}$ (c).

Situation 1. Orthogonal waves ( $\cos \unicode[STIX]{x1D6FE}=0$ ). The source velocity projected onto $\boldsymbol{k}$ is now zero, and the situation is as for $V=0$ . There is always a single solution to the dispersion relation, point  $A$ in figure 4(a).

Situation 2. Sternward waves ( $\cos \unicode[STIX]{x1D6FE}<0$ ). Since $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}<0$ , we denote this situation somewhat roughly as having ‘rearward-pointing’ wave vectors. There are two solutions corresponding to the points  $B$ and  $C$ in figure 4(a). When comparing to situation 1 ( $\mathit{Fr}\cos \unicode[STIX]{x1D6FE}=0$ at point  $A$ ), the wave represented by $B$ is lengthened since $K_{B}<K_{A}$ , and intrinsic phase and group velocities are increased, and a second solution $C$ also occurs with short wavelength and smaller (absolute) velocity. The slope of the straight line intersecting $B$ and $C$ is negative, indicating that source velocity $\boldsymbol{V}$ has a negative component along direction $\boldsymbol{k}$ . Wave $B$ thus travels rapidly in rearward directions. The wave corresponding to point $C$ has negative intrinsic phase and group velocities, meaning that although it is emitted in a rearward direction by the source, it is seen in the lab system to propagate in a forward direction. A $C$ wave of wave vector $\boldsymbol{k}$ has intrinsic phase velocity along direction $-\boldsymbol{k}$ .

Situation 3. Forward waves ( $\cos \unicode[STIX]{x1D6FE}>0$ ). This situation corresponds to ‘forward-directed’ wave vectors, and is the most complicated situation. The pertinent solutions are $\unicode[STIX]{x1D6F4}_{0+}=\unicode[STIX]{x1D6FA}_{0}+\mathit{Fr}K\cos \unicode[STIX]{x1D6FE}$ , and there are now three different subcases as illustrated in figure 4(a). When $\mathit{Fr}\cos \unicode[STIX]{x1D6FE}$ is sufficiently large, no waves exist (Case 1 in figure 4). This situation can occur provided the non-dimensional frequency parameter $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D6FA}_{0}\mathit{Fr}$ , defined in (1.1), exceeds a critical value $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ , the smallest Doppler resonant frequency, to be discussed in § 4.5.1. At a critical value of $\mathit{Fr}\cos \unicode[STIX]{x1D6FE}$ , only a single wave $F$ of wavenumber $K_{F}$ exists (Case 2), corresponding to a double root. The critical values $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ where this occurs can be found by noting that radial group velocity equals projected source velocity at this point:

(3.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{+}}{\unicode[STIX]{x2202}K}(K_{F},\unicode[STIX]{x1D6FE}_{\mathit{excl}})=\mathit{Fr}\cos \unicode[STIX]{x1D6FE}_{\mathit{excl}},\end{eqnarray}$$

where the notation means that the derivative is evaluated at point $(K_{F},\unicode[STIX]{x1D6FE}_{\mathit{excl}})$ . For supercritical values of $\unicode[STIX]{x1D70F}$ , there exists at least one sector $\unicode[STIX]{x1D6FE}_{\mathit{excl}}^{-}<\unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D6FE}_{\mathit{excl}}^{+}$ where Case 1 occurs, although as we shall detail in the case of deep water, as many as three such exclusion sectors may exist, depending on $\unicode[STIX]{x1D6FD}$ and $\mathit{Fr}_{s}$ .

For $\cos \unicode[STIX]{x1D6FE}$ smaller than $\cos \unicode[STIX]{x1D6FE}_{\mathit{excl}}$ , there are two possible solutions corresponding to points  $D$ and  $E$ , denoted Case 3 in figure 4. When $\unicode[STIX]{x1D70F}<\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ , this is the only possible case for $\cos \unicode[STIX]{x1D6FE}>0$ . Point  $D$ corresponds to the faster and longer of the two waves, and since its radial group velocity is greater than the source velocity projected onto the same direction, $\boldsymbol{k}$ , this wave is the only one which might be found in front of the moving source. Wave $E$ has shorter wavelength and moves more slowly, and though propagating in a forward direction, is left behind by the moving source.

In figure 4(b), we illustrate the behaviour of the intrinsic frequency $\unicode[STIX]{x1D6F4}_{+}$ as the water depth $H$ increases; in the figure $\unicode[STIX]{x1D6F4}_{+}(K,\unicode[STIX]{x1D703})$ is plotted as a function of $K$ , presuming $\cos \unicode[STIX]{x1D703}<0$ (for $\cos \unicode[STIX]{x1D703}>0$ the situation is mirrored about the abscissa). As $H\rightarrow \infty$ , the graph of $\unicode[STIX]{x1D6F4}_{+}(K)$ obtains the shape of a straight vertical line from the origin to the point $-\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}$ , thence following a curved shape concave towards the abscissa. This behaviour, leading to the phenomenon of cut-off as discussed in Ellingsen & Tyvand (Reference Ellingsen and Tyvand2016a ,Reference Ellingsen and Tyvand b ) and Tyvand & Lepperød (Reference Tyvand and Lepperød2015), becomes important when next considering the deep-water case.

3.3 Dispersion relation in infinite water depth

When assuming water depth to be infinite, the situation becomes at one time both simpler, in that explicit solutions to the dispersion relation may now be found, and more complicated. Concerning the graphical solutions of the dispersion relation, equation (3.2), the curve of $\unicode[STIX]{x1D6F4}_{+}$ or $\unicode[STIX]{x1D6F4}_{-}$ obtains a vertical section from the origin to the value $-\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}$ as illustrated in figure 4(b). The graphical solution situation for infinite water depth is shown schematically in figure 5. We will distinguish between what we term weak and strong shear situations.

Weak shear. When $\unicode[STIX]{x1D6FA}_{0}>\mathit{Fr}_{\mathit{sb}}$ or, equivalently, $\unicode[STIX]{x1D714}_{0}>S$ , the discussion of which far-field waves occur is qualitatively identical to that for finite water depth given above. Only the situations shown in figures 5(a) and 5(b) can occur in this case.

Strong shear. When $\unicode[STIX]{x1D714}_{0}<S$ , however, the appearance of the vertical section of the graph of $\unicode[STIX]{x1D6F4}_{+}$ (along which $K=0$ ) when $\cos \unicode[STIX]{x1D703}<0$ means that a new situation will arise, not found in finite water depth. In this, strong-shear case, a sector of angles $\unicode[STIX]{x1D703}$ exists centred at $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ , within which waves of type $A,B,$ and $D$ all have wavenumber $K\rightarrow 0$ as $H\rightarrow \infty$ . The situation within this sector is depicted in figure 5(c). This is the phenomenon of ‘cut-off’ discussed for the 2D case in Tyvand & Lepperød (Reference Tyvand and Lepperød2014, Reference Tyvand and Lepperød2015) and Ellingsen & Tyvand (Reference Ellingsen and Tyvand2016a ) and briefly in 3D with $V=0$ in Ellingsen & Tyvand (Reference Ellingsen and Tyvand2016b ). It is shown in Ellingsen & Tyvand (Reference Ellingsen and Tyvand2016a ) that these $K=0$ modes carry no energy and may simply be disregarded in the far field, and we will not consider them to be solutions. Cut-off of $A,B,$ and $D$ waves occurs in a sector $\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}_{0}<\unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}+\unicode[STIX]{x1D703}_{0}$ , where

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{0}=\arccos (\unicode[STIX]{x1D714}_{0}/S).\end{eqnarray}$$

The dispersion relation at infinite water depth may be found by taking $KH\rightarrow \infty$ in (3.2),

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{0}+K\mathit{Fr}\cos \unicode[STIX]{x1D6FE}=\pm \sqrt{K+\left({\textstyle \frac{1}{2}}\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}\right)^{2}}-{\textstyle \frac{1}{2}}\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703},\end{eqnarray}$$

which is valid for $K$ strictly greater than $0$ . Unlike for finite depth, this dispersion relation permits the analytical solutions

(3.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle K_{C,E}={\displaystyle \frac{1-\mathit{Fr}_{s}\cos \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D6FE}-2\unicode[STIX]{x1D70F}\cos \unicode[STIX]{x1D6FE}+\sqrt{\unicode[STIX]{x1D6E5}}}{2\mathit{Fr}^{2}\cos ^{2}\unicode[STIX]{x1D6FE}}}, & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle K_{B,D}={\displaystyle \frac{1-\mathit{Fr}_{s}\cos \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D6FE}-2\unicode[STIX]{x1D70F}\cos \unicode[STIX]{x1D6FE}-\sqrt{\unicode[STIX]{x1D6E5}}}{2\mathit{Fr}^{2}\cos ^{2}\unicode[STIX]{x1D6FE}}}, & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}=(1-\mathit{Fr}_{s}\cos \unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703})^{2}-4\unicode[STIX]{x1D70F}\cos \unicode[STIX]{x1D6FE},\end{eqnarray}$$

which agree with Wehausen & Laitone (Reference Wehausen, Laitone and Flügge1960) and Dagan & Miloh (Reference Dagan and Miloh1982) without shear current. Subscripts $B,C,D,$ and $E$ refer to labels on graphical solutions in figure 5; waves $B,C$ are obtained if $\cos \unicode[STIX]{x1D6FE}<0$ , waves $D,E$ if $\cos \unicode[STIX]{x1D6FE}>0$ . We shall make use of these solutions towards analysing Doppler resonances in § 4.

Figure 6.

Examples showing different far-field waves occurring in different sectors of the $\unicode[STIX]{x1D6FE}$ plane for $\unicode[STIX]{x1D6FD}=0$ (a,d), $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/2$ (b,e), and $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}$ (c,f). Above (a–c): $\unicode[STIX]{x1D70F}<\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ and $\unicode[STIX]{x1D714}_{0}<S$ . A cut-off sector of total angle $2\unicode[STIX]{x1D703}_{0}$ appears within which there are no $A,B,$ or $D$ waves. Below (d–f): $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ and $\unicode[STIX]{x1D714}_{0}>S$ . For $\unicode[STIX]{x1D70F}$ above the Doppler resonance, no waves exist within a sector of forward directions. In all figures the source’s direction of motion, $\unicode[STIX]{x1D6FE}=0$ , is towards the right. Labels within each sector refer to graphical solutions of the dispersion relation as shown in figure 5.

In summary, which and how many far-field waves are found when the propagation angle $\unicode[STIX]{x1D6FE}$ (or  $\unicode[STIX]{x1D703}$ ) is varied between $-\unicode[STIX]{x03C0}$ and $\unicode[STIX]{x03C0}$ is now determined by two criteria: whether or not $\unicode[STIX]{x1D70F}$ exceeds the smallest Doppler resonance frequency $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ , and whether or not $\unicode[STIX]{x1D714}_{0}$ exceeds $S$ . With reference to figure 5, we show some examples in figure 6 of which far-field waves appear in which sectors of the $\boldsymbol{k}$ plane when $\unicode[STIX]{x1D6FE}$ is varied through a full circle. In figure 6(a–c), we assume that $\unicode[STIX]{x1D714}_{0}<S$ and $\unicode[STIX]{x1D70F}<\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ , so there is a cut-off sector symmetrical about $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ . In figure 6(d–f), we assume that $\unicode[STIX]{x1D714}_{0}>S$ and $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ . Now there is no cut-off, but a sector $\unicode[STIX]{x1D6FE}_{\mathit{excl}}^{-}<\unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D6FE}_{\mathit{excl}}^{+}$ appears within which no propagating wave solutions exist since $\unicode[STIX]{x1D6E5}<0$ in (3.9).

4.1 Excluded sectors

Before studying the phenomenon of Doppler resonance, we will regard sectors of propagation directions $\unicode[STIX]{x1D6FE}$ in which $D$ and $E$ waves become evanescent and do not appear in the far field. In deep water it is clear to see from the explicit solutions (3.9) that this situation is characterized by $\unicode[STIX]{x1D6E5}<0$ , that is,

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D70F}>\frac{[1-\mathit{Fr}_{s}\cos (\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D6FD})\cos \unicode[STIX]{x1D6FE}]^{2}}{4\cos \unicode[STIX]{x1D6FE}}\equiv \unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE}).\end{eqnarray}$$

The phenomenon is well known also without shear, in which case exclusion of a single sector $|\unicode[STIX]{x1D6FE}|<\arccos (1/4\unicode[STIX]{x1D70F})$ occurs whenever $\unicode[STIX]{x1D70F}>1/4$ . In the presence of shear, however, there can be as many as three separate excluded sectors. Figure 7(a) shows the no-shear case with a single excluded sector. For moderate values of $\mathit{Fr}_{s}$ , two excluded regions can occur, as illustrated in figure 7(b). Note that the excluded sectors do not include $\unicode[STIX]{x1D6FE}=0$ , so although $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ , the source still has a wave travelling ahead of it. For large values of $\mathit{Fr}_{s}$ , it is even possible for a third sector to appear, as shown in figure 7(c). The values of $\mathit{Fr}_{s}$ and $\unicode[STIX]{x1D70F}$ involved are so large that this case is of doubtful practical significance.

4.2 Criterion of the Doppler resonance

Physically, a Doppler resonance refers to the case where wave energy is held stationary in space from the perspective of the moving source, i.e. zero relative group velocity. We obtain $\boldsymbol{C}_{g}^{R}$ in a polar $\boldsymbol{k}$ -coordinate system from (3.5a,b ) as

(4.2) $$\begin{eqnarray}\displaystyle \boldsymbol{C}_{g}^{R} & = & \displaystyle \left[\pm \frac{1}{2}(K+\mathit{Fr}_{sb}^{2}\cos ^{2}\unicode[STIX]{x1D703}/4)^{-1/2}-\mathit{Fr}\cos \unicode[STIX]{x1D6FE}\right]\boldsymbol{e}_{k}\nonumber\\ \displaystyle & & \displaystyle +\left[\mathit{Fr}\sin \unicode[STIX]{x1D6FE}+{\displaystyle \frac{\mathit{Fr}_{sb}\sin \unicode[STIX]{x1D703}}{2K}}\mp \frac{1}{8K}\mathit{Fr}_{sb}^{2}(K+\mathit{Fr}_{sb}^{2}\cos ^{2}\unicode[STIX]{x1D703}/4)^{-1/2}\sin 2\unicode[STIX]{x1D703}\right]\boldsymbol{e}_{\unicode[STIX]{x1D703}}.\end{eqnarray}$$

The Doppler resonance occurs when $\boldsymbol{C}_{g}^{R}=0$ , which, after eliminating $K$ , may be written as

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{\prime }(\unicode[STIX]{x1D6FE})={\displaystyle \frac{(1-\mathit{Fr}_{s}\cos \unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703})[(1-\mathit{Fr}_{s}\cos \unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703})\tan \unicode[STIX]{x1D6FE}+2\mathit{Fr}_{s}\sin (\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D703})]}{4\cos ^{2}\unicode[STIX]{x1D6FE}}}=0,\end{eqnarray}$$

where $-\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D6FE}<\unicode[STIX]{x03C0}/2$ is assumed. For each $\unicode[STIX]{x1D6FE}$ solving (4.3), there exists a resonant (not necessarily distinct) value of $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ . We will see that this criterion is identical to the condition for infinite amplitudes to be possible, namely that $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6E5}^{\prime }(\unicode[STIX]{x1D6FE})=0$ .

Figure 7.

Sectors of propagation directions $\unicode[STIX]{x1D6FE}$ with no wave solutions. The graphs are plots of $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})$ , the horizontal lines show the (arbitrarily chosen) value of $\unicode[STIX]{x1D70F}$ ; sectors where $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})<\unicode[STIX]{x1D70F}$ are excluded. Pie charts next to each graph illustrate excluded sectors (included sectors are shaded). We show in § 4.5.2 that a resonant value of $\unicode[STIX]{x1D70F}$ is associated with each local maximum or minimum of $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})$ ; if $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})$ has a local extremum at $\unicode[STIX]{x1D6FE}_{\mathit{Res}}$ , then $\unicode[STIX]{x1D70F}_{\mathit{Res}}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE}_{\mathit{Res}})$ . (a) The case without shear, $\mathit{Fr}_{s}=0$ , $\unicode[STIX]{x1D70F}=0.4$ . (b) Moderate case with two excluded sectors, $\mathit{Fr}_{s}=2,\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}$ , $\unicode[STIX]{x1D70F}=1.3$ . (c) Extreme case with three excluded sectors, $\mathit{Fr}_{s}=20$ , $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/2$ , $\unicode[STIX]{x1D70F}=28$ .

4.3 Wave amplitude of $F$ -type wave

We will now consider the wave amplitude due to waves of type $D$ or $E$ (see figure 5) when these flow together in a single point $F$ at some value $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ .

The surface elevation (2.2) is in the form of an integral over $\boldsymbol{k}$ . As is typical of wave descriptions with periodic or stationary time dependence, this integral is not well defined until a radiation condition has been applied. Using the procedure of e.g. § 4.9 of Lighthill (Reference Lighthill1978), we replace $\unicode[STIX]{x1D6FA}_{0}\rightarrow \unicode[STIX]{x1D6FA}_{0}+\text{i}\unicode[STIX]{x1D716}$ , where $\unicode[STIX]{x1D716}=0^{+}$ , i.e.  $\unicode[STIX]{x1D6FA}_{0}$ is given a small positive imaginary part, so that

(4.4) $$\begin{eqnarray}\text{e}^{-\text{i}\unicode[STIX]{x1D6FA}_{0}T}\longrightarrow \text{e}^{-\text{i}\unicode[STIX]{x1D6FA}_{0}T+\unicode[STIX]{x1D716}T}.\end{eqnarray}$$

The procedure is closely related to that for ship waves in Li & Ellingsen (Reference Li and Ellingsen2016), where more a detailed discussion may be found. The introduction of the small quantity $\unicode[STIX]{x1D716}$ now moves the poles where $\unicode[STIX]{x1D6FA}_{\pm }=0$ slightly into the complex $K$ plane, and the integral is well defined.

Henceforth, let us assume infinite water depth to allow more explicit analysis. Using partial fractions, (2.2) may be written in the form

(4.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}(\boldsymbol{R},T)/b=\text{e}^{-\text{i}\unicode[STIX]{x1D6FA}_{0}T}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D6FE}[I_{+}(\unicode[STIX]{x1D6FE})-I_{-}(\unicode[STIX]{x1D6FE})], & \displaystyle\end{eqnarray}$$

(4.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle I_{\pm }(\unicode[STIX]{x1D6FE})=\lim _{\unicode[STIX]{x1D716}\rightarrow 0}\int _{0}^{\infty }\,\text{d}K\frac{f(\boldsymbol{K})\text{e}^{\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}}}{\unicode[STIX]{x1D6FA}_{\pm }(K,\unicode[STIX]{x1D6FE})+\text{i}\unicode[STIX]{x1D716}}, & \displaystyle\end{eqnarray}$$

(4.5c ) $$\begin{eqnarray}\displaystyle & \displaystyle f(\boldsymbol{K})=\frac{P_{\mathit{ext}}(\boldsymbol{K})K^{2}}{2\sqrt{K+\left({\textstyle \frac{1}{2}}\mathit{Fr}_{\mathit{sb}}\cos \unicode[STIX]{x1D703}\right)^{2}}}. & \displaystyle\end{eqnarray}$$

$K$

It is well established (e.g. Lighthill (Reference Lighthill1978) and Li & Ellingsen (Reference Li and Ellingsen2016)) that the leading order contribution to $\unicode[STIX]{x1D701}$ far from the source comes from the contribution to the $\boldsymbol{K}$ integral from the poles where $\unicode[STIX]{x1D6FA}_{\pm }(K,\unicode[STIX]{x1D6FE})=0$ . We call this contribution the far field, and it consists of the waves of type $A$ to $E$ as shown in figure 5 for deep water.

We assume that $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ , and consider $\unicode[STIX]{x1D6FE}$ close to (but just outside) a sector wherein $\unicode[STIX]{x1D6E5}<0$ , as illustrated in figure 7. For simplicity we assume that there is a single such sector delimited by $\unicode[STIX]{x1D6FE}_{\mathit{excl}}^{\pm }$ . In this sector, waves of type $D$ and $E$ become evanescent and do not contribute to the far field. Since $\unicode[STIX]{x1D6FE}$ is close to some $\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ , far-field waves of type $D$ and $E$ in figure 5 have almost, but not exactly, the same wavenumber, and the quantity $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FE})$ from (3.10) is small, but positive. We want the leading order contributions to the far-field integrals as $\unicode[STIX]{x1D6FE}$ approaches $\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ from the side where $D,E$ waves exist. The $D$ and $E$ waves are forward-propagating ( $\cos \unicode[STIX]{x1D6FE}>0$ ), so far-field waves are solutions of $\unicode[STIX]{x1D6FA}_{+}(K,\unicode[STIX]{x1D6FE})=0$ . Since the wave $D$ has group velocity greater than $V\cos \unicode[STIX]{x1D6FE}$ , this wave is found in the far field in front of the source, $\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}>0$ , while the $E$ wave is found behind, so for a single position $\boldsymbol{R}$ in the far field only one of these waves can ever contribute, and the two waves do not interfere with each other (mathematically, the poles corresponding to $D$ and $E$ waves lie on opposite sides of the real $K$  axis).

We consider a $D$ wave (wavenumber $K_{D}$ ) for definiteness. The far-field $D$ -wave surface elevation is now found as

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{\mathit{f.f.}}^{D}/b=\text{e}^{-\text{i}\unicode[STIX]{x1D6FA}_{0}T}\left[\int _{\unicode[STIX]{x1D6FE}_{0}}^{\unicode[STIX]{x1D6FE}_{\mathit{excl}}^{-}}+\int _{\unicode[STIX]{x1D6FE}_{\mathit{excl}}^{+}}^{\unicode[STIX]{x1D6FE}_{1}}\right]\text{d}\unicode[STIX]{x1D6FE}I_{+}^{D}(\unicode[STIX]{x1D6FE}),\end{eqnarray}$$

where $I_{+}^{D}$ is now approximated by the contribution from the pole near $K=K_{D}$ only. The limits $\unicode[STIX]{x1D6FE}_{0}$ and $\unicode[STIX]{x1D6FE}_{1}$ are non-singular and can give a finite contribution only.

Assume that the pole is simple so that

(4.7) $$\begin{eqnarray}\text{Res}_{K=K_{D}}\frac{f(\boldsymbol{K})\text{e}^{\text{i}\boldsymbol{K}\boldsymbol{\cdot }\boldsymbol{R}}}{\unicode[STIX]{x1D6FA}_{+}(K,\unicode[STIX]{x1D6FE})}=\frac{f[K_{D}(\unicode[STIX]{x1D6FE}),\unicode[STIX]{x1D6FE}]\text{e}^{\text{i}K_{D}(\unicode[STIX]{x1D6FE})R\cos (\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FD}})}}{\unicode[STIX]{x1D6FA}_{+}^{\prime }[K_{D}(\unicode[STIX]{x1D6FE}),\unicode[STIX]{x1D6FE}]},\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FD}}$ is the angle between $\boldsymbol{R}$ and $\boldsymbol{V}$ and a prime denotes $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}K$ . Define $K_{F}(\unicode[STIX]{x1D6FE})$ as the value of $K$ so that $\unicode[STIX]{x1D6FA}^{\prime }(K_{F},\unicode[STIX]{x1D6FE})=0$ , found by assuming that $\unicode[STIX]{x1D6E5}=0$ in (3.9),

(4.8) $$\begin{eqnarray}K_{F}(\unicode[STIX]{x1D6FE})=\frac{1-(\mathit{Fr}_{s}\cos \unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703})^{2}}{4\mathit{Fr}^{2}\cos ^{2}\unicode[STIX]{x1D6FE}},\end{eqnarray}$$

which also solves $\unicode[STIX]{x1D6FA}_{+}=0$ if $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ . Then $\unicode[STIX]{x1D6FA}_{+}^{\prime }(K_{D},\unicode[STIX]{x1D6FE})\approx (K_{D}-K_{F})\unicode[STIX]{x1D6FA}_{+}^{\prime \prime }(K_{F},\unicode[STIX]{x1D6FE})$ , and evaluating the contribution from the residue of the pole at $K=K_{D}$ we obtain

(4.9) $$\begin{eqnarray}I_{+}^{D}(\unicode[STIX]{x1D6FE})\approx 2\unicode[STIX]{x03C0}\text{i}\frac{f(K_{F})\text{e}^{K_{F}R\cos (\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FD}})}}{(K_{D}-K_{F})\unicode[STIX]{x1D6FA}_{+}^{\prime \prime }(K_{F},\unicode[STIX]{x1D6FE})}\end{eqnarray}$$

when $\unicode[STIX]{x1D6FE}$ is close to $\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ .

From (3.9) and (3.10), we find that

(4.10) $$\begin{eqnarray}K_{D}-K_{F}=-\frac{\sqrt{\unicode[STIX]{x1D6E5}}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6E5}}{2\mathit{Fr}^{2}\cos ^{2}\unicode[STIX]{x1D6FE}}\approx -\frac{\sqrt{\unicode[STIX]{x1D6E5}}}{2\mathit{Fr}^{2}\cos ^{2}\unicode[STIX]{x1D6FE}},\end{eqnarray}$$

since $\unicode[STIX]{x1D6E5}$ tends to zero as $\unicode[STIX]{x1D6FE}\rightarrow \unicode[STIX]{x1D6FE}_{\mathit{excl}}$ . From $\unicode[STIX]{x1D6FA}_{+}^{\prime }(K_{F},\unicode[STIX]{x1D6FE})=0$ , we obtain $\unicode[STIX]{x1D6FA}_{+}^{\prime \prime }(K_{F},\unicode[STIX]{x1D6FE})=2(\mathit{Fr}\cos \unicode[STIX]{x1D6FE})^{3}$ , yielding

(4.11) $$\begin{eqnarray}I_{+}^{D}(\unicode[STIX]{x1D6FE})\approx -2\unicode[STIX]{x03C0}\text{i}\frac{f(K_{F})\text{e}^{K_{F}R\cos (\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FD}})}}{\mathit{Fr}\cos \unicode[STIX]{x1D6FE}\sqrt{\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FE})}}\quad \text{for}~\unicode[STIX]{x1D6FE}\rightarrow \unicode[STIX]{x1D6FE}_{\mathit{excl}}.\end{eqnarray}$$

We know that $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FE}_{\mathit{excl}})=0$ . Provided the root of $\unicode[STIX]{x1D6E5}$ at $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ is single, the singularity at $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ is of order $(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{\mathit{excl}})^{-1/2}$ and is integrable. The wave amplitude thus remains finite as long as $\unicode[STIX]{x1D6E5}$ has a simple root at $\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ .

Infinite amplitudes are possible when $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FE}_{\mathit{excl}})=\unicode[STIX]{x1D6E5}^{\prime }(\unicode[STIX]{x1D6FE}_{\mathit{excl}})=0$ , which is to say that $\unicode[STIX]{x1D6E5}$ has a double root at $\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ . A little algebra reveals that this exactly matches the criterion (4.3) for a Doppler resonance to exist. This situation occurs for some value of $\unicode[STIX]{x1D6FE}$ when $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ , yet, in the presence of shear, other Doppler resonances can occur as well.

4.4 Diverging amplitude at resonance

Let us consider the leading order contribution to the wave amplitude from (as an example) a wave of type $D$ when $\unicode[STIX]{x1D70F}$ is a very small, but non-zero, distance from $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ . When $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{\mathit{Res}}$ , $\unicode[STIX]{x1D6E5}$ has a double root at $\unicode[STIX]{x1D6FE}=\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}}$ . At the actual value $\unicode[STIX]{x1D70F}$ , $\unicode[STIX]{x1D6E5}$ has a simple pole at $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{excl}}$ , and we consider $\unicode[STIX]{x1D70F}\rightarrow \unicode[STIX]{x1D70F}_{\mathit{Res}}$ and hence $\unicode[STIX]{x1D6FE}_{\mathit{excl}}\rightarrow \tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}}$ .

Considering $\unicode[STIX]{x1D6E5}$ as a function of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D70F}$ , we may Taylor expand,

(4.12) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}\,\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D70F}}(\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}},\unicode[STIX]{x1D70F}_{\mathit{Res}})+{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE})^{2}\,\unicode[STIX]{x1D6E5}^{\prime \prime }(\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}},\unicode[STIX]{x1D70F}_{\mathit{Res}})+\cdots ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}-\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}}$ , $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{\mathit{Res}}$ , and we used that $\unicode[STIX]{x1D6E5}(\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}},\unicode[STIX]{x1D70F}_{\mathit{Res}})=\unicode[STIX]{x1D6E5}^{\prime }(\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}},\unicode[STIX]{x1D70F}_{\mathit{Res}})=0$ and neglected subleading orders. A prime denotes differentiation with respect to $\unicode[STIX]{x1D6FE}$ , a subscript $\unicode[STIX]{x1D70F}$ differentiation with respect to  $\unicode[STIX]{x1D70F}$ . Inserting this into (4.6) and (4.11) gives the leading order contribution to the far-field wave as

(4.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D701}_{\mathit{f.f.}}^{D}}{b}\propto \int ^{\unicode[STIX]{x1D6FE}_{\mathit{excl}}}\frac{\text{d}\unicode[STIX]{x1D6FE}}{\sqrt{(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{\mathit{excl }})^{2}+2{\displaystyle \frac{\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D70F}}(\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl }},\unicode[STIX]{x1D70F}_{\mathit{Res }})}{\unicode[STIX]{x1D6E5}^{\prime \prime }(\tilde{\unicode[STIX]{x1D6FE}}_{\mathit{excl}},\unicode[STIX]{x1D70F}_{\mathit{Res}})}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}}}\propto \ln (\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F})+\cdots .\end{eqnarray}$$

We have shown that the Doppler resonance gives a logarithmically diverging wave amplitude, which is in agreement with the findings of Dagan & Miloh (Reference Dagan, Miloh and Inui1980, Reference Dagan and Miloh1982).

4.5 Position of resonance frequencies

In the following, we determine the resonant values of $\unicode[STIX]{x1D70F}$ for different values of $\unicode[STIX]{x1D6FD}$ and  $\mathit{Fr}_{s}$ .

4.5.1 The minimal resonance $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$

The smallest non-dimensional resonance frequency $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ is the smallest value of $\unicode[STIX]{x1D70F}$ so that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})$ (see (4.1)) has a solution, i.e.

(4.14) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}=\min _{\unicode[STIX]{x1D6FE}}\{\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})\}.\end{eqnarray}$$

The notation denotes that the minimum value is found with respect to $\unicode[STIX]{x1D6FE}$ in the sector $(-\unicode[STIX]{x03C0}/2,\unicode[STIX]{x03C0}/2)$ .

We shall see in the following that while $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ is the smallest value at which infinite wave amplitudes may occur, it is not necessarily the only value. When shear current is not present, the well-known resonance found in the classical literature (Wehausen & Laitone Reference Wehausen, Laitone and Flügge1960) is obtained, i.e.  $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}=1/4$ , which is then the only resonance.

Based on (4.14), figure 8 presents the smallest resonance frequency $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ for various $\mathit{Fr}_{s}$ and $\unicode[STIX]{x1D6FD}$ . The resonance frequency reaches its peak value at $\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0}$ for given $\mathit{Fr}_{s}$ , and its minimum for $\unicode[STIX]{x1D6FD}=0$ . Moreover, the shear vorticity represented by $\mathit{Fr}_{s}$ tends to decrease the resonance frequency for $-\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D6FD}<\unicode[STIX]{x03C0}/2$ . Notably, $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ tends to zero for increasing $\mathit{Fr}_{s}$ when $-\unicode[STIX]{x03C0}/2<\unicode[STIX]{x1D6FD}<\unicode[STIX]{x03C0}/2$ , an observation which could well have implications for the heave and pitch of marine vessels in the presence of shear current, since resonance (often corresponding to sudden increase in wave loads) could occur at a much lower frequency than on still water or uniform current.

Figure 8.

Smallest dimensionless resonance frequency $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ as a function of $\mathit{Fr}_{s}$ and $\unicode[STIX]{x1D6FD}$ , (a) 3D plot and (b) contour plot. The shaded region satisfies (4.18), and here $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}=0$ .

Resonance conditions in 2D were worked out by Tyvand & Lepperød (Reference Tyvand and Lepperød2015) for shear-assisted motion (corresponding to $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}$ in 3D), in which case they obtained

(4.15) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}=(1+\mathit{Fr}_{s})^{2}/4.\end{eqnarray}$$

The 2D geometry with the source travelling in the shear-inhibited direction ( $\unicode[STIX]{x1D6FD}=0$ in 3D) was not considered by Tyvand & Lepperød (Reference Tyvand and Lepperød2015).

We find that the 2D result (4.15) is also a resonance of a 3D source moving in the direction $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}$ , but it is not necessarily the smallest one. An explicit expression for $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ in 3D is not available in general, but may be found from (4.14) when $\unicode[STIX]{x1D6FD}=0,\pm \unicode[STIX]{x03C0}$ :

(4.16a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}(\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0})=\left\{\begin{array}{@{}ll@{}}{\textstyle \frac{1}{4}}(1+\mathit{Fr}_{s})^{2}, & 0\leqslant \mathit{Fr}_{s}\leqslant {\textstyle \frac{1}{3}},\\ {\textstyle \frac{4}{3}}\sqrt{{\textstyle \frac{1}{3}}\mathit{Fr}_{s}}, & \mathit{Fr}_{s}>{\textstyle \frac{1}{3}},\end{array}\right. & \displaystyle\end{eqnarray}$$

(4.16b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}(\unicode[STIX]{x1D6FD}=0)=\left\{\begin{array}{@{}ll@{}}{\textstyle \frac{1}{4}}(1-\mathit{Fr}_{s})^{2}, & 0\leqslant \mathit{Fr}_{s}\leqslant 1,\\ 0, & \mathit{Fr}_{s}>1.\end{array}\right. & \displaystyle\end{eqnarray}$$

The smallest resonance frequency for $\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0}$ is smaller than the 2D result (4.15) when $\mathit{Fr}_{s}>1/3$ . The reason is that for $\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0}$ , $\mathit{Fr}_{s}>1/3$ the resonance condition $\unicode[STIX]{x1D6E5}=0$ is first satisfied for a partial wave in directions

(4.17) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\mathit{Res}}=\pm \arctan \sqrt{3\mathit{Fr }_{s}-1}\end{eqnarray}$$

rather than $\unicode[STIX]{x1D6FE}=0$ , as illustrated in figure 7(b).

We plot the resonance frequencies as a function of $\mathit{Fr}_{s}$ for different values of $\unicode[STIX]{x1D6FD}$ in figure 9 for moderate values of $\mathit{Fr}_{s}$ . Some higher-than-minimal values of $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ also appear in the figure, to be discussed in the next section. Note that the highest branch of resonant $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ values at $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}$ (marked with a circle in the figure) is the 2D result from Tyvand & Lepperød (Reference Tyvand and Lepperød2015), and that two different resonant values can occur quite close to each other in this case.

Figure 9.

Resonance frequencies $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ in different directions of motion $\unicode[STIX]{x1D6FD}$ for moderate values of $\mathit{Fr}_{s}$ . The highest branch of resonant values for $\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0}$ (marked with a circle) corresponds to the 2D result of Tyvand & Lepperød (Reference Tyvand and Lepperød2015).

Except for the special case $\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0}$ , the resonance frequency $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ will always drop to zero for sufficiently high values of $\mathit{Fr}_{s}$ , because the numerator of (4.14) will be zero for some $\unicode[STIX]{x1D6FE}$ . A bit of algebra shows that this is the case, and $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}=0$ , provided

(4.18) $$\begin{eqnarray}\mathit{Fr}_{s}\geqslant 1/\cos ^{2}(\unicode[STIX]{x1D6FD}/2).\end{eqnarray}$$

This is exactly the criterion found for critical velocity to occur for ship waves (i.e.  $\unicode[STIX]{x1D714}_{0}=0$ ) in deep water by Ellingsen (Reference Ellingsen2014b ). In figure 8(b), this region is shaded. As indicated by the $\unicode[STIX]{x1D6FD}=0$ graph in figure 9, when $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ hits zero when $\mathit{Fr}_{s}$ increases, it stays zero for all higher values, but a second branch of resonances also appears with increasing $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ as a function of  $\mathit{Fr}_{s}$ .

We finally note that for $\mathit{Fr}_{s}\ll 1$ , $\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}$ behaves like

(4.19) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\mathit{Res},\mathit{min}}\sim {\textstyle \frac{1}{4}}\left[1-2\mathit{Fr}_{s}\cos \unicode[STIX]{x1D6FD}+{\textstyle \frac{1}{2}}\mathit{Fr}_{s}^{2}(3\cos 2\unicode[STIX]{x1D6FD}-1)+\cdots \right].\end{eqnarray}$$

Thus, the minimal resonance changes linearly as a function of $\mathit{Fr}_{s}$ for small shear, except when $\boldsymbol{V}$ is orthogonal to the shear current ( $\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0}/2$ ) when the behaviour is quadratic.

4.5.2 Additional resonances

Figure 10.

Solutions of $\unicode[STIX]{x1D6F7}^{\prime }(\unicode[STIX]{x1D6FE})=0$ from (4.3). Shaded areas indicate $\unicode[STIX]{x1D6F7}^{\prime }(\unicode[STIX]{x1D6FE})>0$ . To each solution of $\unicode[STIX]{x1D6F7}^{\prime }(\unicode[STIX]{x1D6FE})=0$ (at $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{Res}}$ , say) for a given $\mathit{Fr}_{s}$ there exists a corresponding (not necessarily distinct) resonant $\unicode[STIX]{x1D70F}$ value, $\unicode[STIX]{x1D70F}_{\mathit{Res}}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE}_{\mathit{Res}})$ , as plotted in figure 11.

Figure 11.

Doppler resonant frequencies $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ as a function of $\mathit{Fr}_{s}$ for different directions of motion, $\unicode[STIX]{x1D6FD}$ . Where two panels show the same $\unicode[STIX]{x1D6FD}$ ((b,c), (d,e), (f,g)), the first is a zoom of behaviour for small and moderate $\mathit{Fr}_{s}$ while the second shows the full picture appearing at large $\mathit{Fr}_{s}$ .

As found in § 4.2, the criterion for a Doppler resonance to exist is $\unicode[STIX]{x1D6F7}^{\prime }(\unicode[STIX]{x1D6FE})=0$ when $|\unicode[STIX]{x1D6FE}|<\unicode[STIX]{x03C0}/2$ . In other words, for some value of $\mathit{Fr}_{s}$ , there is a resonant (but not necessarily distinct) value $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})$ associated with each local maximum or minimum of $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})$ ; if $\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE})$ has a local extremum at $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{\mathit{Res}}$ , then $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE}_{\mathit{Res}})$ is a resonant value. In the absence of shear, resonance can only occur at $\unicode[STIX]{x1D6FE}=0$ . Figure 10 shows that the introduction of a shear current results in a much richer resonance situation, where group velocity can vanish for different  $\unicode[STIX]{x1D6FE}$ .

By mapping the zeros of $\unicode[STIX]{x1D6F7}^{\prime }(\unicode[STIX]{x1D6FE})$ onto the $\unicode[STIX]{x1D70F}$ axis by requiring $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FE}_{\mathit{Res}},\unicode[STIX]{x1D70F}_{\mathit{Res}})=0$ , we plot resonance frequencies $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ as a function of $\mathit{Fr}_{s}$ for different directions of motion, $\unicode[STIX]{x1D6FD}$ , in figure 11 (compare also with figure 9). Panels (a,b,d,f,h) show the structure of resonances for low to moderate $\mathit{Fr}_{s}$ . At larger values of $\mathit{Fr}_{s}$ , a more complex picture emerges, as shown clearly e.g. in figure 11(e). For $\mathit{Fr}_{s}\gtrsim 4.2$ , four different resonant values of $\unicode[STIX]{x1D70F}$ can be identified (one of which is zero).

Figure 12.

Doppler resonant frequencies $\unicode[STIX]{x1D70F}_{\mathit{Res}}$ as a function of $\mathit{Fr}_{s}$ as $\unicode[STIX]{x1D6FD}$ approaches $\unicode[STIX]{x03C0}$ (or  $-\unicode[STIX]{x03C0}$ ).

Of particular interest is the situation for directions close to $\unicode[STIX]{x1D6FD}=\pm \unicode[STIX]{x03C0}$ , as shown in figure 12. In figure 12, we plot the structure of resonances for directions close to this maximally shear-inhibited direction. As $\unicode[STIX]{x1D6FD}\rightarrow \unicode[STIX]{x03C0}$ , in the example shown, a pair of higher resonances begins to appear at ever lower $\mathit{Fr}_{s}$ , the smaller of which eventually merges with the smallest resonance with the branch point reaching $\mathit{Fr}_{s}=1/3$ .