4.1 Waves within the Kelvin wedge $|\unicode[STIX]{x1D6FE}|<\unicode[STIX]{x1D6FE}_{0}$

When the azimuthal angle of the flow-field point $|\unicode[STIX]{x1D6FE}|$ is less than the Kelvin angle  $\unicode[STIX]{x1D6FE}_{0}$ , the phase function is purely real and thus Kelvin’s method of stationary phase (Wong Reference Wong1989) can be applied. Then, expression (4.7) mainly stems from the contribution of stationary points $\unicode[STIX]{x1D705}_{-}$ and  $\unicode[STIX]{x1D705}_{+}$ , and it is simplified to

(4.11) $$\begin{eqnarray}\displaystyle G^{W} & {\approx} & \displaystyle -4H(x)\sqrt{\frac{2\unicode[STIX]{x03C0}}{R|\unicode[STIX]{x1D713}_{-}^{\prime \prime }|}}\text{Im}\left\{{\mathcal{A}}_{-}\exp \left[\text{i}R\unicode[STIX]{x1D713}_{-}+\text{sgn}(\unicode[STIX]{x1D713}_{-}^{\prime \prime })\frac{\unicode[STIX]{x03C0}}{4}\right]\right\}\nonumber\\ \displaystyle & & \displaystyle -\,4H(x)\sqrt{\frac{2\unicode[STIX]{x03C0}}{R|\unicode[STIX]{x1D713}_{+}^{\prime \prime }|}}\text{Im}\left\{{\mathcal{A}}_{+}\exp \left[\text{i}R\unicode[STIX]{x1D713}_{+}+\text{sgn}(\unicode[STIX]{x1D713}_{+}^{\prime \prime })\frac{\unicode[STIX]{x03C0}}{4}\right]\right\},\end{eqnarray}$$

where ‘prime’ means differentiation with respect to the variable  $\unicode[STIX]{x1D705}$ , $\unicode[STIX]{x1D713}_{\pm }=\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705}_{\pm },\unicode[STIX]{x1D6FE})$ , $\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }=\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705}_{\pm },\unicode[STIX]{x1D6FE})/\unicode[STIX]{x2202}\unicode[STIX]{x1D705}^{2}$ , ${\mathcal{A}}_{\pm }={\mathcal{A}}(\unicode[STIX]{x1D705}_{\pm })$ and the second derivative of the phase function is given by

(4.12) $$\begin{eqnarray}\unicode[STIX]{x1D713}^{\prime \prime }(\unicode[STIX]{x1D705},\unicode[STIX]{x1D6FE})=\frac{-2\unicode[STIX]{x1D705}^{3}\sin \unicode[STIX]{x1D6FE}-3\unicode[STIX]{x1D705}\sin \unicode[STIX]{x1D6FE}+\cos \unicode[STIX]{x1D6FE}}{(1+\unicode[STIX]{x1D705}^{2})^{3/2}}.\end{eqnarray}$$

At stationary points $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{\pm }$ , expression (4.12) is written as (Ursell Reference Ursell1960):

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }=\mp \frac{4}{\sqrt{3}}\frac{Q\sin \unicode[STIX]{x1D6FE}}{\sqrt{(1\pm Q)(1\mp Q/3)}},\end{eqnarray}$$

where $Q=\sqrt{1-8\tan ^{2}\unicode[STIX]{x1D6FE}}$ . In (4.11), the first term is associated with transverse waves, and the second one is with divergent waves. In addition, $\unicode[STIX]{x1D713}_{-}^{\prime \prime }$ is greater than zero, whereas $\unicode[STIX]{x1D713}_{+}^{\prime \prime }$ is less than zero.

4.2 Behaviours near the singular axis

Within the pure gravity wave theory, the ship-wave Green function is singular when both source and flow-field points are located on the free surface $d=0$ and the flow-field point approaches the track of the disturbance $y=0$ (Ursell Reference Ursell1960; Clarisse & Newman Reference Clarisse and Newman1994; Chen & Wu Reference Chen and Wu2001). The singular behaviour stems from the divergent waves corresponding to the open dispersion curve leading to $k\rightarrow \infty$ .

We consider a flow-field point in the downstream close to the track of the disturbance $y=\unicode[STIX]{x1D700}R>0$ with $\unicode[STIX]{x1D700}$ being a small parameter $\unicode[STIX]{x1D700}<1$ . This indicates that

(4.14a,b ) $$\begin{eqnarray}\sin \unicode[STIX]{x1D6FE}=\frac{y}{R}=\unicode[STIX]{x1D700}\quad \text{and}\quad \cos \unicode[STIX]{x1D6FE}=\frac{x}{R}\approx 1-\frac{\unicode[STIX]{x1D700}^{2}}{2}.\end{eqnarray}$$

Then, two stationary points given by (4.10) are approximately equal to

(4.15a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{-}=\unicode[STIX]{x1D700}+{\displaystyle \frac{5}{2}}\unicode[STIX]{x1D700}^{3}+O(\unicode[STIX]{x1D700}^{5})\quad \text{and}\quad \unicode[STIX]{x1D705}_{+}=\frac{1}{2\unicode[STIX]{x1D700}}-\frac{5}{4}\unicode[STIX]{x1D700}+O(\unicode[STIX]{x1D700}^{3}).\end{eqnarray}$$

Accordingly, the phase function and its second derivative at stationary points are given by

(4.16a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{-}=1-\unicode[STIX]{x1D700}^{2}+O(\unicode[STIX]{x1D700}^{4})\quad \text{and}\quad \unicode[STIX]{x1D713}_{+}=\frac{1}{4\unicode[STIX]{x1D700}}+\frac{\unicode[STIX]{x1D700}}{4}+O(\unicode[STIX]{x1D700}^{3}),\end{eqnarray}$$

(4.16c,d ) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{-}^{\prime \prime }\approx 1-5\unicode[STIX]{x1D700}^{2}+O(\unicode[STIX]{x1D700}^{4})\quad \text{and}\quad \unicode[STIX]{x1D713}_{+}^{\prime \prime }\approx -2\unicode[STIX]{x1D700}+8\unicode[STIX]{x1D700}^{3}+O(\unicode[STIX]{x1D700}^{5}).\end{eqnarray}$$

By introducing (4.15) and (4.16) into formulation (4.11), we have the expression for the asymptotic behaviour of the wave component near the singular axis

(4.17) $$\begin{eqnarray}\displaystyle G^{W} & {\approx} & \displaystyle -4H(x)\sqrt{\frac{2\unicode[STIX]{x03C0}}{R}}\exp [-4\unicode[STIX]{x1D716}x(1+\unicode[STIX]{x1D700}^{2})]\sin \left[R\left(1-\frac{\unicode[STIX]{x1D700}^{2}}{2}\right)+\frac{\unicode[STIX]{x03C0}}{4}\right]\nonumber\\ \displaystyle & & \displaystyle -\,4H(x)\sqrt{\frac{\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D700}R}}\exp \left[-\unicode[STIX]{x1D716}x\left(\frac{1}{8\unicode[STIX]{x1D700}^{4}}-\frac{5}{4}\right)\right]\sin \left(\frac{x^{2}+y^{2}}{4y}+\frac{y}{4}-\frac{\unicode[STIX]{x03C0}}{4}\right).\end{eqnarray}$$

Due to viscous effects, the wave component $G^{W}$ given by (4.17) at $d=0$ and $y\rightarrow 0$ is no longer singular. The divergent waves associated with large wavenumbers die away near the track ( $y\rightarrow 0$ ). Besides, waves along the track of the disturbance decay with the distance in the form of $\text{e}^{-4\unicode[STIX]{x1D716}x}/\sqrt{x}$ (the first term associated with transverse waves).

When viscous effects are ignored, the second expression in (4.17) associated with divergent waves is consistent with the asymptotic representation of the pure gravity ship-wave problem given by (Ursell Reference Ursell1960; Chen & Wu Reference Chen and Wu2001)

(4.18) $$\begin{eqnarray}G_{0}^{W}\approx -4H(x)\sqrt{\frac{\unicode[STIX]{x03C0}}{y}}\sin \left(\frac{x^{2}}{4y}+\frac{y}{2}-\frac{\unicode[STIX]{x03C0}}{4}\right),\end{eqnarray}$$

which is singular at the track of the disturbance $y=0$ , and this singular behaviour is embedded in divergent waves.

4.3 Waves at the cusp line $|\unicode[STIX]{x1D6FE}|=\unicode[STIX]{x1D6FE}_{0}$

As the flow-field point is located at the cusp line of the Kelvin wedge, two stationary points coalesce

(4.19) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{-}=\unicode[STIX]{x1D705}_{+}=\unicode[STIX]{x1D705}_{0}=\frac{\sqrt{2}}{2},\end{eqnarray}$$

and then we apply Havelock’s approximation (Havelock Reference Havelock1908) by expanding the phase function $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D6FE}_{0})$ as

(4.20) $$\begin{eqnarray}\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D6FE}_{0})\approx \unicode[STIX]{x1D713}(\unicode[STIX]{x1D705}_{0},\unicode[STIX]{x1D6FE}_{0})+{\textstyle \frac{1}{6}}\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }(\unicode[STIX]{x1D705}-\unicode[STIX]{x1D705}_{0})^{3}+\cdots ,\end{eqnarray}$$

where the third derivative of the phase function is

(4.21) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }=\unicode[STIX]{x1D713}^{\prime \prime \prime }(\unicode[STIX]{x1D705}_{0},\unicode[STIX]{x1D6FE}_{0})=\frac{-3\unicode[STIX]{x1D705}_{0}\cos \unicode[STIX]{x1D6FE}_{0}-3\sin \unicode[STIX]{x1D6FE}_{0}}{(1+\unicode[STIX]{x1D705}_{0}^{2})^{5/2}}.\end{eqnarray}$$

Then, the asymptotic expression along the cusp line is written as

(4.22) $$\begin{eqnarray}G^{W}\approx -4H(x)\text{Im}\left[\frac{\unicode[STIX]{x1D6E4}(1/3)}{\sqrt{3}}{\mathcal{A}}_{0}\exp (\text{i}R\unicode[STIX]{x1D713}_{0})\left(\frac{6}{R|\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }|}\right)^{1/3}\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D713}_{0}=\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705}_{0},\unicode[STIX]{x1D6FE}_{0})$ , ${\mathcal{A}}_{0}={\mathcal{A}}(\unicode[STIX]{x1D705}_{0})$ and $\unicode[STIX]{x1D6E4}(\cdot )$ denotes the gamma function (Abramowitz & Stegun Reference Abramowitz and Stegun1964).

4.4 Waves outside the Kelvin wedge $|\unicode[STIX]{x1D6FE}|>\unicode[STIX]{x1D6FE}_{0}$

When the flow-field point is located outside the Kelvin wedge $|\unicode[STIX]{x1D6FE}|>\unicode[STIX]{x1D6FE}_{0}$ , two stationary points turn to two complex saddle points

(4.23a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{-}=\frac{\cot \unicode[STIX]{x1D6FE}}{4}(1-\text{i}\sqrt{8\tan ^{2}\unicode[STIX]{x1D6FE}-1})\quad \text{and}\quad \unicode[STIX]{x1D705}_{+}=\frac{\cot \unicode[STIX]{x1D6FE}}{4}(1+\text{i}\sqrt{\tan ^{2}\unicode[STIX]{x1D6FE}-1}).\end{eqnarray}$$

Then, Peters’ approximation (Peters Reference Peters1949) based on the method of steepest descent is applied. In the present problem, $\unicode[STIX]{x1D713}_{-}^{\prime \prime }$ and $\unicode[STIX]{x1D713}_{+}^{\prime \prime }$ are a complex conjugate pair with $\text{Im}(\unicode[STIX]{x1D713}_{-}^{\prime \prime })>0$ and $\text{Im}(\unicode[STIX]{x1D713}_{+}^{\prime \prime })<0$ . By deforming the integration path, only the contribution of the saddle point $\unicode[STIX]{x1D705}_{-}$ matters. The phase function is written as

(4.24) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{-}^{\prime \prime }=|\unicode[STIX]{x1D713}_{-}^{\prime \prime }|\text{e}^{\text{i}\unicode[STIX]{x1D6FC}_{-}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{-}$ denote the argument of $\unicode[STIX]{x1D713}_{-}^{\prime \prime }$ which is complex when $|\unicode[STIX]{x1D6FE}|>\unicode[STIX]{x1D6FE}_{0}$ . Then, the approximation to the wave component $G^{W}$ outside the Kelvin wedge is written as

(4.25) $$\begin{eqnarray}G^{W}\approx -4H(x)\text{Im}\left\{\sqrt{\frac{2\unicode[STIX]{x03C0}}{R|\unicode[STIX]{x1D713}_{-}^{\prime \prime }|}}{\mathcal{A}}_{-}\exp \left[\text{i}R\unicode[STIX]{x1D713}_{-}+\text{i}\left(\frac{\unicode[STIX]{x03C0}}{4}-\frac{\unicode[STIX]{x1D6FC}_{-}}{2}\right)\right]\right\}.\end{eqnarray}$$

5.1 CFU approximation

The CFU approximation was originally proposed by Chester et al. (Reference Chester, Friedman and Ursell1957) and it has been applied by Ursell (Reference Ursell1960) and Dai (Reference Dai and Kashiwagi2014) to study the pure gravity and interfacial ship waves generated by a concentrated pressure point. As an elegant method uniformly valid in the fluid domain, the CFU method is straightforward in principle, but it is complicated in practice due to the ambiguity of the multiple-value problem (Wong Reference Wong1989). The CFU approximation is applied here to approximate the wave component (4.7), and the multiple-value problem, which is missing in Ursell (Reference Ursell1960), is discussed.

Following Chester et al. (Reference Chester, Friedman and Ursell1957), the phase function is represented by a cubic polynomial in  $u$

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D6FE})=-u^{3}/3+p(\unicode[STIX]{x1D6FE})u+q(\unicode[STIX]{x1D6FE}).\end{eqnarray}$$

Differentiating the phase function given by (5.1) with respect to $u$ yields

(5.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D6FE})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}=-u^{2}+p(\unicode[STIX]{x1D6FE}).\end{eqnarray}$$

The left-hand side vanishes when $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{\pm }$ , so does the right-hand side when $u=\pm \sqrt{p}$ . Here, $\unicode[STIX]{x1D705}$ and $u$ are in one-to-one correspondence. Therefore, $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{+}$ and $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{-}$ correspond to $u=+\sqrt{p}$ and $u=-\sqrt{p}$ , respectively. By substituting $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{\pm }$ and $u=\pm \sqrt{p}$ into (5.1), we can obtain $p$ and $q$ (Ursell Reference Ursell1960)

(5.3a ) $$\begin{eqnarray}\displaystyle p^{3/2} & = & \displaystyle \frac{3}{4}(\unicode[STIX]{x1D713}_{+}-\unicode[STIX]{x1D713}_{-})=\frac{3^{5/2}}{64}\frac{\cos ^{2}\unicode[STIX]{x1D6FE}}{\sin \unicode[STIX]{x1D6FE}}(\mathscr{Q}_{+}-\mathscr{Q}_{-}),\end{eqnarray}$$

(5.3b ) $$\begin{eqnarray}\displaystyle q & = & \displaystyle -\frac{1}{2}(\unicode[STIX]{x1D713}_{+}+\unicode[STIX]{x1D713}_{-})=-\frac{3^{3/2}}{32}\frac{\cos ^{2}\unicode[STIX]{x1D6FE}}{\sin \unicode[STIX]{x1D6FE}}(\mathscr{Q}_{+}+\mathscr{Q}_{-}),\quad\end{eqnarray}$$

$\mathscr{Q}_{+}=\sqrt{1+Q}(1-Q/3)^{3/2}$

$\mathscr{Q}_{-}=\sqrt{1-Q}(1+Q/3)^{3/2}$

$q$

$p$

$p$

$\unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D6FE}_{0}$

$p$

$\unicode[STIX]{x1D6FE}$

$\unicode[STIX]{x1D6FE}_{0}$

$\unicode[STIX]{x1D713}(\unicode[STIX]{x1D705},\unicode[STIX]{x1D6FE})$

$\unicode[STIX]{x1D713}_{-}$

$\unicode[STIX]{x1D713}_{+}$

$p$

$p$

$q$

$\unicode[STIX]{x1D700}_{-}=\unicode[STIX]{x1D6FE}_{0}-\unicode[STIX]{x1D6FE}\geqslant 0$

$Q$

(5.4) $$\begin{eqnarray}Q=\sqrt{1-8\tan ^{2}(\unicode[STIX]{x1D6FE}_{0}-\unicode[STIX]{x1D700}_{-})}\approx {\textstyle \frac{3}{2^{1/4}}}\cdot \unicode[STIX]{x1D700}_{-}^{1/2}-{\textstyle \frac{33}{16}}\cdot 2^{1/4}\cdot \unicode[STIX]{x1D700}_{-}^{3/2},\end{eqnarray}$$

and then $p$ and $q$ are written as

(5.5a ) $$\begin{eqnarray}\displaystyle \displaystyle p & = & \displaystyle \left[{\textstyle \frac{3^{5/2}}{64}}({\textstyle \frac{8}{3}}+{\textstyle \frac{20}{3}}\sqrt{2}\unicode[STIX]{x1D700}_{-}+\cdots )\left(4\cdot 2^{1/4}\cdot \unicode[STIX]{x1D700}_{-}^{3/2}-{\textstyle \frac{9}{4}}\cdot 2^{3/4}\cdot \unicode[STIX]{x1D700}_{-}^{5/2}+\cdots \right)\right]^{2/3}\nonumber\\ \displaystyle & {\approx} & \displaystyle {\textstyle \frac{3}{2^{1/2}}}\cdot \unicode[STIX]{x1D700}_{-}+O(\unicode[STIX]{x1D700}_{-}^{2}),\end{eqnarray}$$

(5.5b ) $$\begin{eqnarray}\displaystyle q=-{\textstyle \frac{3^{3/2}}{32}}({\textstyle \frac{8}{3}}+{\textstyle \frac{20}{3}}\sqrt{2}\unicode[STIX]{x1D700}_{-}+\cdots )(2-3\sqrt{2}\unicode[STIX]{x1D700}_{-}+\cdots )\approx -{\textstyle \frac{3^{1/2}}{2}}-{\textstyle \frac{3^{1/2}}{2^{1/2}}}\cdot \unicode[STIX]{x1D700}_{-}+O(\unicode[STIX]{x1D700}_{-}^{2}).\end{eqnarray}$$

Therefore, near the cusp line, $p$ and $q$ have the following asymptotic expansions:

(5.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle p\approx -{\textstyle \frac{3}{2^{1/2}}}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{0})+O[(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{0})^{2}], & \displaystyle\end{eqnarray}$$

(5.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle q\approx -{\textstyle \frac{3^{1/2}}{2}}+{\textstyle \frac{3^{1/2}}{2^{1/2}}}(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{0})+O[(\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{0})^{2}]. & \displaystyle\end{eqnarray}$$

When $\unicode[STIX]{x1D6FE}>\unicode[STIX]{x1D6FE}_{0}$ , $p$ must be consistent with the value inside and at the cusp, and hence $p$ can be uniquely determined. So, we consider $\unicode[STIX]{x1D700}_{+}=\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{0}>0$ , and then $p^{3/2}$ in (5.3a ) can be expanded into

(5.7) $$\begin{eqnarray}p^{3/2}\approx -{\textstyle \frac{3^{3/2}}{2^{3/4}}}\text{i}\unicode[STIX]{x1D700}_{+}^{3/2}+O(\unicode[STIX]{x1D700}_{+}^{5/2}).\end{eqnarray}$$

In (5.7), $-\text{i}$ can be expressed as $\exp (-\text{i}\unicode[STIX]{x03C0}/2+\text{i}\cdot 2n\unicode[STIX]{x03C0})$ with $n=0,1,2,\ldots$ leading to multiple values. Therefore, $(-\text{i})^{2/3}$ has three values. Nevertheless, only one of them is physically realistic. According to the asymptotic relation given by (5.6a ), $n$ should be taken to be 1. As a consequence, the expression of $p$ for $\unicode[STIX]{x1D6FE}>\unicode[STIX]{x1D6FE}_{0}$ is written as

(5.8) $$\begin{eqnarray}p=-\left\{\frac{3^{5/2}}{64}\frac{\cos ^{2}\unicode[STIX]{x1D6FE}}{\sin \unicode[STIX]{x1D6FE}}\text{Im}[\mathscr{Q}_{-}-\mathscr{Q}_{+}]\right\}^{2/3}.\end{eqnarray}$$

Then, the ultimate expression of $p$ is written as

(5.9) $$\begin{eqnarray}p=\left\{\begin{array}{@{}ll@{}}[3(\unicode[STIX]{x1D713}_{+}-\unicode[STIX]{x1D713}_{-})/4]^{2/3}\quad & \unicode[STIX]{x1D6FE}\leqslant \unicode[STIX]{x1D6FE}_{0},\\ -[3\text{Im}(\unicode[STIX]{x1D713}_{-}-\unicode[STIX]{x1D713}_{+})/4]^{2/3}\quad & \unicode[STIX]{x1D6FE}\geqslant \unicode[STIX]{x1D6FE}_{0}.\end{array}\right.\end{eqnarray}$$

To obtain the final asymptotic approximation, the amplitude function is represented by a Bleistein sequence (Wong Reference Wong1989)

(5.10) $$\begin{eqnarray}{\mathcal{A}}(\unicode[STIX]{x1D705})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}=a_{0}+b_{0}u+(u^{2}-p){\mathcal{H}},\end{eqnarray}$$

where the last term on the right-hand side is associated with the high-order term, and it is disregarded here. To calculate the coefficients $a_{0}$ and  $b_{0}$ , we enforce (5.10) at two saddle points $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{\pm }$ written as

(5.11) $$\begin{eqnarray}{\mathcal{A}}_{\pm }\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{\pm }={\mathcal{A}}(\unicode[STIX]{x1D705}_{\pm })\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{u=\pm \sqrt{p}}=a_{0}\pm \sqrt{p}b_{0}.\end{eqnarray}$$

Then, we can obtain the coefficient $a_{0}$ and $b_{0}$

(5.12a ) $$\begin{eqnarray}\displaystyle & \displaystyle a_{0}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{2}\left[{\mathcal{A}}_{-}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{-}+{\mathcal{A}}_{+}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{+}\right]\quad & \unicode[STIX]{x1D6FE}\neq \unicode[STIX]{x1D6FE}_{0},\\ \displaystyle {\mathcal{A}}_{0}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{u=0}\quad & \unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0},\end{array}\right. & \displaystyle\end{eqnarray}$$

(5.12b ) $$\begin{eqnarray}\displaystyle & \displaystyle b_{0}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{2\sqrt{p}}\left[{\mathcal{A}}_{+}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{+}-{\mathcal{A}}_{-}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{-}\right]\quad & \unicode[STIX]{x1D6FE}\neq \unicode[STIX]{x1D6FE}_{0},\\ \displaystyle {\mathcal{A}}_{0}^{\prime }\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{u=0}^{2}+{\mathcal{A}}_{0}\left(\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u^{2}}\right)_{u=0}\quad & \unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}.\end{array}\right. & \displaystyle\end{eqnarray}$$

To obtain the expression $b_{0}$ at $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}$ given by (5.12b ), the Bleistein sequence (5.10) is differentiated with respect to $u$ at $u=0$ . In (5.12), ${\mathcal{A}}(\unicode[STIX]{x1D705})$ is the amplitude function, which is already known. Then, the task is to calculate derivatives $\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}u$ and $\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}u^{2}$ . To calculate these derivatives in (5.12), we differentiate the phase function as well as the corresponding cubic polynomial given by (5.1) and enforcing the resultant expression at two saddle points $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{\pm }$ gives rise to

(5.13) $$\begin{eqnarray}\unicode[STIX]{x1D713}^{\prime \prime }\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)^{2}+\unicode[STIX]{x1D713}^{\prime }\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u^{2}}=-2u\quad \text{with}~\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{\pm }~\text{and}~u=\pm \sqrt{p}.\end{eqnarray}$$

Then, $\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}u$ at two saddle points is expressed as

(5.14) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{\pm }=\sqrt{\frac{\mp 2\sqrt{p}}{\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }}}=\sqrt{\frac{\sqrt{3}}{2}\frac{\sqrt{p}}{\sin \unicode[STIX]{x1D6FE}}Q^{-1}(1\pm Q)^{1/2}(1\mp Q/3)^{1/2}}.\end{eqnarray}$$

In this problem, $\unicode[STIX]{x1D713}_{-}^{\prime \prime }>0$ and $\unicode[STIX]{x1D713}_{+}^{\prime \prime }<0$ when $\unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D6FE}_{0}$ . Hence, expression (5.14) is real. When $u=0$ , we further differentiate (5.13) which yields

(5.15) $$\begin{eqnarray}\unicode[STIX]{x1D713}^{\prime \prime \prime }\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)^{3}+3\unicode[STIX]{x1D713}^{\prime \prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u^{2}}+\unicode[STIX]{x1D713}^{\prime }\frac{\unicode[STIX]{x2202}^{3}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u^{3}}=-2.\end{eqnarray}$$

At the cusp line $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}$ , we have $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{0}$ and $u=0$ as well as $\unicode[STIX]{x1D713}_{0}^{\prime }=\unicode[STIX]{x1D713}_{0}^{\prime \prime }=0$ . Then, $\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}u$ becomes,

(5.16) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{0}=\left(\frac{-2}{\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }}\right)^{1/3}=\sqrt{\frac{3}{2}}.\end{eqnarray}$$

When $\unicode[STIX]{x1D6FE}>\unicode[STIX]{x1D6FE}_{0}$ , there are two saddle points which are a complex conjugate pair, and the corresponding second derivatives are also a conjugate pair. So, $\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }$ can be written as

(5.17) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }=\unicode[STIX]{x1D6F9}^{\prime \prime }\exp (\pm \text{i}\unicode[STIX]{x1D6FC}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F9}^{\prime \prime }$ is real and positive, and $\unicode[STIX]{x1D6FC}$ is the argument of $\unicode[STIX]{x1D713}_{+}^{\prime \prime }$ . Then, expression $\text{d}\unicode[STIX]{x1D705}/\text{d}u$ for $\unicode[STIX]{x1D6FE}>\unicode[STIX]{x1D6FE}_{0}$ is written as

(5.18) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u}\right)_{\pm }=\sqrt{\frac{2\sqrt{-p}}{\unicode[STIX]{x1D6F9}^{\prime \prime }}}\exp \left\{\mp \text{i}\left[\frac{\unicode[STIX]{x03C0}}{4}+{m\atop n}\unicode[STIX]{x03C0}+\frac{\unicode[STIX]{x1D6FC}}{2}\right]\right\}\quad \text{with}~m,n=0,1,2,\ldots .\end{eqnarray}$$

Again, the multiple-value problem occurs. To determine it uniquely, we consider $\unicode[STIX]{x1D700}_{+}=\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D6FE}_{0}>0$ once again, and $\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }$ can be expanded as

(5.19) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }=\mp {\textstyle \frac{2^{7/4}}{3^{1/2}}}\text{i}\cdot \unicode[STIX]{x1D700}_{+}^{1/2}-{\textstyle \frac{2^{3/2}}{3^{1/2}}}\cdot \unicode[STIX]{x1D700}_{+}+O(\unicode[STIX]{x1D700}_{+}^{3/2}).\end{eqnarray}$$

To be consistent with expression (5.16) when $\unicode[STIX]{x1D700}_{+}$ approaches zero, $m$ and $n$ must be equal to zero, so that $\unicode[STIX]{x2202}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}u$ is uniquely determined. So far, $a_{0}$ can be determined, and no multiple-value problem is suffered from. To determine  $b_{0}$ , one needs to evaluate $\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}u^{2}$ . To achieve this goal, we further differentiate expression (5.15) with respect to  $u$ , and then $\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}/\unicode[STIX]{x2202}u^{2}$ at the cusp line is written as

(5.20) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D705}}{\unicode[STIX]{x2202}u^{2}}\right)_{u=0}=-\frac{\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime \prime }}{6\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }}\left(\frac{-2}{\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }}\right)^{2/3}=\frac{\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime \prime }}{(-54^{1/5}\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime })^{5/3}}.\end{eqnarray}$$

In Dai (Reference Dai and Kashiwagi2014), the power is wrongly written as $3/5$ . Once $a_{0}$ and $b_{0}$ are determined, the uniform approximation to the wave component can be expressed as (Chester et al. Reference Chester, Friedman and Ursell1957)

(5.21) $$\begin{eqnarray}G^{W}=-4\,\text{Im}\left\{2\unicode[STIX]{x03C0}\exp (-\text{i}Rq)\left[\frac{a_{0}}{R^{1/3}}\text{Ai}(-R^{2/3}p)+\frac{\text{i}b_{0}}{R^{2/3}}\text{Ai}^{\prime }(-R^{2/3}p)\right]\right\},\end{eqnarray}$$

where $\text{Ai}(\cdot )$ and $\text{Ai}^{\prime }(\cdot )$ denote the Airy function and its derivative, respectively (Abramowitz & Stegun Reference Abramowitz and Stegun1964).

5.2 KHP approximation

The KHP approximation developed by Wu et al. (Reference Wu, He, Zhu and Noblesse2018) is based on the approximation to the Airy function $\text{Ai}(u)$ and its derivative $\text{Ai}^{\prime }(u)$ , which are oscillatory for $u<0$ and decay exponentially for $u>0$ (Abramowitz & Stegun Reference Abramowitz and Stegun1964). Their asymptotic approximations are present for both $u<0$ and $u>0$ . Nevertheless, the asymptotic approximations are invalid near $u=0$ (the asymptotic approximation of $\text{Ai}(u)$ is infinite at $u=0$ , while that of $\text{Ai}^{\prime }(u)$ at $u=0$ is equal to zero). To treat this issue, a blending function $\exp (-u^{2})$ is introduced so that the Airy function and its derivative are uniformly approximated using the asymptotic expressions as well as the values of $\text{Ai}(0)\approx 0.355$ and $\text{Ai}^{\prime }(0)\approx -0.259$ .

Inspired by the approximation to the Airy function and its derivative, a uniform approximation to the wave component $G^{W}$ can be obtained in the same manner through combining three classical approximations (Kelvin, Havelock and Peters) introduced in § 4. The blending factor $u$ is thus determined via the CFU approximation containing both the Airy function and its derivative.

Then, following the work by Wu et al. (Reference Wu, He, Zhu and Noblesse2018), the approximation combining expressions by Kelvin and Havelock is written as

(5.22) $$\begin{eqnarray}\displaystyle G_{KHP}^{W} & = & \displaystyle -4H(x)F_{KH}^{-}\text{Im}\left[{\mathcal{A}}_{-}\exp \left(\text{i}R\unicode[STIX]{x1D713}_{-}+\frac{\unicode[STIX]{x03C0}}{4}\right)\right]\nonumber\\ \displaystyle & & \displaystyle -\,4H(x)F_{KH}^{+}\text{Im}\left[{\mathcal{A}}_{+}\exp \left(\text{i}R\unicode[STIX]{x1D713}_{+}-\frac{\unicode[STIX]{x03C0}}{4}\right)\right]\quad \text{for}~\unicode[STIX]{x1D6FE}\leqslant \unicode[STIX]{x1D6FE}_{0},\end{eqnarray}$$

where the function $F_{KH}^{\pm }$ combining both Kelvin’s and Havelock’s approximations is expressed as

(5.23) $$\begin{eqnarray}F_{KH}^{\pm }=(1-\unicode[STIX]{x1D707})\sqrt{\frac{2\unicode[STIX]{x03C0}}{R|\unicode[STIX]{x1D713}_{\pm }^{\prime \prime }|}}+\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x1D6E4}(1/3)}{\sqrt{6}}\left(\frac{6}{R|\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }|}\right)^{1/3},\end{eqnarray}$$

with the blending function given by

(5.24) $$\begin{eqnarray}\unicode[STIX]{x1D707}=\exp [-(-R^{2/3}p)^{2}]=\exp \{-[3R(\unicode[STIX]{x1D713}_{+}-\unicode[STIX]{x1D713}_{-})/4]^{4/3}\}.\end{eqnarray}$$

In (5.24), expression (5.9) has been applied. Representation (5.24) is uniformly valid for $\unicode[STIX]{x1D6FE}\leqslant \unicode[STIX]{x1D6FE}_{0}$ , and it is consistent with Havelock’s approximation at the cusp line $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}$ . In the same manner, a uniformly valid approximation for $\unicode[STIX]{x1D6FE}\geqslant \unicode[STIX]{x1D6FE}_{0}$ can be obtained by combining expressions (4.22) and (4.25) due to Havelock and Peters

(5.25) $$\begin{eqnarray}G_{KHP}^{W}=-4H(x)\text{Im}\left[F_{HP}^{-}{\mathcal{A}}_{-}\exp \left(\text{i}R\unicode[STIX]{x1D713}_{-}+\text{i}\frac{\unicode[STIX]{x03C0}}{4}\right)\right]\quad \text{for}~\unicode[STIX]{x1D6FE}\geqslant \unicode[STIX]{x1D6FE}_{0},\end{eqnarray}$$

where the function $F_{HP}^{-}$ is expressed as

(5.26) $$\begin{eqnarray}F_{HP}^{-}=(1-\unicode[STIX]{x1D707}^{\prime })\sqrt{\frac{2\unicode[STIX]{x03C0}}{R|\unicode[STIX]{x1D713}_{-}^{\prime \prime }|}}\exp \left(-\frac{\text{i}\unicode[STIX]{x1D6FC}}{2}\right)+\unicode[STIX]{x1D707}^{\prime }\frac{(1-\text{i})\unicode[STIX]{x1D6E4}(1/3)}{\sqrt{6}}\left(\frac{6}{R|\unicode[STIX]{x1D713}_{0}^{\prime \prime \prime }|}\right)^{1/3}.\end{eqnarray}$$

According to expression (5.9), the blending function $\unicode[STIX]{x1D707}^{\prime }$ in (5.26) is defined as

(5.27) $$\begin{eqnarray}\unicode[STIX]{x1D707}^{\prime }=\exp \{-[3R\text{Im}(\unicode[STIX]{x1D713}_{-}-\unicode[STIX]{x1D713}_{+})/4]^{4/3}\}.\end{eqnarray}$$

As a consequence, the KHP approximation including (5.22) and (5.25), which are applicable for $\unicode[STIX]{x1D6FE}\leqslant \unicode[STIX]{x1D6FE}_{0}$ and $\unicode[STIX]{x1D6FE}\geqslant \unicode[STIX]{x1D6FE}_{0}$ , respectively, is obtained. It is uniform in the entire fluid domain and is consistent with Havelock’s approximation at the cusp line $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}$ .

5.3 Comments on the CFU and KHP approximations

In this section, the CFU and KHP approximations, which are uniformly valid in the flow field, are considered. The classical CFU approximation contains the Airy function and its derivative, while the KHP approximation contains elementary functions only, and is thus fully analytical. Although the KHP approximation is less accurate than the CFU approximation, as noted by Wu et al. (Reference Wu, He, Zhu and Noblesse2018), yet the KHP approximation that results in explicit expressions facilitates the analysis of the flow physics associated with the viscous-ship-wave Green function. In addition, the KHP approximation provides a formal decomposition of ship waves into transverse and divergent waves from (5.22). This feature is favourable in the study of flow physics.

6.1 Numerical results

The results based on the numerical evaluation of the viscous-ship-wave Green function are considered. Figures 2 and 3 depict the free-surface term $G^{F}$ varying along a transverse cut in the downstream at $x=10.0$ generated by a fully submerged source $d=0.1$ and a slightly submerged source $d=1.0\times 10^{-5}$ with unit strength. The translating speed of the point source is $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ . Here, both the pure gravity ship-wave Green function and the viscous-ship-wave Green function are displayed. Besides, two different non-dimensional viscosity coefficients $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ and $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ corresponding to the kinematic fluid viscosities $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ and $\unicode[STIX]{x1D708}=10^{-4}~\text{m}^{2}~\text{s}^{-1}$ are considered.

Figure 2.

The free-surface term $G^{F}$ induced by a source with the submergence $d=gD/\mathscr{U}^{2}=0.1$ and the translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ varying along a transverse cut at $x=gX/\mathscr{U}^{2}=10.0$ . (a) Pure gravity ship-wave Green function; (b) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ corresponding to $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ ; (c) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ corresponding to $\unicode[STIX]{x1D708}=10^{-4}~\text{m}^{2}~\text{s}^{-1}$ .

Figure 3.

The free-surface term $G^{F}$ induced by a slightly submerged source at $d=gD/\mathscr{U}^{2}=1.0\times 10^{-5}$ and the forward speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ varying along a transverse cut at $x=gX/\mathscr{U}^{2}=10.0$ . (a) Pure gravity ship-wave Green function; (b) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ corresponding to $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ ; (c) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ corresponding to $\unicode[STIX]{x1D708}=10^{-4}~\text{m}^{2}~\text{s}^{-1}$ .

When the singularity is fully submerged $d=0.1$ , as shown in figure 2, viscous effects do not make a difference because there is no obvious difference between the three panels. As for a slightly submerged source $d=1.0\times 10^{-5}$ , as exhibited in figure 3, the difference is evident. In the top panel, in which viscous effects are ignored, the free-surface term $G^{F}$ is strongly oscillatory, and the wave height near the track $y=0$ is very large. When only the molecular viscosity $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ is accounted for in the middle panel, waves are greatly damped, but rapid oscillations still exist. When the non-dimensional viscosity coefficient is taken to be $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ in the bottom panel, waves become very smooth and highly oscillatory behaviours are severely damped due to viscous effects. It is noted that viscous effects only attenuate the short waves near the track associated with large wavenumbers. In the region with a relatively large azimuthal angle where short waves do not appear, waves are merely affected by viscous effects.

6.2 Non-uniform asymptotic approximations

Then, we consider the asymptotic approximations to the wave component $G^{W}$ due to Kelvin, Havelock and Peters as presented in § 4. For a flow-field point located inside ( $|\unicode[STIX]{x1D6FE}|<\unicode[STIX]{x1D6FE}_{0}$ ), at ( $|\unicode[STIX]{x1D6FE}|=\unicode[STIX]{x1D6FE}_{0}$ ) and outside ( $|\unicode[STIX]{x1D6FE}|>\unicode[STIX]{x1D6FE}_{0}$ ) the Kelvin wedge, expressions (4.11), (4.22) and (4.25) are applied, respectively. Figure 4 shows the comparison between the numerical evaluation and three asymptotic approximations for a slightly submerged source with the submergence $d=1.0\times 10^{-5}$ and translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ considered in figure 3. Here, the numerical results are represented by solid lines, while the asymptotic approximations due to Kelvin, Havelock and Peters are plotted by dashed lines, hollow symbols and dash–dotted lines, respectively. The Kelvin’s and Peters’ expressions approximate the wave component $G^{W}$ well when the azimuthal angle $\unicode[STIX]{x1D6FE}$ deviates from the cusp angle  $\unicode[STIX]{x1D6FE}_{0}$ , and highly oscillatory behaviours of the pure gravity ship-wave Green function are well reflected by the Kelvin’s approximation. At the cusp line, the approximations due to Kelvin and Peters are singular, whereas Havelock’s approximation is well finite and agrees well with the numerical evaluation.

Figure 4.

The wave component $G^{W}$ induced by a slightly submerged source with the translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ at $d=1.0\times 10^{-5}$ varying along a transverse cut at $x=10.0$ . The comparison is made between the numerical evaluation by solid lines (——) and the asymptotic approximations due to Kelvin, Havelock and Peters represented by the dashed lines (– – – –), hollow scatters (○) and dash–dotted lines (– ⋅ – ⋅ –), respectively. (a) Pure gravity ship-wave Green function; (b) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ ; (c) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ . The vertical line $y\approx 3.5$ corresponds to the cusp of Kelvin wedge.

6.3 Uniform approximations

Uniform approximations based on the CFU approximation and KHP approximation elucidated in § 5 are considered here. Figure 5 exhibits the wave component $G^{W}$ induced by a slightly submerged translating source with the submergence $d=1.0\times 10^{-5}$ and translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ considered in figure 3. Results obtained from the CFU approximation and the KHP approximation are compared with the direct numerical evaluation. Generally, these three groups of results are in close agreement. Nevertheless, a slight difference is witnessed between the KHP approximation and the two others near the cusp line $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}$ . It has already been pointed out by Wu et al. (Reference Wu, He, Zhu and Noblesse2018) that the KHP approximation is not as accurate as the CFU approximation in proximity to cusps. The KHP approximation is consistent with Havelock’s expression at the cusp line. It is noted that the reminder of Havelock’s approximation (4.22) is of the order of $O(R^{-2/3})$ . However, the error of the CFU approximation is of the order of $O(R^{-4/3})$ . Therefore, the CFU approximation is more accurate, but the accuracy of the KHP approximation can be improved by introducing high-order asymptotic expressions. The refined KHP approximation has been recently studied by Liang et al. (Reference Liang, Wu, He and Noblesse2020).

Figure 5.

The wave component $G^{W}$ induced by a slightly submerged source with the translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ at $d=gD/\mathscr{U}^{2}=1.0\times 10^{-5}$ varying along a transverse cut at $x=gX/\mathscr{U}^{2}=10.0$ . The comparison is made between the numerical evaluation by solid lines (——) and the uniform asymptotic approximations based on the CFU approximation by dashed lines (– – – –) and KHP approximation by dash-dotted lines (– ⋅ – ⋅ –). (a) Pure gravity ship-wave Green function; (b) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ ; (c) viscous-ship-wave Green function with $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ . The vertical line $y\approx 3.5$ corresponds to the cusp of Kelvin wedge.

Figure 6 depicts the wave component $G^{W}$ induced by a source exactly located at the free surface $d=0$ with the translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ varying along a transverse cut at $x=10.0$ . Besides the pure gravity ship-wave Green function, the viscous-ship-wave Green functions with viscosity coefficients $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ and $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ are exhibited. Results determined by the CFU approximation and KHP approximation are displayed in the top and bottom panels, respectively. It provides strong evidence that singular and highly oscillatory behaviours in the pure gravity ship-wave Green function are eliminated and damped by including viscous effects. Furthermore, steep short waves still exist if only molecular viscosity is accounted for, and can be largely removed via taking a proper viscosity coefficient, i.e. 100 times the molecular viscosity coefficient as suggested by Dias et al. (Reference Dias, Dyachenko and Zakharov2008) and Horsley & Forbes (Reference Horsley and Forbes2013).

Figure 6.

The wave component $G^{W}$ induced by a source exactly at the free surface $d=gD/\mathscr{U}^{2}=0$ with the translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ varying along a transverse cut at $x=gX/\mathscr{U}^{2}=10.0$ . Three groups of results are displayed including the pure gravity ship-wave Green function by solid lines (——), viscous-ship-wave Green functions with $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ ( $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ ) plotted by dashed lines (– – – –) and $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ ( $\unicode[STIX]{x1D708}=10^{-4}~\text{m}^{2}~\text{s}^{-1}$ ) by dash-dotted lines (– ⋅ – ⋅ –). (a) CFU approximation; (b) KHP approximation.

Figure 7 shows the plan view of the free-surface pattern generated by a steadily translating source with the submergence $d=5.0\times 10^{-2}$ and travelling speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ in a rectangular region $10\leqslant x\leqslant 20$ , $0\leqslant y\leqslant 8$ . Results obtained from the numerical evaluation, the CFU approximation and KHP approximation are placed in the top row, centre row and bottom row, respectively. The pure gravity ship waves as well as viscous ship waves with the viscosity coefficients $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ and $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ are shown in the left column, middle column and right column, respectively. Discrepancies of wave patterns determined via three methods are indistinguishable. When viscous effects are ignored or only the molecular viscosity is accounted for, short waves are present inside the Kelvin wedge. In the right column in which the kinematic fluid viscosity coefficient is $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ , short waves die out.

Figure 7.

Plan view of the free-surface elevation generated by a steadily translating source with the submergence $d=gD/\mathscr{U}^{2}=5.0\times 10^{-2}$ and the travelling speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ . Numerical evaluations are shown in (a–c), results determined by the CFU approximation in (d–f) and results by the KHP approximation in (g–i). Pure gravity ship waves as well as viscous ship waves with viscosity coefficients $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ and $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ corresponding to $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ and $\unicode[STIX]{x1D708}=10^{-4}~\text{m}^{2}~\text{s}^{-1}$ , respectively, are placed from left to right. The white dashed line corresponds to the cusp of the Kelvin wedge.

7.1 Decomposition of transverse and divergent waves

The uniform KHP approximation enables us to determine the individual contributions of transverse waves and divergent waves. Figure 8 shows the free-surface elevations induced by a translating source with the submergence $d=0.01$ and the translating speed $\mathscr{U}=3.132~\text{m}~\text{s}^{-1}$ varying with the ratio between the azimuthal angle and the Kelvin angle $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ at a constant radius $R=10$ , and where the transverse waves and divergent waves are represented by solid lines and dashed lines, respectively. The envelope of divergent waves is also exhibited and is plotted by dash-dotted lines. In this case, the transverse waves with a small amplitude are mildly varying with the angle, but the divergent waves are oscillatory with an amplitude much higher than that of the transverse waves. Therefore, the divergent waves dominate the wave pattern at small $d$ , corresponding to a small submergence or a high travelling speed. In addition, viscous effects hardly affect the transverse waves, but attenuate the amplitude of the divergent waves near the track. Furthermore, there is a narrow region near the track where the amplitude of divergent waves is almost zero when the source is submerged. The middle and bottom panels show that viscous effects make the region wider.

Figure 8.

Decomposition of transverse and divergent waves inside the Kelvin wedge. The transverse waves are represented by solid lines (——), divergent waves by dashed lines (– – – –) and the envelope of divergent waves by dash-dotted lines (– ⋅ – ⋅ –). (a) Pure gravity ship waves; (b) viscous ship waves with $\unicode[STIX]{x1D716}=3.2\times 10^{-7}$ ; (c) viscous ship waves with $\unicode[STIX]{x1D716}=3.2\times 10^{-5}$ .

7.2 Apparent wake angle

In order to study the apparent angle of free-surface waves generated by a translating source, the maximum of the envelope of divergent waves is considered. According to § 4.1, the envelope of divergent waves induced by a point source is

(7.1) $$\begin{eqnarray}E=\frac{1}{\unicode[STIX]{x03C0}}\sqrt{1+\unicode[STIX]{x1D705}_{+}^{2}}\exp \left[-(1+\unicode[STIX]{x1D705}_{+}^{2})d-4\unicode[STIX]{x1D716}x\frac{(\unicode[STIX]{x1D705}_{+}^{2}+1)^{3}}{2\unicode[STIX]{x1D705}_{+}^{2}+1}\right]\sqrt{\frac{2\unicode[STIX]{x03C0}}{R|\unicode[STIX]{x1D713}_{+}^{\prime \prime }|}},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{+}$ and $\unicode[STIX]{x1D713}_{+}^{\prime \prime }$ are given by

(7.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{+}\approx \frac{1}{2\unicode[STIX]{x1D6FE}}-\frac{7}{6}\unicode[STIX]{x1D6FE}+O(\unicode[STIX]{x1D6FE}^{3})\quad \text{and}\quad \unicode[STIX]{x1D713}_{+}^{\prime \prime }\approx -2\unicode[STIX]{x1D6FE}+\frac{25}{3}\unicode[STIX]{x1D6FE}^{3}+O(\unicode[STIX]{x1D6FE}^{5}),\end{eqnarray}$$

based on the small azimuthal angle approximation. Following Darmon et al. (Reference Darmon, Benzaquen and Raphaël2014), Pethiyagoda et al. (Reference Pethiyagoda, McCue and Moroney2014a ), Miao & Liu (Reference Miao and Liu2015), the envelope of the divergent waves is expanded as

(7.3) $$\begin{eqnarray}\displaystyle E & {\approx} & \displaystyle \frac{1}{\sqrt{\unicode[STIX]{x03C0}R}}\left(\frac{1}{2}\unicode[STIX]{x1D6FE}^{-3/2}+\frac{7}{8}\unicode[STIX]{x1D6FE}^{1/2}+\cdots \right)\nonumber\\ \displaystyle & & \displaystyle \times \exp \left[-\frac{\unicode[STIX]{x1D716}x}{8}\unicode[STIX]{x1D6FE}^{-4}-\left(\frac{\unicode[STIX]{x1D716}x}{12}+\frac{d}{4}\right)\unicode[STIX]{x1D6FE}^{-2}+\frac{107}{180}\unicode[STIX]{x1D716}x+\frac{d}{6}+\cdots \right].\end{eqnarray}$$

By differentiating $E$ with respect to $\unicode[STIX]{x1D6FE}$ and setting it to zero $\unicode[STIX]{x2202}E/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}=0$ , we can obtain the maximum wake angle subject to

(7.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D716}x}{2}\unicode[STIX]{x1D6FE}^{-4}+\left(\frac{25}{24}\unicode[STIX]{x1D716}x+\frac{d}{2}\right)\unicode[STIX]{x1D6FE}^{-2}-\frac{3}{2}=0,\end{eqnarray}$$

resulting in the apparent wake angle in a simple expression:

(7.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{app}\approx \frac{1}{\sqrt{3}}\sqrt{d}+\frac{\sqrt{3}}{2}\frac{\unicode[STIX]{x1D716}x}{d^{3/2}}=\frac{\sqrt{gD}}{\sqrt{3}\mathscr{U}}+\frac{\sqrt{3}}{2}\frac{\unicode[STIX]{x1D708}X\sqrt{g}}{D^{3/2}\mathscr{U}^{2}}.\end{eqnarray}$$

When viscous effects are disregarded, expression (7.5) is consistent with the one given by Pethiyagoda et al. (Reference Pethiyagoda, McCue and Moroney2014a ). Nevertheless, the apparent wake angle under the influence of viscous effects is dependent on the distance along the track from the point source, and it is associated with both $\mathscr{U}^{-1}$ and $\mathscr{U}^{-2}$ . Furthermore, viscous effects play an increasingly important role with increasing distance. Figures 9 and 10 show the apparent wake angle induced by a translating source as a function of the submergence Froude number $F_{d}=\mathscr{U}/\sqrt{gD}$ at a fixed immersion depth $D=0.01$  m and at a constant speed $\mathscr{U}=31.3~\text{m}~\text{s}^{-1}$ . The asymptotic expression is exhibited by solid symbols, and the direct measurements are by solid lines. At a high submergence Froude number, viscous effects make a difference, and the apparent wake angle associated with viscous ship waves is larger than the pure gravity counterpart.

Figure 9.

Apparent wake angle induced by a source with the submergence $D=0.01$  m as a function of the submergence Froude number $F_{d}=\mathscr{U}/\sqrt{gD}$ . The direct measurements are represented by solid lines, and the results determined by the asymptotic approximation (7.5) by solid symbols. The dashed line corresponds to the Kelvin angle.

Figure 10.

Apparent wake angle induced by a source at a constant speed $\mathscr{U}=31.3~\text{m}~\text{s}^{-1}$ as a function of the submergence Froude number $F_{d}=\mathscr{U}/\sqrt{gD}$ . The direct measurements are represented by solid lines, and the results determined by the asymptotic approximation (7.5) by solid symbols. The dashed line corresponds to the Kelvin angle.