3.1 Soliton solutions and notations

One soliton solution is obtained by the ${\it\tau}$ function with $M=2$ and $N=1$ , i.e. ${\it\tau}=f_{1}=a_{11}E_{1}+a_{12}E_{2}$ with $\unicode[STIX]{x1D63C}=(a_{11}~a_{12})$ . Since the solution $u$ is given by (3.2), one can take $a_{11}=1$ and denote $a_{12}=a>0$ . Then we have

of which the solution is localized along the line ${\it\theta}_{1}-{\it\theta}_{2}=\ln a$ . Note that the line soliton appears at the boundary of two regions in the $xy$ plane where either $E_{1}$ or $E_{2}$ is the dominant exponential term, and because of this we call this soliton a [1,2]-soliton solution. Likewise, we refer to each of these asymptotic line solitons as the  $[i,j]$ -soliton with $i<j$ whose ${\it\tau}$ function around the soliton can be expressed by ${\it\tau}\approx {\it\Delta}_{I}E_{I}+{\it\Delta}_{J}E_{J}$ with the index sets $I$ containing $i$ and $J$ containing $j$ , which satisfy $I\setminus \{i\}=J\setminus \{j\}$ . The $[i,j]$ -soliton solution has the same local structure as the one-soliton solution and can be described as follows

with some constant ${\it\Theta}_{[i,j]}^{0}$ . The amplitude $A_{[i,j]}$ , the wavenumber vector $\boldsymbol{K}_{[i,j]}$ and the frequency ${\it\Omega}_{[i,j]}$ are expressed by

The direction of the wavenumber vector $\boldsymbol{K}_{[i,j]}=(K_{[i,j]}^{x},K_{[i,j]}^{y})$ is measured in the counterclockwise direction from the $y$ axis and it is given by

that is, ${\it\Psi}_{[i,j]}$ gives the angle between the line $\boldsymbol{K}_{[i,j]}\boldsymbol{\cdot }\boldsymbol{x}=\text{const.}$ and the $y$ axis (see figure 1). Note that $\boldsymbol{K}_{[i,j]}$ and ${\it\Omega}_{[i,j]}$ satisfy the soliton-dispersion relation,

The crest line of $[i,j]$ -soliton is given by $\boldsymbol{K}_{[i,j]}\boldsymbol{\cdot }\boldsymbol{x}-{\it\Omega}_{[i,j]}t+{\it\Theta}_{[i,j]}^{0}=0$ , i.e.

for some constant $x_{0}$ . Note $V_{[i,j]}>0$ , that is, each soliton propagates in the positive $x$ direction, which is important when we set-up the coordinates for a laboratory or numerical experiment of shallow-water waves.

Figure 1.

One line-soliton solution of $[i,j]$ -type and the corresponding chord diagram. The upper oriented chord represents the part of $[i,j]$ -soliton for $y\gg 0$ and the lower one for $y\ll 0$ . The middle line of the chord diagram shows the coordinate axis for the ${\it\kappa}$ parameters. Notice that the wave amplitude is represented by $A_{[i,j]}=({\it\kappa}_{j}-{\it\kappa}_{i})^{2}/2$ and the counterclockwise shift of the wave propagation is expressed by $\tan {\it\Psi}_{[i,j]}={\it\kappa}_{i}+{\it\kappa}_{j}$ , which is positive for this example.

In figure 1, we illustrate one line-soliton solution of $[i,j]$ -type. In the right panel of this figure, we show a chord diagram which represents this single line-soliton solution. The chord diagram is a product of the KP analysis made by Chakravarty & Kodama (Reference Chakravarty and Kodama2009), which represents the asymptotic soliton solutions for $y\rightarrow \pm \infty$ .

Now we describe some exact soliton solutions of the KP equation for two line solitons. For two interacting line solitons, the matrix $\unicode[STIX]{x1D63C}=(a_{ij})$ is a constant $2\times 4$ matrix and each solution $u(x,y,t)$ is determined by the $\unicode[STIX]{x1D63C}$ matrix and the ${\it\kappa}$ parameters. From (3.7), the ${\it\tau}$ function can be written as

where ${\it\Delta}_{i,j}(A)$ is the $2\times 2$ minors consisting of $i$ th and $j$ th columns of the $2\times 4$ $\unicode[STIX]{x1D63C}$ -matrix, and $E_{i,j}=Wr(e^{{\it\theta}_{i}},e^{{\it\theta}_{j}})=({\it\kappa}_{j}-{\it\kappa}_{i})e^{{\it\theta}_{i}+{\it\theta}_{j}}$ (note $E_{i,j}>0$ with the order ${\it\kappa}_{i}<{\it\kappa}_{j}$ ). As shown by Chakravarty & Kodama (Reference Chakravarty and Kodama2009), ${\it\tau}$ function (3.15) generates a soliton solution that consists of two line solitons for $y\rightarrow \pm \infty$ . We call this type soliton solution a (2,2)-soliton solution. In general, an $(N_{-},N_{+})$ soliton means that the solution has $N_{-}$ asymptotic line solitons for $y\ll 0$ and $N_{+}$ asymptotic line solitons for $y\gg 0$ . Those asymptotic soliton solutions can be parameterized by the permutations, which lead to the chord diagram to express the classification of soliton solutions as a chord joining a pair of parameters ${\it\kappa}_{j}$ following the permutation. Figure 2 shows seven possible types of $(2,2)$ -soliton solutions for $2\times 4$ matrix $\unicode[STIX]{x1D63C}$ . For example, the chord diagram of $(3142)$ -type soliton shows that there are [1,3]- and [3,4]-solitons for $y\gg 0$ and [1,2]- and [2,4]-solitons for $y\ll 0$ . Detailed mathematical descriptions are given by Chakravarty & Kodama (Reference Chakravarty and Kodama2009).

Figure 2.

The chord diagrams for seven different types of (2, 2)-soliton solutions. The number in the parenthesis represents the permutation. After Chakravarty & Kodama (Reference Chakravarty and Kodama2009).

4.1 KP theory

Figure 4 illustrates the contour plots of (2143)-type and (3142)-type solutions of the KP theory in the $xy$ plane which are symmetric with respect to the $x$ axis and the corresponding chord diagrams, respectively. Recall that the upper chords represent the asymptotic solitons $[i,j]$ for $y\gg 0$ and the lower chords the asymptotic solitons $[i,j]$ for $y\ll 0$ . Let us fix the amplitudes $A_{[i,j]}$ and the angles ${\it\Psi}_{[i,j]}$ of the solitons in the positive $x$ regions for both (2143)- and (3142)-types, so that those solutions are symmetric with respect to the $x$ axis:

Figure 4.

Contour plots and chord diagrams. (a) (2143)-type solution. (b) (3142)-type solution. The length of [1,4]-soliton in the right figure changes in $t$ .

Then we express the ${\it\kappa}$ parameters in terms of $A_{0}$ and $\tan {\it\Psi}_{0}$ with ${\it\kappa}_{1}=-{\it\kappa}_{4}$ and ${\it\kappa}_{2}=-{\it\kappa}_{3}$ (due to the symmetry). In the case of (2143)-type. With the use of (3.11a ) and (3.12), we have

The ordering ${\it\kappa}_{2}<{\it\kappa}_{3}=-{\it\kappa}_{2}$ then implies $\tan {\it\Psi}_{0}>\sqrt{2A_{0}}$ . On the other hand, for (3142)-type, we have

The ordering ${\it\kappa}_{2}<{\it\kappa}_{3}$ implies $\tan {\it\Psi}_{0}<\sqrt{2A_{0}}$ . Thus, if all the solitons in the positive $x$ region have the same amplitude $A_{0}$ for both (2143)- and (3142)-types, then an (2143)-type solution arises when $\tan {\it\Psi}_{0}>\sqrt{2A_{0}}$ and a (3142)-type when $\tan {\it\Psi}_{0}<\sqrt{2A_{0}}$ .

For (3142)-type, the solitons in the negative $x$ region are smaller than those in the positive region, i.e. $A_{[3,4]}=A_{[1,2]}=(1/2)({\it\kappa}_{1}-{\it\kappa}_{2})^{2}=(1/2)\tan ^{2}{\it\Psi}_{0}<A_{0}$ with $A_{0}=A_{[1,3]}=A_{[2,4]}$ . Also, the angles of those waves depend only on the amplitude of the waves in the positive $x$ region, i.e. $\tan {\it\Psi}_{[3,4]}=\sqrt{2A_{0}}=-\tan \,{\it\Psi}_{[1,2]}$ . Two sets of three solitons $\{[1,3],[1,4],[3,4]\}$ in $y>0$ and $\{[2,4],[1,4],[1,2]\}$ in $y<0$ show soliton resonances forming Y-shaped waves. Then the maximum amplitude of (3142)-type soliton is given by the amplitude of the [1,4] soliton, that is,

where $A_{0}=A_{[1,3]}=A_{[2,4]}$ and the parameter $k$ is given by

We call this parameter the KP parameter. In § 5, we will express $A_{0}$ in terms of the physical amplitude $a_{0}$ (see (2.14) and note that in the leading-order approximation, we have $a_{0}=2A_{0}/3$ ). One should note that the amplification formula ${\it\alpha}_{w}$ in (4.1) is the same as that in (4.7), but the $k$ parameter in (4.1) is now replaced by (4.8). Thus the (3142)-soliton appears in the case with $k<1$ . In this soliton profile, the [1,4]-soliton can be identified as the Mach stem wave and the maximum wave amplification is given by ${\it\alpha}_{w}=A_{[1,4]}/A_{0}=(1+k)^{2}$ . The [1,2]-soliton (and [3,4]-soliton) in the profile is then identified as the reflected wave, and the reflected wave amplification is given by ${\it\alpha}_{r}=A_{[1,2]}/A_{0}=k^{2}$ with $k$ in (4.8).

The (2143)-type solitons in the positive and negative $x$ directions are identical in amplitude and symmetric with respect to the $y$ axis; hence, it forms a regular reflection. One should however note that the intersection of the ‘X’ shape causes the negative phase shift, which is given by

where $A_{0}=A_{[1,2]}$ and $k$ is given by (4.8) (see Chakravarty & Kodama Reference Chakravarty and Kodama2009). The phase shift appears as a stem-like wave formation at the wall. Although $|{\rm\Delta}x_{[1,2]}|$ approaches infinity as $k\rightarrow 1^{+}$ , such a large phase shift can be obtained only for the case of $k$ being very close to $1$ : note that the logarithmic function in (4.9) is extremely sensitive near $(1-k^{-2})\rightarrow 0^{+}$ . The contour plot of (2143)-type shown in figure 4 is the case of $k=1.04$ , which exhibits a very small phase shift. Even for $k=1.02$ (very close to 1.0), we obtain ${\rm\Delta}x_{[1,2]}\approx -1.5{\it\lambda}_{0}$ , where ${\it\lambda}_{0}(=\sqrt{2/A_{0}})$ represents the breadth of the soliton and the magnitude of the phase shift is comparable to the soliton breadth (i.e. not large). Because the KP equation is accurate up to $O({\it\epsilon})$ , the prediction of a large phase shift in such a narrow range of $k({\approx}1^{+})$ may not be realizable for the KP approximation (see Chakravarty & Kodama Reference Chakravarty and Kodama2014). Therefore, the (2143)-type solitons can only create a minute amount of phase shift; a definitive formation of the stem wave often observed in laboratories and the field must be considered as a consequence of the (3142)-type solitons.

Figure 5.

Laboratory realization of (2, 2)-soliton solutions: (a–c) (2143)-type wave-profile evolution with incidence wave amplitudes $a_{0}=0.083$ with ${\it\Psi}_{0}=0.52$  rad and (d–f) (3142)-type $a_{0}=0.322$ with the same angle ${\it\Psi}_{0}=0.52$  rad; (a,d)  $x=10.2$ , (b,e) 40.7 and (c,f) 71.1.

Figure 6.

(2143)-type and $(3142)$ -type KP solutions corresponding to the waves in figure 5. The ${\it\kappa}$ parameters are (a) $({\it\kappa}_{1},\ldots ,{\it\kappa}_{4})=(-0.495,-0.082,0.082,0.495)$ for (2143)-type soliton and (b) $({\it\kappa}_{1},\ldots ,{\it\kappa}_{4})=(-0.683,-0.106,0.106,0.683)$ for $(3142)$ -soliton.

Unlike [1,4]-soliton of (3142)-type, the phase shift is steady and it does not correspond to a soliton. Those distinct behaviours for (2143)-type and (3142)-type are verified experimentally in the laboratory wave tank, as shown in figure 5. Note that the laboratory data in the figure are obtained with the use of the same apparatus and procedure used by Li, Yeh & Kodama (Reference Li, Yeh and Kodama2011). In figures 5 and 6, at the location $x=10.2$ that is immediately after the incident soliton meets the vertical wall, the wave patterns of (2143)-type and the (3142)-type are similar: the region of wave–wave intersection is small and the reflected wave angle is larger than that of the incident wave. This is because both (2143)-type and (3142)-type interactions are in the development stage. The larger reflected wave angle is a consequence of undeveloped wave reflection: note that, as depicted in figure 3, only the incident wave was generated in the laboratory. Once wave reflection is developed, the (2143)-type exhibits the stem-like wave formation but its length remains short and constant; this stem-like appearance is a consequence of the phase shift arising from the wave–wave intersection. For the (2143)-type reflection, the reflected wave angle becomes the same as the incident wave angle ${\it\Psi}_{0}$ , resulting in symmetrical wave formation, as shown in figure 5(c). On the other hand, the (3142)-type shows continual growth of the stem wave. As shown in figure 5(f), the reflected wave angle remains larger than that of the incident wave. This wave pattern is consistent with the theoretical prediction shown in figure 4. With the experimental parameters, $a_{0}$ and ${\it\Psi}_{0}$ , the exact KP solitons are shown in figure 6. The KP parameters ${\it\kappa}_{i}$ values and the matrix $\unicode[STIX]{x1D63C}$ are calculated explicitly from the laboratory setting. In particular, the ${\it\kappa}$ parameters are obtained using the KP amplitude $A_{0}$ from (5.14) that includes the higher-order corrections (the higher-order corrections will be discussed in § 5).

Chakravarty & Kodama (Reference Chakravarty and Kodama2009) found that the maximum amplitude of a (2143)-type soliton at the intersection is

which is equivalent to (4.1). One should again note that the $k$ parameter here is given by (4.8) instead of the Miles parameter (4.2).

The limiting value at ${\it\kappa}_{2}={\it\kappa}_{3}$ $(=0)$ defines the critical angle ${\it\Psi}_{c}$ ,

This limit corresponds to the critical cases of (2143)-type and (3142)-type, and at this critical state, the phase shift (4.9) for the (2143)-type soliton becomes infinite. This means, as depicted in figure 7, that the (degenerate) chord diagram in the limit splits into two diagrams for $y\gg 0$ and $y\ll 0$ describing two sets of resonant Y-shape solitons in those regions. Notice that at the critical angle, the ${\it\tau}$ function has only three exponential terms, i.e. $E_{12}=E_{13},E_{14}$ and $E_{24}=E_{34}$ ( $E_{23}=0$ ), resulting in the resonant Y-shape solution. This is equivalent to Miles’s 4-fold amplification of the stem wave at the critical angle ${\it\Psi}_{c}$ or at $k=1$ in (4.1). The 4-fold amplification is interpreted from the cord diagram in figure 7: $A_{[1,4]}=(1/2)({\it\kappa}_{4}-{\it\kappa}_{1})^{2}=4((1/2)({\it\kappa}_{3}-{\it\kappa}_{1})^{2})=4((1/2)({\it\kappa}_{4}-{\it\kappa}_{2})^{2})$ . According to (4.11), the critical state is at $k=1$ in which $k$ in the KP theory is given by (4.8). Note that the Miles parameter $k$ expressed in (4.2) is for small angle ${\it\Psi}_{0}\approx \tan {\it\Psi}_{0}$ . One should however note that the parameter $k$ was obtained under the assumption $\sin {\it\Psi}_{0}\approx {\it\Psi}_{0}$ , and, as will be discussed in § 5.3, the difference between $\tan {\it\Psi}_{0}$ and $\sin {\it\Psi}_{0}$ is quite significant for larger angle ${\it\Psi}_{0}$ .

Figure 7.

Splitting the degenerate chord diagram at the critical angle.

Figure 8.

Experimental results of the amplification factor ${\it\alpha}_{w}$ versus Miles’s parameter $k$ . (a) Laboratory results. The open circles $\circ$ show the results of Perroud (Reference Perroud1957), the open squares ▫ show Melville (Reference Melville1980) and the solid circles ● are the data by Li et al. (Reference Li, Yeh and Kodama2011) with ${\it\Psi}_{0}=0.35$  rad, solid triangle ▴ with ${\it\Psi}_{0}=0.52$  rad, and solid square ▪ with ${\it\Psi}_{0}=0.70$  rad. (b) Numerical results. The crosses $\times$ show the results of Funakoshi (Reference Funakoshi1980) and the open triangles ▵ show the results of Tanaka (Reference Tanaka1993).

4.2 Previous laboratory and numerical results

Preceding Miles’s analysis by 20 years, Perroud (Reference Perroud1957) was the first to experimentally investigate soliton reflection with oblique incidence. Perroud used a small tank (6.1 m long, 1.1 m wide, 0.13 m deep) to generate a soliton by displacing a paddle driven by a weight with a pulley system. The water depths $\tilde{h}_{0}$ used for the experiments were 4 and 6 cm, and the wave amplitudes were $a_{0}=\tilde{a}_{0}/\tilde{h}_{0}=0.05{-}0.43$ . A few years after Miles’s analysis, Melville (Reference Melville1980) conducted the laboratory experiments specifically to attempt to validate Miles’s theory. The experiments were carried out in a wave basin (18.3 m long, 6.2 m wide with water of 20 and 30 cm depth). Figure 8(a) compares Perroud and Melville’s experimental data with Miles’s prediction (4.1). Melville’s data show that the maximum amplification ${\it\alpha}_{w}=a_{max}/a_{0}$ at the wall was 2.0 (simple linear superposition) when the Miles parameter $k=1.43$ ; this amplification is smaller than Perroud’s observation ${\it\alpha}_{w}=2.4$ . Both Perroud and Melville’s data indicate that the measured amplification monotonically increases with the Miles parameter $k$ , which is qualitatively in disagreement with the theory that predicts the formation of peak amplification at $k=1$ .

Li et al. (Reference Li, Yeh and Kodama2011) revisited the problem, performing laboratory experiments with a wave basin (7.3 m long and 3.6 m wide with water depth of 6.0 cm). They found that discrepancies between the previous laboratory results (Perroud Reference Perroud1957; Melville Reference Melville1980) and Miles’s prediction (4.1) are attributed partly to the insufficient propagation distance in the laboratory experiments so that the asymptotic state could have not been reached. To circumvent this shortcoming, Li et al. (Reference Li, Yeh and Kodama2011) conducted so called ‘patched’ experimental runs: taking advantage of the precision laboratory apparatus, wave measurements at a long propagation distance were accomplished by generating the observed waveform from the parent experiment with a wavemaker and patching the data with those from the subsequent extended experiment. Figure 8(a) shows the measured data, effectively at $x=121.1$ with Miles’s parameter $k$ . The result exhibits a peak in amplification at the Miles parameter $k=0.753$ . The measured maximum amplification factor is ${\it\alpha}_{w}=2.92$ , that is, less than Miles’s 4-fold prediction, but substantially more than linear superposition.

Funakoshi (Reference Funakoshi1980) conducted the numerical experiments: see figure 8(b), Funakoshi’s results deviate from Miles’s theory in spite of identical assumptions and approximations used in his model as were used in Miles’s analysis. The asymptotic state could not be attained in the numerical simulations at $k\approx 1.0$ ; consequently, he could not demonstrate the critical amplification factor of ${\it\alpha}_{w}=4.0$ . The numerical experiments performed by Tanaka (Reference Tanaka1993) were based on the Euler formulation with the use of the higher-order spectral method developed by Dommermuth & Yue (Reference Dommermuth and Yue1987). This higher-order model allowed him to study conditions less restricted in the nonlinearity parameter ${\it\alpha}=(\tilde{a}_{0}/\tilde{h}_{0})$ and the incident wave angle ${\it\Psi}_{0}$ . As shown in figure 8(b), Tanaka’s higher-order numerical results deviate more than those calculated by Funakoshi (Reference Funakoshi1980) – note that both Tanaka and Funakoshi’s numerical results show the pattern of amplification being shifted to the lower value of $k$ . The value of $k$ where the maximum amplification ${\it\alpha}_{w}$ = 2.897 is, however, in good agreement with the laboratory results of Li et al. (Reference Li, Yeh and Kodama2011), presented in figure 8(a): ${\it\alpha}_{w}=2.92$ at $k=0.753$ .

According to Li et al. (Reference Li, Yeh and Kodama2011), the discrepancy in amplification ${\it\alpha}_{w}$ must stem from the assumptions in the theory: particularly, small incident wave angle ${\it\Psi}_{0}^{2}=O({\it\varepsilon})$ . This condition is clearly violated in Tanaka’s numerical experiments: Tanaka (Reference Tanaka1993) used $a_{0}=0.3$ in his numerical experiments so that ${\it\Psi}_{0}=0.95$  rad at $k=1$ that is hardly ‘small’. On the other hand, Funakoshi (Reference Funakoshi1980) used $a_{0}=0.05$ in his numerical experiments so that ${\it\Psi}_{0}=0.39$  rad at $k=1$ . This must be the reason why the agreement of Funakoshi’s results with the theory is better than Tanaka’s (see figure 8 b). The incident wave angles ${\it\Psi}_{0}$ used in the laboratory experiments are also not small: the experiments by Perroud (Reference Perroud1957) were done with ${\it\Psi}_{0}=0.13\sim {\rm\pi}/2$ ; Melville (Reference Melville1980) used ${\it\Psi}_{0}=0.17{-}0.79$  rad and Li et al. (Reference Li, Yeh and Kodama2011) used ${\it\Psi}_{0}=0.35{-}0.70$  rad. In addition to the need for a long propagation distance (or computation time) to achieve the 4-fold amplification in numerical and laboratory experiments, we recognize that the critical condition $k=1$ is unstable: according to (4.1), the gradient of ${\it\alpha}_{w}$ becomes $-\infty$ at $k\rightarrow 1^{+}$ . A gradual viscous attenuation would prevent the wave from sustaining the critical condition at $k=1$ . A similar problem exists in the numerical experiments due to small but unavoidable numerical noise, including numerical dissipation and dispersion.

One should note here that the previous numerical studies used the Miles parameters (4.2) for analysis, even for the cases where the approximation ${\it\Psi}_{0}\approx \tan {\it\Psi}_{0}$ fails. Therefore, their analyses lead to the discrepancy between the Miles prediction and the numerical results. In § 5, we will derive a new $k$ parameter to include the higher-order approximations of quasi-two-dimensionality and weak nonlinearity, and re-evaluate those results based on the new parameter.

4.3 Remark on the quasi-two-dimensional approximation

In order to remedy the discrepancy caused by a finite value of ${\it\Psi}_{0}$ , Li et al. (Reference Li, Yeh and Kodama2011) proposed a heuristic fix for the KP soliton paradox. Recall that, in terms of physical coordinates, the KP equation is written as

As an exact solution of (4.12), we find a soliton solution in the coordinate perpendicular to the wave crest, $\tilde{{\it\chi}}=\tilde{x}\cos {\it\Psi}_{0}+{\tilde{y}}\sin {\it\Psi}_{0}$ :

Recall that the KP equation is derived with the assumption of quasi-two-dimensionality, that is ${\it\gamma}=\tan ^{2}{\it\Psi}_{0}=O({\it\epsilon})$ . It is recognized that the solution (4.13) is unphysical in the case of a finite angle ${\it\Psi}_{0}\neq 0$ . While (4.13) is an exact solution to the KP equation (4.12), the breadth of the wave profile depends on the propagation direction, ${\it\Psi}_{0}$ . More precisely, the width of the soliton artificially changes by rotating the coordinate system. To correct this artefact, a heuristic fix is proposed by defining a new amplitude parameter $\hat{a}_{0}=\tilde{a}_{0}/\cos ^{2}{\it\psi}=\tilde{a}_{0}(1+\tan ^{2}{\it\psi})=\tilde{a}_{0}(1+O({\it\varepsilon}))$ . Using this re-defined amplitude $\hat{a}_{0}$ , (4.13) can be reduced to the KdV soliton:

It is remarked that this heuristic modification makes the solution no longer exact to the KP equation (4.12). With the newly defined wave amplitude $\hat{a}_{0}$ , Miles’s parameter (4.2) is modified for a finite-but-small value of ${\it\Psi}_{0}$ as

Comparing this with (4.8), we have

This relation between the physical wave amplitude and the KP wave amplitude gives the higher-order correction to the quasi-two-dimensional approximation, as will be shown in (5.14). In the next section, we show how it can be formally corrected using a normal form theory concerning the higher-order correction to the KP equation.

5.1 The normal form for the KdV equation

Before discussing a normal form for (2.9), we give a brief summary of the result for the case of the KdV equation. Taking $\partial _{Y}u=0$ in (2.12), we have the KdV equation with higher-order corrections:

(As stated earlier, the lower case letters $(x,y,z)$ are used instead of $(X,Y,T)$ for succinct expression.) According to Kodama (Reference Kodama1985), one can transform (5.1) with a formal change of variable,

into the equation which we refer to as the normal form of (5.1):

Note that the higher-order term of $O({\it\epsilon}^{7/2})$ in this equation is the fifth-order symmetry of the KdV equation, and hence the normal form is integrable up to $O({\it\epsilon}^{7/2})$ . That is, we have an integrability not only at the KdV of $O({\it\epsilon}^{5/2})$ but also at the next-order correction of $O({\it\epsilon}^{7/2})$ . This means that the solitary waves are robust and their interaction properties remain the same as the KdV solitons even including the next-order correction.

This is true even for the general form of the higher-order correction,

with arbitrary coefficients ${\it\alpha}_{1},\ldots ,{\it\alpha}_{4}$ . This implies that any weakly nonlinear long-wave equation whose leading order is approximated by the KdV equation is asymptotically integrable up to the next-order approximation. Then, using the transformation (5.2), one can find higher-order corrections of the KdV solitons including the phase shifts due to the soliton interactions (see Hiraoka & Kodama Reference Hiraoka and Kodama2009).

Note that the normal form up to $O({\it\epsilon}^{7/2})$ admits a one-soliton solution in the form,

where $x_{0}(t)$ is determined by $(\text{d}x_{0}/\text{d}t):=(1/2)A_{0}+(19/60)A_{0}^{2}+O({\it\epsilon}^{3})$ . Then the solution of the higher-order KdV equation (5.1) is given by the transformation (5.2):

where $S:=\,\text{sech}\sqrt{A_{0}/2}(x+x_{0})$ . Note that the water surface ${\it\eta}$ of (2.10) for the KdV case is given by

This formula is consistent with the one obtained by Grimshaw (Reference Grimshaw1971), who derived the correction up to the third order.

5.2 Normal form of the KP equation

Now we define a normal form of (2.12) using the following transformation,

where the ${\it\beta}_{j}$ values are determined in such a way that the transformed equation (normal form) has a ‘good’ property. First, we require that the normal form is reduced to the KdV normal form when we have $\partial _{y}u=0$ . That is, from (5.2), we have ${\it\beta}_{1}=1,{\it\beta}_{2}=(4/3),{\it\beta}_{3}=(1/2)$ , but ${\it\beta}_{4}$ remains undetermined. Then we require that the normal form admits a solitary wave in the form of one-soliton solution,

This determines uniquely ${\it\beta}_{4}=1/2$ and the normal form of (2.12) is now given by

The phase $x_{0}(y,t)$ satisfies

Setting the angle ${\it\Psi}_{0}$ of the soliton with $(\partial x_{0}/\partial y)=\tan {\it\Psi}_{0}$ , we have the velocity $(\partial x_{0}/\partial t)$ of the KP soliton with the higher-order corrections. The corresponding solitary wave solution $u$ is then given by

and the water surface ${\it\eta}$ of (2.10) is given by

where $[A_{0}]:=A_{0}(1+\tan ^{2}{\it\Psi}_{0})=A_{0}/\cos ^{2}{\it\Psi}_{0}$ (see § 4.3). It is interesting to note that the correction to the quasi-two-dimensional approximation is entirely absorbed into the amplitude of the KdV equation (cf. (5.7)). This formula gives the relation between the observed amplitude $a_{0}$ from numerical or laboratory experiments of a shallow-water wave system and the KP amplitude $A_{0}$ . More explicitly, the observed amplitude $a_{0}$ of one soliton up to $O({\it\epsilon}^{2})$ is given by

This is a key equation when we apply the KP theory to laboratory experiments of shallow-water waves with finite values of ${\it\Psi}_{0}$ and $a_{0}$ .

5.3 Re-evaluation of the previous results based on the normal form

Recall that the KP equation is derived under the assumptions of quasi-two-dimensionality, weak dispersion and weak nonlinearity. Here, we show that it is necessary to include a higher-order correction for physical applications of the KP theory. Specifically, the observed wave amplitude in the laboratory and numerical experiments should be converted to the corresponding KP amplitude via the formula (5.14) that was derived from the normal form of the higher-order KP equation (5.10). Then, the KP parameter $k$ of (4.8) can be expressed in terms of the observed incident-wave amplitude $a_{0}$ and the incident angle ${\it\Psi}_{0}$ by

Note that, for small $a_{0}$ , the KP parameter $k$ , including the higher-order corrections, reduces to the heuristically derived parameter in (4.15). The maximum wave amplification $\hat{{\it\alpha}}$ in the KP theory can be obtained as follows.

For the case with $k<1$ (the Mach reflection), we have

where we have used (5.14) to evaluate the stem wave amplitudes $A_{[1,4]}$ and the incident wave amplitude $A_{0}=A_{[1,3]}=A_{[2,4]}$ . It is emphasized that (5.16) shows the relation between the physical wave amplification ${\it\alpha}_{w}$ and the KP amplification $\hat{{\it\alpha}}$ .

For the case $k>1$ (regular reflection), we use the same formula in (4.1) with (5.15) which gives

Note here that the difference between (4.3) and this formula is $(1/2)\sin ^{2}{\it\Psi}_{0}$ , which is negligible for this approximation.

The numerical and laboratory results are now re-evaluated with the new formulae (5.16) and (5.17) with (5.15), and the results are shown in figure 9(b). For comparison, the original prediction of Miles is also shown in figure 9(a). Laboratory data by Perroud (Reference Perroud1957) and Melville (Reference Melville1980) are not presented here because the propagation distance in their experiments was too short to achieve the asymptotic state.

Since the numerical simulations of Funakoshi (Reference Funakoshi1980) were conducted with small incident waves ( $a_{0}=0.05$ ), his results match the KP predictions very well; the slightly shifted deviation to the left in $k$ with the original Miles’s theory (figure 9 a) is now corrected. Tanaka (Reference Tanaka1993)’s results also match the theory; the higher-order correction substantially shifts his numerical data to the positive $k$ value. Note that Tanaka’s numerical experiments were undertaken with a relatively large value of $a_{0}$ and a large value of the incidence angle ${\it\Psi}_{0}$ . The higher-order correction of the amplification factors (5.16) and (5.17) is very effective, although the agreement of Tanaka’s results with the KP theory is not as good as Funakoshi’s, especially for the data near $k=1$ . This is because the KP theory, even at higher order, still assumes small values of $a_{0}$ and $\tan {\it\Psi}_{0}$ . (Recall that Tanaka’s results were with $a_{0}=0.3$ and ${\it\Psi}_{0}\approx 1$ near $k=1$ .)

As shown in figure 9, the higher-order KP theory results in a striking improvement in agreement of the laboratory data with the theory. While the laboratory data with the different incident wave angles appear in the separate datasets in the plot with Miles’s $k$ (in figure 9 a), the plot of the same datasets with the higher-order KP theory (figure 9 b) emerges as a single coherent dataset. In particular, the data with ${\it\Psi}_{0}=0.35$  rad exhibit substantial improvement in the amplification factor.

To obtain the maximum amplification near $k\approx 1$ , Tanaka (Reference Tanaka1993) used the incident wave amplitude $a_{0}=0.3$ in his numerical experiments and our laboratory experiments used the amplitude $a_{0}=0.277$ . The resulting amplifications for both numerical and laboratory results are $\hat{{\it\alpha}}\approx 3$ . Because the wave amplitude along the wall is close to unity ( $\neq O({\it\varepsilon})$ ), such not-so-small finite-amplitude conditions clearly violate the assumption of weak nonlinearity. Nonetheless, all of the numerical and laboratory data shown in figure 9 are in good agreement with the higher-order KP theory. In spite of the violations of the small parameters $a_{0}$ and ${\it\Psi}_{0}$ , this rather unexpected performance of the theory is remarkable.

Figure 9.

Numerical and experimental results of (a) the amplification factor ${\it\alpha}_{w}$ versus $k$ , the Miles parameter (4.2) and (b) the KP amplification factor $\hat{{\it\alpha}}$ versus the new higher-order KP parameter (5.15). The crosses $\times$ show the numerical results by Funakoshi (Reference Funakoshi1980), the hollow triangles ▵ show the numerical results by Tanaka (Reference Tanaka1993), the solid circles ● are the laboratory data by Li et al. (Reference Li, Yeh and Kodama2011) with ${\it\Psi}_{0}=0.35$  rad, solid triangles ▴ with ${\it\Psi}_{0}=0.52$  rad and solid squares ▪ with ${\it\Psi}_{0}=0.70$  rad.