Inclusion of dispersive terms in the averaged Lagrangian method: turning to the complex amplitude of envelope

Abstract

Whitham’s method of averaged Lagrangian is extended to include dispersive terms on the example of Stokes waves on the surface of a layer of ideal fluid. We derive a Lagrangian with explicit ${\mathcal {A}}_ {x}\overline{\mathcal {A}}_ {x}$ and ${\mathcal {A}}_ {x}\overline{\mathcal {A}}$ terms partially present in Whitham’s Lagrangian in implicit form in the term with nonlinear frequency, ${\mathcal {A}}$ being the complex-valued amplitude of the wave envelope. The ${\mathcal {A}}_ {x} \overline{\mathcal {A}}_ {x}$ term generates the $a_x^2$ term in the dispersive part of the Lagrangian, in addition to Whitham’s terms with $a^2$ and $a^4$ ($a$ being the real wave amplitude). The use of complex amplitude of envelope in the averaged Lagrangian allowed us to avoid the introduction of the nonlinear dispersion relation in the Whitham’s method. The variation of the generalized Lagrangian gives the correct evolution equations for the wave envelope and velocity potential.

Appendix

The coefficients of equation (25) before averaging over $\theta _{0}$ are ($E=\exp i\theta _{0}$)

$$\begin{aligned}&m_{1} =\varepsilon \frac{iE}{2k_{0}}\hbox {sinh}\xi ,\quad m_{2}=\varepsilon ^{2}\frac{iE^{2}}{2k_{0}}\hbox {sinh}2\xi ,\\&m_{11} =\frac{1}{8}\varepsilon ^{2}k_{0}\hbox {sinh}2\xi ,\quad m_{11}^{a}=-\varepsilon ^{2}\frac{E^{2}}{8}k_{0}\xi , \\&m_{12} ={\frac{E^{-1}}{6}}\varepsilon ^{3}k_{0}\hbox {sinh}3\xi ,\quad m_{12}^{a}=-\frac{1}{2}\varepsilon ^{3}k_{0}E^{3}\hbox {sinh}\xi ,\\&m_{22} =\frac{1}{4}\varepsilon ^{4}k_{0}\hbox {sinh}4\xi ,\quad m_{22}^{a}=-\frac{1}{2}\varepsilon ^{4}k_{0}E^{4}\xi ,\\&n_{1}=\varepsilon ^{2}\frac{iE}{2k_{0}^{2}}(\left( 1+k_{0}h\sigma \right) \hbox {sinh}\xi -\xi \hbox {cosh}\xi ),\\&n_{2} =\varepsilon ^{3}\frac{1}{8}(k_{0}h\sigma {\hbox {sinh}}\hbox {2} \xi -\xi \hbox {cosh}2\xi +\xi ),\\&n_{2}^{a} =-\varepsilon ^{3}\frac{E^{2}}{16}({\hbox {sinh}}\hbox {2}\xi +4k_{0}h\sigma \xi -2\xi ),\\&n_{3} =\varepsilon ^{4}\frac{E}{72}(9\hbox {sinh}\xi -\hbox {sinh3}\xi +6k_{0}h\sigma {\hbox {sinh}}\hbox {3}\xi -6\xi \hbox {cosh}3\xi ), \\&n_{3}^{a} =\varepsilon ^{4}\frac{E^{3}}{24}(9\hbox {sinh}\xi -\hbox { sinh3}\xi -6k_{0}h\sigma \hbox {sinh}\xi -6\xi \hbox {cosh}\xi ),\\&n_{4} =\varepsilon ^{4}{\frac{1}{16k_{0}}}\left( \left( {2k_{0}^{2}h^{2}\sigma ^{2}+2\xi ^{2}+1}\right) {\hbox {sinh}2\xi }\right. \\&\quad \left. -\,{4}k_{0}h\sigma \xi \hbox {cosh}2\xi {+}2\xi \left( {2}k_{0}h\sigma -1\right) \right) , \\&n_{4}^{a} =\varepsilon ^{4}{\frac{E^{2}}{48k_{0}}(-3k_{0}h\sigma \hbox { sinh}2\xi +3\xi \hbox {cosh}2\xi } \\&\quad -\,{6k_{0}^{2}h^{2}\sigma ^{2}\xi +6k_{0}h\sigma \xi +2\xi ^{3}-3\xi )},\\&n_{5}=\varepsilon ^{2}\frac{E}{2{k_{0}}}\hbox {sinh}\xi ,\quad n_{6}=\varepsilon ^{3}\frac{E^{2}}{4{k_{0}}}\hbox {sinh}2\xi ,\quad n_{7}=-i\varepsilon n_{1},\\&n_{j}^{a} =E^{2}n_{j},\quad j=8\div 11,\\&n_{12}^{a} =-E^{2}n_{12},\quad n_{13}^{a}=E^{2}n_{13},\\&n_{9} =\varepsilon \frac{1}{ik_{0}}n_{8}=\varepsilon ^{4}\frac{1}{16k_{0}} (\sinh 2\xi +2\xi ),\\&n_{10} =-2n_{12}=\varepsilon ^{4}\frac{iE}{12}(\sinh 3\xi +3\sinh \xi ),\\&n_{11}=\nu _{13}=\varepsilon ^{4}\frac{i}{32k_{0}}(\left( 2k_{0}h\sigma +1\right) \hbox {sinh}2\xi \\&\quad -\,2\,\xi \hbox {cosh}2\xi +4k_{0}h\sigma \xi ),\quad n_{14}=\varepsilon ^{2}n_{5}. \end{aligned}$$

Note that $\theta _{0}$ is also present in $\xi =k_{0}(h+\eta )$, where $\eta$ is given by Eq. (20).