HYDROMECHANICS

R. M. GARIPOV

ON THE ASYMPTOTICS OF WAVES IN A LIQUID OF FINITE DEPTH, CAUSED BY AN ARBITRARY INITIAL ELEVATION OF THE FREE SURFACE

(Presented by Academician M. A. Lavrent’ev, 29 V 1962)

If the free surface of a heavy liquid had the initial elevation \(\eta(x,0)=f(x)\), then, according to the linear theory,

\[ \eta(x,t)=\frac{1}{4\pi}\left\{\int_{-\infty}^{\infty} F(s)e^{i(sx-\omega t)}\,ds+ \int_{-\infty}^{\infty} F(s)e^{i(sx-\omega t)}\,ds\right\}; \tag{1} \]

\[ \omega=\sqrt{s\,\operatorname{th}s}\,\operatorname{sgn}s; \]
\(F(s)\) is the Fourier transform of the function \(f(x)\); the depth \(h\) is taken as the unit of length, and \(\sqrt{h/g}\) as the unit of time (\(g\) is the acceleration of gravity).

We shall consider waves propagating to the right from the place of disturbance \((x>0)\); then the second integral in (1) will be small in comparison with the first. We shall denote the first integral by \(I\); in it we replace \(\omega(s)\) and \(F(s)\) by their approximate values in a neighborhood of zero,
\[ \omega_0=s-\frac{s^3}{6},\qquad F_0(s)=|s|^p(a+ib\,\operatorname{sgn}s), \]
and denote the resulting expression by \(I_0\).

The asymptotic formulas for \(\eta(x,t)\), obtained by the method of stationary phase as \(t\to\infty\), are unsuitable for describing waves moving with maximal velocity, but the idea of the method suggests taking \(I_0\) as the asymptotic expression in this case. In work \((^1)\), for example, \(I_0\) (in the case \(p=1/2\)) was used for calculating the profile of the very first wave in a small neighborhood of its crest. But it turns out that the range of applicability of \(I_0\) is much wider; namely, the following assertion is valid:

If \(F(s)\) in \((-\infty,\infty)\) has absolutely integrable derivatives up to order \(m\), inclusive, and in \((-\infty,-\Delta)\) and \((\Delta,\infty)\), for arbitrarily small \(\Delta\), up to order \(n\), where \(n\ge p/3+2\), and for \(0<|s|\ll 1\),

\[ \left|F^{(k)}-F_0^{(k)}\right|\le \text{const.}\, |s|^{q-k},\qquad q\ge p+2\ge 2,\qquad k=0,1,\ldots,n, \tag{2} \]

then for values of \(x\)

\[ x-t\ge -\frac{c^2}{2}\,t^{1/3} \tag{3} \]

the inequality holds

\[ t^{\frac{p+1}{3}}\left|\eta(x,t)- \frac{1}{2}\left(\frac{2}{t}\right)^{-\frac{p+1}{3}} \{aA_p(\xi)+bB_p(\xi)\}\right| \le \frac{C(F)}{1-c^2}\left(t^{-\varepsilon}+t^{-\varepsilon_0}\right), \tag{4} \]

\[ \varepsilon_0=m-\frac{p+1}{3},\qquad \xi=(x-t)\left(\frac{2}{t}\right)^{1/3}, \]

where the integrals

\[ A_p(\xi)=\frac{1}{\pi}\int_0^\infty \sigma^p \cos\left(\xi\sigma+\frac{\sigma^3}{3}\right)\,d\sigma,\qquad B_p(\xi)=\frac{1}{\pi}\int_0^\infty \sigma^p \sin\left(\xi\sigma+\frac{\sigma^3}{3}\right)\,d\sigma \]

are to be understood in the Abel sense if they diverge; \(\varepsilon\) and \(\nu\) are related by

\[ \varepsilon=\frac{2}{3}-\frac{p+6}{2}\,\nu . \]

Let us explain that in \(I_0\) we made the change of integration variable

\[ s=\left(\frac{2}{t}\right)^{1/3}\sigma . \]

We introduce the notation
\[ \varphi=\frac{x}{t}s-\omega,\quad \varphi_0=\frac{x}{t}s-\omega_0,\quad DF=\frac{d}{ds}\frac{F}{\varphi'},\quad D_0F=\frac{d}{ds}\frac{F}{\varphi_0'}, \]
and outside \([-\delta,\delta]\) integrate \(I\) by parts \(n\) times:

\[ 4\pi I= \int_{|s|\leqslant\delta} F e^{i\varphi t}\,ds +\left.\sum_{k=0}^{n-1}\frac{1}{\varphi'}\left(\frac{i}{t}\right)^{k+1}e^{i\varphi t}D^kF\right|_{-\delta}^{\delta} +\left(\frac{i}{t}\right)^n \int_{\delta\leqslant |s|} e^{i\varphi t}D^nF\,ds; \tag{5} \]

replacing in this expression \(F,\omega\) by \(F_0,\omega_0\), denoting the result by \(I_0^*\), we estimate the difference

\[ \begin{aligned} 4\pi |I-I_0^*|\leqslant& \left|\int_{|s|\leqslant\delta}\left(Fe^{i\varphi t}-F_0e^{i\varphi_0t}\right)ds\right| +\left.\sum_{k=0}^{n-1}\left|\frac{e^{i\varphi t}D^kF}{\varphi't^{k+1}} -\frac{e^{i\varphi_0t}D_0^kF_0}{\varphi_0't^{k+1}}\right|\right|_{\pm\delta} \\ &+\frac{1}{t^n}\left|\int_{\delta\leqslant |s|\leqslant \delta_1} \left(e^{i\varphi t}D^nF-e^{i\varphi_0t}D_0^nF_0\right)ds\right| \\ &+\frac{1}{t^n}\left|\int_{\delta_1\leqslant |s|} e^{i\varphi t}D^nF\,ds\right| +\frac{1}{t^n}\left|\int_{\delta_1\leqslant |s|} e^{i\varphi_0t}D_0^nF_0\,ds\right|. \end{aligned} \tag{6} \]

On the numbers \(\delta_0,\delta,\delta_1\) we impose the restrictions

\[ 0<\delta\leqslant\delta_1\leqslant1,\qquad \frac{\delta_0}{\delta}\leqslant c<1,\qquad \delta^3t\geqslant1 \tag{7} \]

and we shall assume

\[ x\geqslant \omega'(\delta_0)t; \]

then in the interval \(\delta\leqslant |s|\leqslant1\), noting that \(0<\omega'(s)\leqslant1\) is even and is maximal at \(s=0\), we have

\[ \varphi'\geqslant (1-\omega'(1))(1-c^2)s^2, \tag{8} \]

and for \(|s|\geqslant1\),

\[ \varphi'\geqslant (1-\omega'(1))(1-c^2). \tag{8a} \]

Taking this into account, we estimate \(D^rF,\;D^rF-D_0^rF_0\). In detail,

\[ D^rF= \sum_{\substack{k_0+\ldots+jk_j=r\\ k_1+\ldots+k_j=k-r}} \frac{F^{(k_0)}(\varphi'')^{k_1}\ldots(\varphi^{(j+1)})^{k_j}} {(\varphi')^k}; \]

as in the other intermediate calculations, the constant coefficients are omitted here. By virtue of (8a) and the boundedness of \(\omega^{(k)}(s)\), we have, for \(|s|\geqslant1\),

\[ |D^rF|\leqslant \frac{1}{1-c^2}\sum_{k=0}^{r}|F^k|, \tag{9} \]

and for \(\delta \leq |s| \leq 1\), taking into account that \(\varphi^{(2k)}=-\omega^{(2k)}\sim s\) \((k=1,2,\ldots)\), and using (8), we obtain

\[ |D^rF|\leq \frac{1}{1-c^2}\sum |s|^{p-k_0-2k+k_1+k_3+\cdots} \leq \frac{1}{1-c^2}|s|^{p-3r}, \tag{10} \]

since, expressing \(k\) and \(k_0\) in terms of the remaining summation indices, we have

\[ p-k_0-2k+k_1+k_3+\cdots = p-3r+2k_3+\cdots \geq p-3r. \tag{11} \]

It is not difficult to see that \(D_0^rF_0\) satisfies inequality (10) for all \(|s|\geq\delta\), since in this case, because the derivatives of \(\varphi_0\) above third order are equal to zero \((k_3=\cdots=0)\), equality always holds in (11).

Thus, \(D_0^rF_0\) is absolutely integrable for \(r>(p+1)/3\), and \(D^rF\) for \(r\leq n\); therefore the integration by parts carried out in (5) is permissible.

By analogous, but longer, calculations, using (2), (8), and the obvious fact

\[ |s|\leq 1,\qquad |\varphi^{(k)}-\varphi_0^{(k)}| \leq \text{const}\cdot |s|^{5-k},\qquad k=0,1,\ldots, \tag{12} \]

we obtain

\[ \delta\leq |s|\leq 1,\qquad |D^rF-D_0^rF_0| \leq \frac{1}{1-c^2}\left(|s|^{p+2-3r}+|s|^{q-3r}\right). \tag{13} \]

Now let us return to (6). In view of (2) and (12),

\[ |Fe^{i\varphi t}-F_0e^{i\varphi_0t}| = |(F-F_0)e^{i\varphi t}+F_0(e^{i\varphi t}-e^{i\varphi_0t})| \leq |s|^q+t|s|^{p+5}. \]

Proceeding similarly with the remaining terms, taking into account the inequalities (10) and (13) obtained above, and simplifying with the help of the last of inequalities (7), we obtain

\[ |I-I_0^*| \leq \frac{C}{1-c^2} \left\{\delta^{q+1}+t\delta^{p+6} +\left(t\delta_1^{p+6}+\delta_1^{p+1}\right)(t\delta_1^3)^{-n} +t^{-n}\right\}. \tag{14} \]

Next we integrate by parts the second integral in (1) \(m\) times; then

\[ \left|\int_{-\infty}^{\infty} Fe^{i(sx+\omega t)}\,ds\right| \leq Ct^{-m}, \tag{15} \]

where the constant \(C\) in these inequalities depends only on the function \(F(s)\). Now, varying \(\delta,\delta_1\) subject to the conditions (7), we minimize the right-hand side of (14). Put \(\delta_0=c\delta\), \(\delta=t^{-\alpha}\) \((\alpha\leq 1/3)\), \(\delta_1=t^{-\alpha_1}\) \((\alpha\geq \alpha_1\geq 0)\); for brevity we restrict ourselves to the case \(n\geq p/3+2,\ q\geq p+2\) (the result is valid for \(n>(p+1)/3,\ q>p\)); then the first term may be neglected in comparison with the second, while the third is minimal for \(\alpha_1=0\) and is \(\leq t^{-n+1}\), and the second term is \(\geq t^{-(p+3)/3}\geq t^{-n+1}\).

Thus, on the right-hand side of (14) only the second term may be retained, since the remaining terms can be made smaller than its minimal value. Combining (14) and (15), and denoting \(1-2\alpha=1/3+\nu,\ \alpha(p+6)-1=(p+1)/3+\varepsilon\), we complete the proof of the assertion if we show that

\[ I_0^*=\lim_{w\to +0}\int_{-\infty}^{\infty} e^{-w|s|}F_0e^{i\varphi_0t}\,ds. \tag{16} \]

This is obvious if \(I_0\) exists in the ordinary sense; in the opposite case—

where \(I_0(w)\) (\(w>0\)) is first integrated by parts analogously to (5):

\[ 4\pi I_0(w)=\int_{|s|\leqslant 1} e^{-w|s|}F_0 e^{i\varphi_0 t}\,ds +\sum_{k=0}^{n-1}\frac{1}{\varphi_0'} \left(\frac{i}{t}\right)^{k+1} e^{i\varphi_0 t}D^k\left(e^{-w|s|}F_0\right)\Bigg|_{-1}^{1} + \left(\frac{i}{t}\right)^n \int_{1\leqslant |s|} e^{i\varphi_0 t}D_0^n\left(e^{-w|s|}F_0\right)\,ds. \tag{17} \]

Let us verify that the last integral converges uniformly with respect to \(w\geqslant 0\); indeed,

\[ \left|e^{i\varphi_0 t}D^n\left(e^{-w|s|}F_0\right)\right| \leqslant |s|^{p-3n}\sum_{k=0}^{n}|ws|^k e^{-w|s|} \leqslant \left(1+\sum_{k=1}^{n} k^k e^{-k}\right)|s|^{p-3n}. \]

Passing to the limit in expression (17), we obtain (16), since in \(I_0^*\) one may put \(\delta=1\), because it in fact does not depend on \(\delta\).

The functions \(A_p(x)\), \(B_p(x)\) satisfy the differential equation

\[ y'''-xy'-(p+1)y=0 \]

and in certain special cases can be expressed in terms of the well-known Airy functions \({}^{(2)}\),

\[ A_{2k}(x)=(-1)^k\operatorname{Ai}^{(2k)}(x),\qquad B_{2k+1}(x)=(-1)^{k+1}\operatorname{Ai}^{(2k+1)}(x),\qquad k=0,1,\ldots . \]

In conclusion, the author expresses his gratitude to S. Pokhozaev for discussion of this work and for valuable comments.

Institute of Hydromechanics
Siberian Branch of the Academy of Sciences of the USSR

Received
26 V 1962

REFERENCES

\({}^{1}\) Yu. L. Gazaryan, Acoustical Journal, 1, No. 3, 203 (1955).
\({}^{2}\) J. C. P. Miller, The Airy Integral Giving Tables of Solutions of the Differential Equation \(y'''=xy\), Cambridge, 1946.