Euler–Darboux–Poisson Equation in Context of the Traveling Waves in a Strongly Inhomogeneous Media

Abstract

The existence of traveling waves in an inhomogeneous medium is a vital problem, the solution of which can help in modeling the wave propagation over long distances. Such waves can be storm waves or tsunami waves in the seas and oceans. The presence of solutions in the form of traveling waves indicates that the wave propagates without reflection and, therefore, can transfer energy over long distances. Traveling waves within the framework of the 1D variable-coefficient wave equation exist only for certain configurations of an inhomogeneous medium, some of which can be found by transforming the original equation to the Euler–Darboux–Poisson equation. The solution of the last equation for certain parameter values is expressed in elementary functions, which are the sum of waves running in opposite directions. The mathematical features of such a transformation are discussed in this paper.

1. Introduction

In mathematical models describing water dynamics, one of the most important classes of solutions is the so-called traveling waves, that is, solutions represented as $f(x+t)$, where f is an arbitrary function, x is a coordinate, and t is time. The presence of solutions in the form of traveling waves tells us that the wave propagates without reflection and, therefore, without loss of energy and over long distances.

If the medium is smoothly inhomogeneous, the reflection is very small and asymptotic methods are effective here, since there is a small parameter—the ratio of the wavelength to the scale of inhomogeneities. In the simplified case of one-dimensional linear wave propagation in shallow water, the asymptotic methods [1,2,3,4,5] lead to the well-known Green’s law when the wave amplitude changes in direct proportion to $h^{-\frac{1}{4}}$. However, not all waves obey this law. For example, the height of a solitary wave (soliton) in the nonlinear dispersion theory can vary proportionally to $h^{-1}$. Therefore, an interesting and important task for practical purposes is to find solutions in which amplitude varies significantly more (or less) than Green’s law suggests.

The linear system of shallow water equations is chosen as the mathematical model of this paper and described in Section 2. Furthermore, in Section 3, the principle of reducing the wave equation with variable coefficients to the Euler–Poisson–Darboux equation, described in [6], is touched upon. Section 4 contains the second component of the wave field—the velocity averaged over the current depth. Furthermore, in Section 5, a qualitative analysis of the results obtained is given. It was also shown that the water flow rate at the shore takes finite values, despite the infinite speed values. In Section 6 and Section 7, examples of wave fields obtained within the framework of this approach are given. All the results are summarized in the conclusion.

2. Shallow Water Equations for Waves in a Liquid of Variable Depth

Let us consider the dynamics of one-dimensional long waves propagating in a reservoir of variable depth (Figure 1). We assume that the point $x=0$ is the shore, and the depth change occurs along the x axis.

Then, to describe such waves, the system of equations of the linear theory of shallow water [1,3] is applicable, which represents the following two equalities

$$\frac{\partial \eta }{\partial t}+\frac{\partial }{\partial x}\left[h\left(x\right)u\right]=0$$

(1)

$$\frac{\partial u}{\partial t}+g\frac{\partial \eta }{\partial x}=0$$

(2)

where $h\left(x\right)$ is the water depth (the distance from the bottom to the water level in a state of calm), $\eta (x,t)$ is the displacement of the water surface, $u(x,t)$ is the depth-averaged flow velocity, and g is the gravity acceleration.

From Equations (1) and (2), simple transformations can exclude the depth-averaged velocity u; therefore, we obtain a wave equation for the displacement $\eta$ with variable coefficients

$$\frac{{\partial }^2\eta }{\partial t^2}-\frac{\partial }{\partial x}\left[c^2\left(x\right)\frac{\partial \eta }{\partial x}\right]=0$$

(3)

where

$$c\left(x\right)=\sqrt{gh\left(x\right)}$$

(4)

is the variable speed of the long wave propagation. In a similar way, it is possible to exclude the displacement from Equations (1) and (2) while obtaining the wave equation for u

$$\frac{{\partial }^{2}u}{\partial t^2}-\frac{{\partial }^2}{\partial x^2}\left[c^2\left(x\right)u\right]=0$$

(5)

The resulting Equations (3) and (5) for displacement and velocity, generally speaking, are not equivalent; that is, not every Equation (3) solution is a solution of (5) and vice versa. It is worth noting that in classical examples of mathematical physics, Equations (3) and (5) are considered separately, although to describe the wave process, it is necessary to know all the wave field components; that is, both the displacement $\eta$ and the velocity u. By solving these equations separately, we can obtain large classes of mathematically good solutions. However, when finding the second component, some solutions from these classes may turn out not to be physically feasible or give impossible results.

Next, we will focus on finding solutions in the form of traveling waves in Equation (3), after which we will find the component u by expressing it from Equation (2)

$$u(x,t)=-g\frac{\partial }{\partial x}\int \eta (x,t)dt$$

(6)

3. The Method of Reducing the Wave Equation to the Euler–Poisson–Darboux Equation

In this section, we briefly present the method for obtaining a family of non-reflective bottom profiles and, accordingly, solutions to the $\eta$ offset obtained with such profiles $h\left(x\right)$, following [6].

We will look for the Equation (3) solution in the form

$$\eta (x,t)=A\left(x\right)G\left[\tau \right(x),t]$$

(7)

where $A\left(x\right)$, $\tau \left(x\right)$ and $G(\tau ,t)$ are some arbitrary smooth functions. After substituting (7) into Equation (3), we obtain the following expression:

$$A\left[\frac{{\partial }^{2}G}{\partial t^2}-c^2{\left(\frac{d\tau }{dx}\right)}^2\frac{{\partial }^{2}G}{\partial {\tau }^2}\right]-\left[\frac{d}{dx}\left(c^{2}A\frac{d\tau }{dx}\right)+c^2\frac{dA}{dx}\frac{d\tau }{dx}\right]\frac{\partial G}{\partial \tau }-\frac{d}{dx}\left[c^2\frac{dA}{dx}\right]G=0$$

(8)

and, if the following conditions are required,

$$c^2{\left(\frac{d\tau }{dx}\right)}^2=1$$

(9)

$$\frac{d}{dx}\left(c,A\right)+c\frac{dA}{dx}=\frac{2m}{\tau }A$$

(10)

$$\frac{d}{dx}\left(c^2\frac{dA}{dx}\right)=0$$

(11)

Equation (8) turns into the Euler–Poisson–Darboux equation for the function $G(\tau ,t)$

$$\frac{{\partial }^{2}G}{\partial t^2}-\frac{{\partial }^{2}G}{\partial {\tau }^2}-\frac{2m}{\tau }\frac{\partial G}{\partial \tau }=0$$

(12)

which is solved on the semi-axis $\tau >0$. It is well known (see, for example, [7,8,9]) that for a whole m, this equation has a general solution in the form of a finite sum of traveling waves. It can be written for natural m as

$${\tau }^{2m-1}G(\tau ,t)=\sum _{k=0}^{m-1}{a}_k{\tau }^k(f^{\left(k\right)}(\tau +t)+g^{\left(k\right)}(\tau -t))$$

(13)

where the coefficient $a_0$, due to linearity, can be taken equal to 1. Here, it is a fact that $a_1=-a_0=-1$ and further coefficients $a_k$ can be searched not only by the direct substitution of (13) in (12) but also by the following formula:

$$a_k={(-1)}^k\frac{2^{k-1}}{k!}\frac{A_{m-2}^{k-1}}{A_{2m-3}^{k-1}},\mathrm{where}{A}_n^k=\frac{n!}{(n-k)!},m>2$$

(14)

which is a generalization of the record of the coefficients obtained by Euler in [7] (problem 52).

The case of reduction to Equation (12) with a negative integer m is not discussed in this paper, since such a transition results in a fundamentally different wave behavior.

Now, it is necessary to deal with the conditions of (9)–(11). If we suppose that $A\left(x\right)=1$, condition (11) is automatically fulfilled. Condition (9) gives an expression for $\tau$

$$\tau \left(x\right)={\int }_0^x\frac{d\xi }{c\left(\xi \right)}$$

(15)

and by solving Equation (10), the following expressions are obtained (up to the shift of the coordinate x) for $\tau \left(x\right)$ and profiles $h\left(x\right)$

$$\tau \left(x\right)={\tau }_0{\left(\frac{x}{L}\right)}^{\frac{1}{2m+1}}$$

(16)

$$h\left(x\right)=h_0{\left(\frac{x}{L}\right)}^{\frac{4m}{2m+1}}=\frac{h_0}{{\tau }_0^{4m}}{\tau }^{4m}$$

(17)

where L is the characteristic scale of the depth change, $h_0$ is the depth at a distance of L from the edge, and the parameter ${\tau }_0$ characterizes the time of wave movement from a depth of $h_0$ to the shore (L, $h_0$, ${\tau }_0$$\in {\mathbf{R}}_{+}$). From Equation (17), we can immediately say that all such non-reflective profiles $h\left(x\right)$ (up to multiplication by a constant) lie between $x^{\frac{4}{3}}$ and $x^2$.

Thus, we reproduced the calculated equation of the bottom profiles obtained in [6], at which the non-reflective propagation of long waves is possible.

4. Speed Field

Next, we find the depth-averaged velocity u for an incident wave $\eta$ moving towards the shore. The formula for its displacement is written as

$$\eta (\tau ,t)=\frac{1}{{\tau }^{2m-1}}\sum _{k=0}^{m-1}{a}_k{\tau }^k{f}^{\left(k\right)}(\tau +t)$$

(18)

Further, in our work we consider only the displacement and velocity functions from the space $L^2$, which is equivalent to the finiteness of the wave energy in physics. The consequence of this choice is the equality ∀${\tau }_1>0$

$$\underset{t\to \infty }{lim}\eta ({\tau }_1,t)=0$$

(19)

$$\underset{t\to \infty }{lim}u({\tau }_1,t)=0$$

(20)

which can be interpreted as follows. Condition (19) describes the assumption that after the wave passage, the water returns to its original state. That is, the wave passes some arbitrary point ${\tau }_1$, and then, after some time, the water calms down and its level displacement will tend to 0. Before the wave came to this point ${\tau }_1$, there were also no wave disturbances in it. A similar requirement for speed is given by the condition (20), which says that after the passage (and before the appearance) of the wave through a fixed point ${\tau }_1$, no currents should arise and the water should rest at zero speed.

If we express velocity in terms of displacement (formula (6)), there is an indefinite integral in such an entry. However, since we are working with functions from $L^2$, using the causality principle, we can find an arbitrary constant and proceed to an integral with a variable upper limit. Namely, at the time $-\infty$, it is necessary that the speed be zero. Therefore, we have a definite integral from $-\infty$ to t. The expression for speed then takes the form

$$u=-g\frac{\partial }{\partial x}{\int }_{-\infty }^t\sum _{k=0}^{m-1}{a}_k{\tau }^{k+1-2m}{f}^{\left(k\right)}(\xi +\tau )d\xi =$$

(21)

$$=-\frac{g}{c}{\int }_{-\infty }^t\sum _{k=0}^{m-1}{a}_k[(k+1-2m){\tau }^{k-2m}{f}^{\left(k\right)}(\xi +\tau )+{\tau }^{k+1-2m}{f}^{(k+1)}(\xi +\tau )]d\xi$$

$$=-\frac{g}{c}\left[\frac{b_{-1}}{{\tau }^{2m}}\mathsf{\Phi }(t+\tau )+\sum _{k=0}^{m-1}\frac{b_k}{{\tau }^{2m-k-1}}{f}^{\left(k\right)}(t+\tau )+\frac{b_{m-1}}{{\tau }^m}{f}^{(m-1)}(t+\tau )\right]$$

where $\mathsf{\Phi }(t+\tau )={\int }_{-\infty }^{t}f(\xi +\tau )d\xi$, $b_{-1}=a_0(1-2m)$, $b_k=a_k+a_{k+1}(k+2-2m)$, $b_{m-1}=a_{m-1}$. From Formula (21), it can be seen that the speed consists not only of the sum of the first $m-1$ derivatives but also the integral of an arbitrary function f. If we denote $\mathsf{\Phi }(t+\tau )$ as $f^{(-1)}(t+\tau )$, Formula (21) can be written more compactly

$$u=-\frac{g}{c{\tau }^{2m}}\sum _{k=-1}^{m-1}{b}_k{\tau }^{k+1}{f}^{\left(k\right)}(t+\tau )$$

(22)

Expressions (18) and (22) form a hydrodynamic wave field in a traveling wave, the study of which is described below.

5. The Wave Field Analysis of the Traveling Wave

Set by Equation (16), $\tau \left(x\right)$ tends to zero near the shore. Therefore, a wave can reach the shore in a finite time $t={\tau }_0$, and it reaches infinity in an infinite time.

In the incident wave displacement far from the shore, that is, at $\tau \to +\infty$, as can be seen from Formula (18), the last term prevails, the amplitude of which is directly proportional to $h^{-\frac{1}{4}}$; therefore, far from the shore, the wave satisfies the well-known Green’s law. Furthermore, when the wave approaches the shore, the terms with smaller derivatives prevail over the others (except for in the case of $m=1$, when the solution of $\eta$ consists of one term). Approaching the shore, the wave amplitude is already proportional to $h^{-\frac{1}{2}+\frac{1}{4m}}$.

Next, we describe the speed behavior. Away from the shore ($\tau \to +\infty$), its formula is also dominated by the last term, which for any m, is directly proportional to $h^{-\frac{3}{4}}$ (Green’s law for velocity), and near the shore, where the first term prevails, the velocity is directly proportional to $h^{-1}$.

Both displacement and velocity are singular at the point $x=0$. From a physical point of view, it is clear why $x=0$ for our equation is a singularity. $x=0$ is the shore, and the equation describes the water movement, and we cannot jump from the equation for the liquid to the equation for the shore (or emptiness).

As mentioned earlier, in Formula (18), away from the shore, the most recent term prevails. We consider the wave to be set at some distance from the shore (far enough away). Its shape is described by an oscillogram (mareogram) $\eta (t,{\tau }_0)=a_{m-1}\phi (t+{\tau }_0)/{\tau }_0^m$, where $\phi \left(t\right)$ is some continuous on $\mathbf{R}$ function. Then, due to the smallness of the remaining terms, we can assume that this waveform describes the behavior of the $m-1$ derivative $a_{m-1}{f}^{m-1}(t,{\tau }_0)/{\tau }_0^m$, and it is convenient to rewrite the solution for (18) for this case in the form of

$$\eta (\tau ,t)=\frac{a_{m-1}}{{\tau }^m}\phi (t+\tau )+\frac{a_{m-2}}{{\tau }^{m+1}}{\int }_{-\infty }^t\phi (t_1+\tau )d{t}_1+\frac{a_{m-2}}{{\tau }^{m+2}}{\int }_{-\infty }^t{\int }_{-\infty }^{t_2}\phi (t_1+\tau )d{t}_{1}d{t}_2+\cdots$$

(23)

$$\cdots +\frac{a_0}{{\tau }^{2m-1}}{\int }_{-\infty }^t\cdots {\int }_{-\infty }^{t_2}\phi (t_1+\tau )d{t}_1\cdots d{t}_{m-1}$$

This formula helps us to understand the main feature of this solution, which is to change the wave shape when it approaches the shore. The wave shape seems to integrate $m-1$ times when approaching the shore.

From entry (23) and Formula (19), the following conditions immediately follow for the class of initial functions (oscillograms) $\phi \left(t\right)$.

$${\int }_{-\infty }^{+\infty }\phi \left(t_1\right)d{t}_1=0$$

(24)

$${\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{t_2}\phi \left(t_1\right)d{t}_{1}d{t}_2=0$$

(25)

$$...$$

$${\int }_{-\infty }^{+\infty }\cdots {\int }_{-\infty }^{t_2}\phi \left(t_1\right)d{t}_1\cdots d{t}_{m-1}=0$$

(26)

Similarly to the offset (23), you can write an expression for the velocity

$$u(\tau ,t)=\frac{b_{m-1}}{{\tau }^m}\phi (t+\tau )+\frac{b_{m-2}}{{\tau }^{m+1}}{\int }_{-\infty }^t\phi (t_1+\tau )d{t}_1+\frac{b_{m-3}}{{\tau }^{m+2}}{\int }_{-\infty }^t{\int }_{-\infty }^{t_2}\phi (t_1+\tau )d{t}_{1}d{t}_2+\cdots$$

(27)

$$\cdots +\frac{b_{-1}}{{\tau }^{2m}}{\int }_{-\infty }^t\cdots {\int }_{-\infty }^{t_2}\phi (t_1+\tau )d{t}_1\cdots d{t}_m$$

with a detailed analysis of which, together with (20), the last condition is immediately added to the waveform function $\phi$.

$${\int }_{-\infty }^{+\infty }\cdots {\int }_{-\infty }^{t_2}\phi \left(t_1\right)d{t}_1\cdots d{t}_m=0$$

(28)

it can be noted that the conditions (24)–(26) and (28) are not only sufficient but also necessary (provided that all integrals converge) to meet the requirements of (19) and (20), since the difference from zero of at least one integral entails the divergence of the subsequent ones following.

Thus, in order to apply the reduction model to the Euler–Poisson–Darboux equation for a wave, of which the $\phi$ waveform we know far from the shore, it is necessary and sufficient that its integrals over the infinite limit (by $\mathbf{R}$) up to the order of m are equal to 0. Alternatively, this can be replaced by a stronger constraint imposed on $\phi$. It is sufficient to require the existence of such a $C^m$ smooth function $F\left(t\right)$ that

$$F^{\left(m\right)}\left(t\right)=\phi \left(t\right)$$

(29)

and

$$\underset{t\to \infty }{lim}F\left(t\right)=0$$

(30)

Let us note that the resulting solution is singular near the shore (displacement and velocity tend to infinity). Such a singularity is characteristic of linear problems and, taking into account the nonlinearity, the wave either turns or rolls onto the shore and is reflected from it. It is worth noting that a physically important characteristic, such as the water flow rate Q, turns out to be finite at the edge, since

$$Q\left(t\right)=\underset{x\to 0}{lim}h\left(x\right)u(x,t)=\underset{x\to 0}{lim}{h}_0{\left(\frac{x}{L}\right)}^{\frac{4m}{2m+1}}\frac{-g}{c\left(x\right){\tau }^{2m}\left(x\right)}\sum _{k=-1}^{m-1}{b}_k{\tau }^{k+1}\left(x\right)f^{\left(k\right)}(t+\tau \left(x\right))=$$

(31)

$$=-\frac{h_{0}g}{{\tau }_0^{2m-1}}\underset{\tau \to 0}{lim}\sum _{k=-1}^{m-1}{b}_k{\tau }^{k+1}{f}^{\left(k\right)}(t+\tau )=-\frac{g{h}_0{b}_{-1}}{{\tau }_0^{2m-1}}{f}^{(-1)}\left(t\right)=\frac{g{h}_0(2m-1)}{{\tau }_0^{2m-1}}{\int }_{-\infty }^{t}f\left(\xi \right)d\xi$$

or, through the known oscillogram $\phi$, the flow rate is written as

$$Q\left(t\right)=\frac{g{h}_0(2m-1)}{{\tau }_0^{2m-1}}{\int }_{-\infty }^t\cdots {\int }_{-\infty }^{t_2}\phi \left(t_1\right)d{t}_1\cdots d{t}_m$$

(32)

from which we have obtained a finite function of water flow from time. Thus, using (32), we can estimate the amount of liquid that goes to the dry shore in the absence of wave reflection from the shore.

6. Solution of Shallow Water Equations at m = 1

As an example, we give here the classic case of the bottom profile shape, well-described in the paper [10]. However, it is worth mentioning that by using slightly different techniques in [6,10], the same non-reflective profile was obtained, which is given by the equation

$$h\left(x\right)=h_0{\left(\frac{x}{L}\right)}^{\frac{4}{3}}$$

(33)

and $\tau \left(x\right)$ is given by the equation

$$\tau \left(x\right)={\tau }_0{\left(\frac{x}{L}\right)}^{\frac{1}{3}}={\int }_0^x\frac{d\xi }{c\left(\xi \right)}=\frac{3L^{\frac{2}{3}}{x}^{\frac{1}{3}}}{\sqrt{g{h}_0}}$$

(34)

Then, the general equation solution for $m=1$ is the function

$$\eta (\tau ,t)=\frac{f(t+\tau )+g(t-\tau )}{\tau }$$

(35)

Let us investigate and illustrate the behavior of the wave running to the left.

$$\eta (\tau ,t)=\frac{f(t+\tau )}{\tau }$$

(36)

obviously, so that

$$\tau \left(x\right)\sim h^{\frac{1}{4}}$$

(37)

therefore, it can be seen from Formula (36) that the wave amplitude is modified in direct proportion to $h^{-\frac{1}{4}}$; that is, it changes as Green’s law suggests. This can be seen in the illustration of the example below (Figure 2).

The expression for the depth-averaged velocity contains 2 terms and appears as:

$$u(x,t)=-\frac{g}{c}\frac{\partial }{\partial \tau }{\int }_{-\infty }^t\frac{f(t+\tau )}{\tau }dt=-\frac{g}{c}\left[\frac{f(t+\tau )}{\tau }-\frac{1}{{\tau }^2}\mathsf{\Phi }(t+\tau )\right]$$

(38)

The amplitude of the first velocity term, which prevails far from the shore, is directly proportional to $h^{-\frac{3}{4}}$, and the second term, which plays a role near the shore, is directly proportional to $h^{-1}$.

Let us consider a specific function as an illustration. At the point ${\tau }_0=\tau \left(x_0\right)$, the oscillogram $\eta \left(t\right)$ is known as:

$$\eta \left(t\right)=-2A\frac{tanh\frac{t}{T}}{cosh\frac{t}{T}}$$

(39)

where the parameter T is responsible for the duration of the pulse passing through the given point $x_0$, and the parameter A is equal to the wave amplitude (take the point ${\tau }_0=100T$).

Let us note that the function in question (39) satisfies all the requirements obtained in the previous part, since

$${\int }_{-\infty }^{+\infty }\eta \left(t\right)dt=-2A{\int }_{-\infty }^{+\infty }\frac{tanh\frac{t}{T}}{cosh\frac{t}{T}}dt=0$$

(40)

in this case, the formula that sets the wave running to the left is represented as this

$$\eta (\tau ,t)=-\frac{2A{\tau }_0}{\tau }\frac{tanh\left(\frac{t+\tau -{\tau }_0}{T}\right)}{cosh\left(\frac{t+\tau -{\tau }_0}{T}\right)}=-\frac{200A}{{\tau }_d}\frac{tanh(t_d+{\tau }_d-100)}{cosh(t_d+{\tau }_d-100)}$$

(41)

where $t_d=t/T$, and ${\tau }_d=\tau /T$ are dimensionless quantities.

The velocity field is then given as:

$$u(x,t)=2gA{\tau }_0{\tau }^{\prime }\left(\frac{tanh(t_d+{\tau }_d-100)}{\tau cosh(t_d+{\tau }_d-100)}+\frac{T}{{\tau }^{2}cosh(t_d+{\tau }_d-100)}\right)$$

(42)

it is known that

$${\tau }^{\prime }\left(x\right)=\frac{1}{c\left(x\right)}=\frac{1}{\sqrt{gh\left(x\right)}}=\frac{1}{x_d^{\frac{2}{3}}\sqrt{g{h}_0}}$$

(43)

where $x_d=x/L$. Assuming that the wave was initially set at ${\tau }_0=100T$, that is, at a distance of $x=L$, we thereby assumed (according to the equality (34)) that

$$\sqrt{g{h}_0}=\frac{3L}{100T}$$

(44)

then we can rewrite the equation for the velocity:

$$u(x,t)=2g_d\frac{L}{T^2}{A}_{d}L·100T·\frac{100T}{3L}{x}_d^{\frac{2}{3}}·\frac{1}{T}\left(\frac{tanh\left(t_d+{\tau }_d-100\right)}{{\tau }_{d}cosh(t_d+{\tau }_d-100)}+\frac{1}{{\tau }_d^{2}cosh(t_d+{\tau }_d-100)}\right)$$

(45)

where the index d means the dimensionlessness of the corresponding quantities. After the reduction, we really obtain the velocity dimension—$\frac{L}{T}$.

$$\frac{3}{20000A_d{g}_d}u(x,t)=x_d^{\frac{2}{3}}\left(\frac{tanh\left(t_d+{\tau }_d-100\right)}{{\tau }_{d}cosh(t_d+{\tau }_d-100)}+\frac{1}{{\tau }_d^{2}cosh(t_d+{\tau }_d-100)}\right)\frac{L}{T}$$

(46)

Let us denote k as the coefficient before the speed $1/A_d{g}_d$. Figure 3 shows that the shape of the wave velocity graph has changed significantly, and near the shore, as mentioned earlier, it resembles the integral of the function away from the shore. The transformation of the transition from one form to another can be seen in Figure 4.

The maximum values of the first and second terms in the speed Formula (38) cannot exactly be reached at the same point, because if the second term reaches its maximum (minimum), the first term is zero.

It is possible to estimate from what distance to the shore the second term (with $\mathsf{\Phi }$) in the velocity formula does not exactly exceed the first (with f). Let the waveform at the point ${\tau }_0$ have the amplitude of A and the wavelength of $\lambda$. Then, we can estimate the value of $\mathsf{\Phi }(\tau +t)<A\lambda$. Then

$$\frac{A\lambda }{{\tau }^2}<\frac{A}{\tau }$$

on condition

$$\tau >\lambda$$

(47)

therefore, for a cardinal transformation of the velocity function shape, it is at least necessary that $\tau$ be less than the length of the initial pulse.

7. Solution of Shallow Water Equations at m = 3

With the value of $m=3$, the incident wave approaching the shore is given by the expression

$$\eta (x,t)=\frac{1}{{\tau }^5}f(t+\tau )-\frac{1}{{\tau }^4}\frac{\partial f(t+\tau )}{\partial t}+\frac{1}{3{\tau }^3}\frac{{\partial }^{2}f(t+\tau )}{\partial t^2}$$

(48)

where $h\left(x\right)$ (up to the offset of the x coordinate):

$$h\left(x\right)=h_0{\left(\frac{x}{L}\right)}^{\frac{12}{7}}$$

(49)

and also $\tau \left(x\right)$

$$\tau \left(x\right)={\tau }_0{\left(\frac{x}{L}\right)}^{\frac{1}{7}}={\int }_0^x\frac{d\xi }{c\left(\xi \right)}=\frac{7L^{\frac{6}{7}}{x}^{\frac{1}{7}}}{\sqrt{g{h}_0}}$$

(50)

The first term in the solution (48) is proportional to $h^{-\frac{5}{12}}$, the second is proportional to $h^{-\frac{1}{3}}$, and the third is proportional to $h^{-\frac{1}{4}}$. Of the total sum (48), only the third term, which prevails in amplitude over the rest only away from the shore, satisfies Green’s law. As the wave approaches the shore, the second and first terms make a significant contribution to the amplitude.

In integral form, the displacement and velocity are rewritten as:

$$\eta (x,t)=\frac{1}{{\tau }^5}{\int }_{-\infty }^t{\int }_{-\infty }^{\xi }\phi (t_1+\tau )d{t}_{1}d\xi -\frac{1}{{\tau }^4}{\int }_{-\infty }^t\phi (t_1+\tau )d{t}_1+\frac{1}{3{\tau }^3}\phi (t+\tau )$$

(51)

$$u=-\frac{g}{c}(-\frac{5}{{\tau }^6}{\int }_{-\infty }^t{\int }_{-\infty }^{\psi }{\int }_{-\infty }^{\xi }\phi (t_1+\tau )d{t}_{1}d\xi d\psi +\frac{5}{{\tau }^5}{\int }_{-\infty }^t{\int }_{-\infty }^{\xi }\phi (t_1+\tau )d{t}_{1}d\xi$$

(52)

$$-\frac{2}{{\tau }^4}{\int }_{-\infty }^t\phi (t_1+\tau )d{t}_1+\frac{1}{3{\tau }^3}\phi (t+\tau ))=$$

As an example of the specific function, let us consider the waveform also at the point ${\tau }_0=100T$

$$\phi \left(t\right)=3A{\tau }_0^3\frac{({sinh}^2\frac{t}{T}-5)sinh\frac{t}{T}}{{cosh}^4\frac{t}{T}}$$

(53)

which is the third derivative of the function $F\left(t\right)$:

$$F\left(t\right)=-\frac{3A{T}^3{\tau }_0^3}{cosh\frac{t}{T}}$$

(54)

Similarly (with the case $m=1$), we construct displacement and velocity graphs (see Figure 5, Figure 6, Figure 7 and Figure 8).

As you can see from the series of drawings and Formulas (18) and (23), the main difference between cases with different m is the number of transformations of the wave when it approaches the shore directly. Furthermore, for large m, the amplitude of the incident wave near the shore is larger (proportional to $h^{-\frac{1}{2}+\frac{1}{4m}}$). Away from the shore, the wave amplitude repeats Green’s law and behaves the same for all m. It is also worth noting that the asymptotics of the velocity away ($h^{-\frac{3}{4}}$) and near ($h^{-1}$) the coast do not depend on m at all (more precisely, they depend implicitly, since h depends on m).

8. Conclusions

In this paper, we discussed the method for obtaining solutions in the form of a traveling wave by reducing the linear shallow water model to the Euler–Poisson–Darboux equation. In this case, the waveform changes with distance, but, nevertheless, propagation occurs without reflection from inhomogeneities. The expression for the depth-averaged velocity is found, and the dynamics of such a wave field are investigated. It is also shown that the amount of water coming to the dry shore is finite, despite the infinite values of the amplitude of the traveling wave and its velocity. In the examples considered, it was illustrated how the waveform changes (integrates) when moving towards the shore. Of course, traveling waves in linear theory become singular in the vicinity of the shore ($x=0$); the wave amplitude, as well as the ratio $\frac{u}{c}$, tend to infinity. Here, we have obtained asymptotics of the solution behavior for displacement ($h^{-\frac{1}{4}}$ away and $h^{-\frac{1}{2}+\frac{1}{4m}}$ near the shore) and velocity ($h^{-\frac{3}{4}}$ away and $h^{-1}$ near the shore). However, nonlinear effects become important near the shore, which can lead to wave breaking, and this process is often observed in nature. Nevertheless, if the wave is long enough, the breaking does not have time to occur, and the wave is reflected from the shore. This process is well known for waves over a constant slope, where a nonlinear problem is solved; see, for example, [11]. The analysis of the wave field at the edge within the framework of the Euler–Darboux–Poisson equation, taking into account reflection and breaking, is an independent task that the authors hope to deal with in the future.