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A class of exact nonlinear traveling wave solutions for shallow water with a non-stationary bottom surface
Accepted Manuscript New exact solutions of nonlinear equations for shallow water with a non-uniform, non-stationary bottom surface F. Kogelbauer, M.B. Rubin
PII: DOI: Reference:
S0997-7546(18)30375-3 https://doi.org/10.1016/j.euromechflu.2018.12.005 EJMFLU 3416
To appear in:
European Journal of Mechanics / B Fluids
Received date : 6 June 2018 Revised date : 24 October 2018 Accepted date : 13 December 2018 Please cite this article as: F. Kogelbauer and M.B. Rubin, New exact solutions of nonlinear equations for shallow water with a non-uniform, non-stationary bottom surface, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
NEW EXACT SOLUTIONS OF NONLINEAR EQUATIONS FOR SHALLOW WATER WITH A NON-UNIFORM, NON-STATIONARY BOTTOM SURFACE F KOGELBAUER AND MB RUBIN Abstract. The GN nonlinear shallow water wave equations developed by Green and Naghdi [5, 6] are valid for a non-uniform pressure on the top surface and a non-stationary, non-uniform bottom surface. In contrast, the S nonlinear shallow water wave equations developed by Serre [12, 13] for uniform depth and later generalized by Seabra-Santos et al. [11] for non-uniform depth are limited to a stationary bottom surface and a uniform pressure applied to the top surface. This paper develops two new classes of exact nonlinear solutions of the GN equations for traveling and steady non-uniform bottom surfaces. One class admits cnoidal and soliton waves while the other class admits only cnoidal waves. Also, the explicit expressions for the pressure acting on the bottom surface and the average pressure through the depth in the GN equations are used to place physical restrictions on the motion which ensure that these pressures remain non-negative preventing cavitation.
December 29, 2018 1. Introduction Nonlinear wave propagation in shallow water is an old topic originally focused on waves in channels, but it has current important applications in coastal engineering. Since the equations for shallow water waves are developed using approximations, there are a number of different formulations that have been developed in the literature. Dias and Milewski [3] discuss the interesting history of various formulations and point out that the equations developed by Serre [12, 13] are identical to those developed by Green and Naghdi [5, 6] for flow in water of constant depth. An important physical feature of these equations is that they are invariant under superposed constant velocity and thus are valid for bi-directional wave propagation. The equations developed by Green and Naghdi are valid for three-dimensional flow, they depend upon two space variables and time and they are valid for 2010 Mathematics Subject Classification. Key words and phrases. Exact solution, nonlinear, shallow water, traveling wave, nonstationary bottom. 1
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F KOGELBAUER AND MB RUBIN
flow over a non-unifom bottom surface. These equations are denoted as GN. Seabra-Santos et al. [11] developed a generalization of the Serre equations that are valid for two-dimensional flow in water over a non-uniform bottom surface, which are denoted by S. Also, Bonneton et al. [1] indicate that ”It is now recognized that the Serre or Green-Naghdi (S-GN) equations represent the relevant system to model highly nonlinear weakly dispersive waves propagating in shallow water“. Some additional discussions of flow in water of variable depth can be found in [2, 4, 7, 10]. It is commonly thought that the S equations are the same as the GN equations. However, it will be shown in the next section that the GN equations are more general than the S equations. Although attention was limited to flow over a stationary bottom surface in [5, 6], this restriction is not necessary for the validity of these theoretical developments. In particular, the resulting equations remain valid without change for a non-uniform, nonstationary bottom surface with the vertical location of the bottom surface being a function of two horizontal coordinates (x, y) and time t α = α(x, y, t).
(1.1)
This observation allows the GN equations to be used for modeling ocean floor motion as a source of Tsunamis. In addition, the GN equations are valid for a non-uniform pressure applied to the top surface of the fluid. It will be shown that the S equations are not valid for either of these situations. The main purpose of this paper is to discuss some new exact solutions of the GN equations for two-dimensional traveling waves over a non-uniform, non-stationary bottom surface and stationary waves over a non-uniform bottom surface. 2. Review and discussion of the S-GN equations The development in the remainder of this work limits attention to twodimensional flow. Specifically, with reference to fixed Cartesian base vectors ei (i=1,2,3), consider a two-dimensional flow in the e1 − e3 plane with the force of gravity g per unit mass acting in the negative e3 -direction. The notation used here is the same as that used in [6]. A material fluid particle has location x in the horizontal e1 direction, z in the vertical e3 direction, and the bottom, middle and top surfaces of the fluid are located by z = (α, ψ, β) respectively, such that 1 ψ = α + φ, β = α + φ, (2.1) 2 where φ is the current depth of the fluid and all functions, including α, depend on (x, t).
NEW EXACT SOLUTIONS
3
The horizontal velocity u, the vertical velocity λ of the middle surface and the rate of change w of the vertical depth are defined by u = x, ˙
˙ λ = ψ,
˙ w = φ,
(2.2)
where a superposed (·) denotes material time differentiation φ˙ = φt + φx u.
(2.3)
Also, subscripts are used to denote partial differentiation with respect to t, x or later ξ. The fluid motion in the GN model, developed in [6], is characterized by the equations w + φux = 0,
(2.4a)
ρ∗ φu˙ = pˆβx − pαx − px ,
(2.4b)
ρ∗ φλ˙ = −ρ∗ gφ + p − pˆ,
1 ∗ 1 p ρ φw˙ = − (p + pˆ) + , 12 2 φ
(2.4c) (2.4d)
with (2.4a) being the conservation of mass, (2.4b) being the horizontal component of linear momentum, (2.4c) being the vertical component of linear momentum and (2.4d) being the balance of director momentum. In these equations, ρ∗ is the constant mass density of the fluid, (p, pˆ) are the pressures acting on the bottom and top surfaces of the fluid, respectively, and p is an integrated average pressure through the depth of the fluid, which is an arbitrary function of (x, t) due to the incompressibility assumption. The S equations, cf. [11], can be written as (2.4a) and an equation for u of the form 1 1 1 1 2 2 2 φu+ ˙ φ g + w˙ + αx u˙ + u αxx +φ g + αx u˙ + αxx u + w˙ αx = 0, 2 3 2 2 x (2.5) where the variables (h, ξ) in the S equations in [11] have been replaced by (φ, α), respectively, to unify the notations. Also, use has been made of the expressions u˙ = ut + ux u,
w = −φux ,
w˙ = φ(u2x − uuxx − uxt ).
(2.6)
The equations (2.4b)-(2.4d) have an advantage over (2.5) in that the values and influence of the pressures (p, p, pˆ) are explicit. This was used in [9] to determine the pressure acting on the top surface of the fluid due to the
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F KOGELBAUER AND MB RUBIN
bottom surface of a planing boat and in [8] to determine an equation for the bottom free surface in the double free region of the solution for a waterfall. To compare the GN and the S equations, it is convenient to solve equations (2.4c) and (2.4d) for the pressures (p, p) to obtain 2 1 1 ∗ 2 ∗ ¨ + w˙ , p = pˆ + ρ φ g + α ¨ + w˙ , p = pˆφ + ρ φ g + α (2.7) 2 2 3 ˙ Then, using (2.1) where use has been made of (2.1) and (2.2) to eliminate λ. and substituting (2.7) into (2.4b) yields 1 1 1 1 1 2 g + w˙ + α ¨ + ∗ pˆx φ + φ g + α φu˙ + φ ¨ + w˙ αx = 0. (2.8) 2 3 2 ρ 2 x Next, use is made of the expression α ¨ = (αtt + 2αtx u) + (αx u˙ + αxx u2 ),
(2.9)
to rewrite (2.8) in the form 1 1 1 1 2 2 2 φu˙ + φ g + w˙ + (αx u˙ + αxx u ) + φ g + αx u˙ + αxx u + w˙ αx 2 3 2 2 x 1 1 = − ∗ pˆx φ − φ2 (αtt + 2αtx u) − φ(αtt + 2αtx u)αx . ρ 2 x (2.10) It therefore follows that the GN equation (2.10) is more general than the S equation (2.5), since the GN equations do not need to limit attention to a uniform top pressure or a stationary bottom surface. The generality of a non-stationary bottom is essential for the new solutions developed in the next section. 3. Traveling wave solutions with a non-stationary bottom surface For the new class of solutions developed in this section, the vertical location of the bottom surface is specified by α = a(φ − H),
(3.1)
where a is a constant to be determined and H is a constant defining a critical value of the depth φ [see (3.15)]. It then follows from (2.1) that β = (1 + a)φ − aH,
1 ψ = [(1 + 2a)φ − 2aH], 2
1 λ = (1 + 2a)w. 2
(3.2)
NEW EXACT SOLUTIONS
5
Using (3.1), the pressures in (2.7) are given by 2 1 1 ∗ 2 ∗ p = pˆ + ρ φ g + + a w˙ , p = pˆφ + ρ φ g + + a w˙ , (3.3) 2 2 3 and equation (2.8) requires 1 1 1 1 1 2 φu˙ + φ g+ + a w˙ + ∗ pˆx φ + aφ g + + a w˙ φx = 0. 2 3 2 ρ 2 x (3.4) In the remainder of this paper, attention is limit to the case of a constant pressure p0 applied to the top surface pˆ = p0 .
(3.5)
Next, for a traveling wave with constant speed C in the positive x direction, all functions except x depend only on the parameter ξ defined by ξ = x − Ct,
f = f (ξ),
f˙ = ft + fx u = (u − C)fξ ,
(3.6)
and the conservation of mass (2.4a) requires (u − C)φ = k1 , where k1 is a constant of integration. Using (3.5) and the facts that 2 k12 φξ 1 k12 k1 2 φξ , w˙ = , , φu˙ = u˙ = −k1 3 = φ 2 φ2 ξ φ ξ φ φ ξ
(3.7)
(3.8)
equation (3.4) can be rewritten in the form " # 1 2 2 φξ φξ 2 (1 + a) 2 3 φ + + +a φ = −a(1+2a)φξ , (3.9) F H φ 3 φ ξ φ ξ ξ
where use has been made of the Froude number F defined by F2 =
k12 . gH 3
(3.10)
It will be shown that when a vanishes in (3.9) and the bottom surface is flat, the solution yields standard solitary and cnoidal waves, as discussed in subsection 3.1. From (3.9) it can also be seen that when (a = − 12 ) a new nontrivial class of solutions can be obtained which is discussed in subsection
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F KOGELBAUER AND MB RUBIN
3.2. To discover a second new class of solutions it is convenient to expand (3.9) and divide the result by 2φ to deduce that " # 2 1 1 φ 1 1 φ 1 1 ξ ξξ (1 + a) 2 3 φ2 + 2 + a2 − + + a = 0. F H 2φ 2 3 φ 3 2 φ ξ
(3.11) in (3.11) is discussed in The class of solutions associated with (a = subsection 3.3. Also, using (3.5), the pressures in (3.3) are given by " # 1 φξ 1 1 +a , p = p0 + ρ∗ gF 2 H 3 φ + 2 3 F H 2 φ φ ξ " (3.12) # 1 ∗ 2 3 2 1 1 φξ 2 p = p0 φ + ρ gF H φ +a . + 2 F 2H 3 3 φ φ ξ − 23 )
3.1. Solutions for a flat bottom (a=0). For a flat bottom a = 0 and (3.9) integrates to deduce that φξ 2 1 H 1 φ2 2 2 φ [k2 −(|F |−1) ]+ = 1− + 2 1 − 2 , (3.13) 3 φ ξ F 2H H φ F H H where k2 is a constant of integration. Then, multiplying (3.13) by φξ /φ2 and integrating yields 1 φ φ φ 1 2 2 φ φ = k2 − (|F | − 1) − 1− 2 −1 , 1− 3 ξ F2 H F H H H (3.14) where the constant of integration has been specified so that φ = H is a critical value of φ, i. e., φξ = 0,
for φ = H.
(3.15)
Moreover, substituting (3.14) into (3.12) yields 2 1 ∗ 1 3 H 3 H2 2 2H p = p0 + ρ gφ + (k2 − (|F | − 1) ) + 3F 2 1 − + , 2 2 2 φ φ 2 φ2 2 1 ∗ 2 H 2 φ 2 p = p0 φ + ρ gH (k2 − (|F | − 1) ) 2 + 1 + 2F 1 − . 2 H φ (3.16) Furthermore, it can be shown that (3.14) can be written in the form 1 2 1 φ − φ1 φ2 − φ φ φ = 2 −1 , (3.17) 3 ξ F H H H
NEW EXACT SOLUTIONS
where the roots (φ1 , φ2 ) are real and non-negative p 4k2 |F | + k22 k2 φ1 = |F | + − ≥ 0, H 2 2 p 4k2 |F | + k22 φ2 k2 = |F | + + ≥ 0, H 2 2
7
(3.18)
for k2 ≥ 0, so the solution of (3.17) corresponds to cnoidal waves. As a special case, (3.17) yields 2 1 2 φ φ φξ = 1 − 2 −1 for k2 = (|F | − 1)2 , (3.19) 3 F H H which is satisfied by the well-known solitary wave, e.g. [10], ! r 2 − 1) ξ 3(F φ(ξ) = H + H(F 2 − 1)sech2 for F 2 ≥ 1. 2 F 2H
(3.20)
3.2. A new class of solutions for (a = − 12 ). For a non-uniform, nonstationary bottom surface with a = − 21 , equation (3.9) integrates to obtain " 2 # 1 φξ 1 1 2 H 1 φ2 φ = k3 − |F | − √ + 1− + 2 1− 2 , 6 φ ξ F 2H H φ 2F H H 2 (3.21) where k3 is a constant of integration. Then, multiplying (3.21) by φξ /φ2 and integrating yields " # 2 ! 1 1 φ φ φ 1 2 φ φ = k3 − |F | − √ + 1− −1 −1 , 12 ξ F2 H 2F 2 H H H 2 (3.22) where the constant of integration has been specified using the condition (3.15). Moreover, substituting (3.21) into (3.12) yields p = p0 + ρ∗ gφ,
2 ! 2 1 H 1 H2 1 ∗ 2 p = p0 φ + ρ gφ k3 − |F | − √ + +1 2 φ2 2 φ2 2 2 H 2H + 2F 2 1 − . φ φ
(3.23)
Furthermore, it can be shown that (3.22) can be written in the form 1 2 1 φ − φ1 φ2 − φ φ φ = 2 −1 , (3.24) 6 ξ F H H H
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F KOGELBAUER AND MB RUBIN
where the roots (φ1 , φ2 ) are real and non-negative, q √ φ1 √ = 2|F | + k3 − 2 2k3 |F | + k32 ≥ 0, H q √ φ2 √ = 2|F | + k3 + 2 2k3 |F | + k32 ≥ 0, H
(3.25)
for k3 ≥ 0, so the solution of (3.24) corresponds to cnoidal waves. As a special case, (3.22) yields 1 2 φ = 12 ξ
1−
φ 2F 2 H
2 φ −1 , H
2 1 , for k3 = |F | − √ 2
(3.26)
which is satisfied by the solitary wave 1 ψ = H, 2
! 6(2F 2 − 1) ξ φ = H + H(2F − 1)sech , F2 2H ! r 2 − 1) ξ 1 6(2F 2 α = − H(2F 2 − 1)sech , 2 F2 2H ! r 2 − 1) ξ 1 6(2F β = H + H(2F 2 − 1)sech2 2 F2 2H 2
2
r
(3.27)
for 2F 2 ≥ 1.
It is interesting to note that for this solution the height of the middle surface of the fluid remains constant, ψ = H2 , so there is no vertical acceleration of this surface. Also, the result (3.23) that the pressure p is purely hydrostatic indicates that the vertical accelerations of the fluid above and below the middle surface compensate for each other. 3.3. A new class of solutions (a = − 32 ). Taking a = − 23 and integrating (3.11) yields 1 2 3 φ 1 φ2 φ φ = 1+ − 2 2 −1 , (3.28) 6 ξ 2 H F H H where the constant of integration has been determined by the condition (3.15). Differentiating this equation, dividing the result by φ2 φξ and using the expression 1 φξ 1 = 3 (φφξξ − φ2ξ ), (3.29) φ φ ξ φ
NEW EXACT SOLUTIONS
9
it can be shown that 1 1 φξ 1 3 1 φ 1 φ2 1 φ3 = − + . − 6 φ φ ξ φ3 2 F 2 H F 2 H 2 2F 2 H 3 Then, substituting (3.29) into (3.12) yields 3 ∗ φ F 2H 2 2 H p = p0 + ρ gH − −1 , + 2 H φ2 3 φ 1 p = p0 φ + ρ∗ gφ2 . 2
(3.30)
(3.31)
Furthermore, it can be shown that (3.28) can be written in the form 1 2 φ2 − φ φ 1 φ − φ1 φ = 2 −1 , (3.32) 6 ξ F H H H where the roots (φ1 , φ2 ) are real, with one the other being non-negative, " r 3F 2 φ1 = 1− 1+ H 4 " r 3F 2 φ2 = 1+ 1+ H 4
of them being non-positive and 8 3F 2 8 3F 2
# #
≤ 0,
(3.33)
≥ 0,
so the solution of (3.32) corresponds to cnoidal waves with either φ2 ≤ φ < H for 3F 2 < 1
or
H ≤ φ ≤ φ2 for 3F 2 ≥ 1.
(3.34)
3.4. Physical restrictions. For these solutions to be physical, the constants of integration must be restricted so that φ2ξ and the pressures (p, p) remain non-negative. To prove existence of solutions defined in subsection 3.1 which are physical take k2 = k(|F | − 1)2 ,
k ≥ 1 and F 2 ≥ 1.
(3.35)
These conditions ensure that φ2 ≥ H ≥ φ1 in (3.18), with cnoidal waves oscillating between the two largest roots, H ≤ φ ≤ φ2 .
(3.36)
The specifications (3.35) and the restriction (3.36) ensure that all terms in the expressions (3.16) are non-negative for p0 ≥ 0 so the pressures (p, p) are non-negative.
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F KOGELBAUER AND MB RUBIN
Similarly, to prove the existence of a class of solutions defined in subsection 3.2 which are physical, take 2 1 , k ≥ 1 and 2F 2 ≥ 1. (3.37) k3 = k |F | − √ 2 These conditions ensure that φ2 ≥ H ≥ φ1 in (3.25) so the solution of (3.24) requires (3.36). The specifications (3.37) and the restriction (3.36) ensure that all terms in the expressions in (3.23) are non-negative for p0 ≥ 0 so the pressures (p, p) are non-negative. Finally, to develop restrictions which ensure that the solution in subsection 3.3 is physical, it is noted that p in (3.31) can be rewritten in the form 3 H2 2 2 φ . p = p0 + ρ∗ gH( 2 )f (x) , f (x) = x3 − x2 + x − F 2 , x = 2 φ 3 3 H
(3.38)
The discriminant of the cubic equation f (x) = 0 is negative for all values of course F 2 , which indicates that f (x) has only one real root. Furthermore, it can be shown that p(φ2 ) ≥ 0 for p0 ≥ 0 and all values of F 2 . Moreover, the value of p(H) is given by 3 p(H) = p0 + ρ∗ gH(1 − F 2 ) . 2
(3.39)
Consequently, the pressures (p, p) in (3.31) will be non-negative for p0 ≥ 0 only when F 2 is bounded above by F2 ≤ 1 +
2p0 . 3ρ∗ gH
(3.40)
It is emphasized that these restrictions on the fluid motion without explicit equations for the pressures (p, p), which are not known in the S equations for motion of shallow water waves. 4. Examples To compare the solitary wave solutions discussed in subsections 3.1 and 3.2, it is noted that for the flat bottom solution, the vertical location of the bottom and top surfaces of the waves are given by s ! 2 3(F − 1) ξ F , αF = 0, βF = φF (ξ) = H + H(FF2 − 1)sech2 FF2 2H (4.1) for FF2 ≥ 1,
NEW EXACT SOLUTIONS
11
where FF is the associated Froude number defined in (3.10). Similarly, the non-uniform, non-stationary, variable bottom solution in subsection 3.2 can be written in the form s ! 2 6(2F − 1) ξ 1 V , αV = − H(2FV2 − 1)sech2 2 FV2 2H (4.2) s ! 2 1 ξ 6(2F − 1) V βV = H + H(2FV2 − 1)sech2 , for 2FV2 ≥ 1, 2 FV2 2H where FV is the associated Froude number. Since the GN equations are invariant under superposed rigid body motions and allow bidirectional wave propagation, the Froude numbers (FF , FV ) can be determined by traveling waves using (3.7), (3.10) and the boundary conditions φ = H, φ = H,
u = 0, u = 0,
CF2 , gH C2 C = CV for ξ → ±∞ =⇒ FV2 = V , gH C = CF for ξ → ±∞ =⇒ FF2 =
(4.3)
or they can be determined by the velocity of the fluid past a stationary depression using (3.7), (3.10) and the boundary conditions φ = H, φ = H,
u = uF , u = uV ,
u2F , gH u2 C = 0 for ξ → ±∞ =⇒ FV2 = V . gH
C = 0 for ξ → ±∞ =⇒ FF2 =
(4.4)
In particular, for the traveling waves (4.3), it can be seen from (4.1) and (4.2) that the bottom of the fluid far from the peak of the traveling wave has the same vertical location for both solutions αF = αV = 0 for ξ → ±∞, and the peaks of the waves (at ξ = 0) are given by 1 2 2 + FV . βF = HFF , βV = H 2
(4.5)
(4.6)
Figure 1 plots the flat bottom solution (F ) and the variable bottom solution (V ) when the waves travel with the same speed FF2 = FV2 = 1.2,
(4.7)
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F KOGELBAUER AND MB RUBIN
and Figure 2 plots these solutions when the waves travel with different speeds, 1 2 2 FF = + FV = 1.7, FV2 = 1.2, (4.8) 2 but have the same peak height (βF = βV ). From these figures, it can be seen that the variable bottom traveling wave has a narrower peak than the flat bottom traveling wave. 5. Conclusions The main conclusions of this paper are summarized as: • The general GN equations describe three-dimensional fluid motion over a non-uniform, non-stationary bottom surface with non-uniform pressure applied to the top surface. In contrast, the generalized S equations are valid only for a stationary bottom surface and a uniform pressure applied to the top surface. • The GN equations include explicit expressions for the pressures (p, pˆ) on the bottom and top surfaces, respectively, and for the average pressure p through the depth of the fluid. Physical restrictions on the solutions have been proposed which require these pressures to remain non-negative to prevent cavitation. These pressures are not determined by the S equations. • Two new classes of exact nonlinear solutions for two-dimensional motion have been obtained for traveling waves over a non-uniform, nonstationary bottom surface. One class admits both solitary and cnoidal waves while the other class admits only cnoidal waves. • Since the equations are invariant under translational velocity, these solutions also describe steady motion over a non-uniform, stationary bottom surface. Acknowledgments. This research was partially supported by MB Rubin’s Gerald Swope Chair in Mechanics. References [1] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche, and M. Tissier. Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes. European Journal of Mechanics - B/Fluids, 30(6):589 – 597, 2011. Special Issue: Nearshore Hydrodynamics.
NEW EXACT SOLUTIONS
Figure 1. Comparison of the solitary wave solutions for a flat bottom (F ) and a variable bottom (V ): Traveling at the same speed (FF2 = FV2 = 1.2)
Figure 2. Comparison of the solitary wave solutions for a flat bottom (F ) and a variable bottom (V ): Traveling at different speeds (FF2 = 1.7 , FV2 = 1.2) but having the same peak height.
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[2] H. Demiray. Weakly nonlinear waves in water of variable depth: variablecoefficient Korteweg–de Vries equation. Computers & Mathematics with Applications, 60(6):1747–1755, 2010. [3] F. Dias and P. Milewski. On the fully-nonlinear shallow-water generalized Serre equations. Physics Letters A, 374(8):1049 – 1053, 2010. [4] D. Dutykh and D. Clamond. Modified shallow water equations for significantly varying seabeds. Applied Mathematical Modelling, 40(23-24):9767–9787, 2016. [5] A. E. Green and P. Naghdi. Directed fluid sheets. Proceedings of the Royal Society of London A, 347(1651):447–473, 1976. [6] A. E. Green and P. M. Naghdi. A derivation of equations for wave propagation in water of variable depth. Journal of Fluid Mechanics, 78(2):237–246, 1976. [7] R. Grimshaw. The solitary wave in water of variable depth. Journal of Fluid Mechanics, 42(3):639–656, 1970. [8] P. Naghdi and M. Rubin. On inviscid flow in a waterfall. Journal of Fluid Mechanics, 103:375–387, 1981. [9] P. Naghdi and M. Rubin. On the transition to planing of a boat. Journal of Fluid Mechanics, 103:345–374, 1981. [10] D. H. Peregrine. Long waves on a beach. Journal of Fluid Mechanics, 27(4):815–827, 1967. [11] F. J. Seabra-Santos, D. P. Renouard, and A. M. Temperville. Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle. Journal of Fluid Mechanics, 176:117–134, 1987. [12] F. Serre. Contribution `a l’´etude des ´ecoulements permanents et variables dans les canaux. La Houille Blanche, (6):830–872, 1953. [13] F. Serre. Contribution `a l’´etude des ´ecoulements permanents et variables dans les canaux. La Houille Blanche, (3):374–388, 1953. ¨ rich, Leonhardstrasse 21, Institute for Mechanical Systems, ETH Zu ¨ 8092 Zurich, Switzerland E-mail address: [email protected] Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel E-mail address: [email protected]
PII: DOI: Reference:
S0997-7546(18)30375-3 https://doi.org/10.1016/j.euromechflu.2018.12.005 EJMFLU 3416
To appear in:
European Journal of Mechanics / B Fluids
Received date : 6 June 2018 Revised date : 24 October 2018 Accepted date : 13 December 2018 Please cite this article as: F. Kogelbauer and M.B. Rubin, New exact solutions of nonlinear equations for shallow water with a non-uniform, non-stationary bottom surface, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
NEW EXACT SOLUTIONS OF NONLINEAR EQUATIONS FOR SHALLOW WATER WITH A NON-UNIFORM, NON-STATIONARY BOTTOM SURFACE F KOGELBAUER AND MB RUBIN Abstract. The GN nonlinear shallow water wave equations developed by Green and Naghdi [5, 6] are valid for a non-uniform pressure on the top surface and a non-stationary, non-uniform bottom surface. In contrast, the S nonlinear shallow water wave equations developed by Serre [12, 13] for uniform depth and later generalized by Seabra-Santos et al. [11] for non-uniform depth are limited to a stationary bottom surface and a uniform pressure applied to the top surface. This paper develops two new classes of exact nonlinear solutions of the GN equations for traveling and steady non-uniform bottom surfaces. One class admits cnoidal and soliton waves while the other class admits only cnoidal waves. Also, the explicit expressions for the pressure acting on the bottom surface and the average pressure through the depth in the GN equations are used to place physical restrictions on the motion which ensure that these pressures remain non-negative preventing cavitation.
December 29, 2018 1. Introduction Nonlinear wave propagation in shallow water is an old topic originally focused on waves in channels, but it has current important applications in coastal engineering. Since the equations for shallow water waves are developed using approximations, there are a number of different formulations that have been developed in the literature. Dias and Milewski [3] discuss the interesting history of various formulations and point out that the equations developed by Serre [12, 13] are identical to those developed by Green and Naghdi [5, 6] for flow in water of constant depth. An important physical feature of these equations is that they are invariant under superposed constant velocity and thus are valid for bi-directional wave propagation. The equations developed by Green and Naghdi are valid for three-dimensional flow, they depend upon two space variables and time and they are valid for 2010 Mathematics Subject Classification. Key words and phrases. Exact solution, nonlinear, shallow water, traveling wave, nonstationary bottom. 1
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flow over a non-unifom bottom surface. These equations are denoted as GN. Seabra-Santos et al. [11] developed a generalization of the Serre equations that are valid for two-dimensional flow in water over a non-uniform bottom surface, which are denoted by S. Also, Bonneton et al. [1] indicate that ”It is now recognized that the Serre or Green-Naghdi (S-GN) equations represent the relevant system to model highly nonlinear weakly dispersive waves propagating in shallow water“. Some additional discussions of flow in water of variable depth can be found in [2, 4, 7, 10]. It is commonly thought that the S equations are the same as the GN equations. However, it will be shown in the next section that the GN equations are more general than the S equations. Although attention was limited to flow over a stationary bottom surface in [5, 6], this restriction is not necessary for the validity of these theoretical developments. In particular, the resulting equations remain valid without change for a non-uniform, nonstationary bottom surface with the vertical location of the bottom surface being a function of two horizontal coordinates (x, y) and time t α = α(x, y, t).
(1.1)
This observation allows the GN equations to be used for modeling ocean floor motion as a source of Tsunamis. In addition, the GN equations are valid for a non-uniform pressure applied to the top surface of the fluid. It will be shown that the S equations are not valid for either of these situations. The main purpose of this paper is to discuss some new exact solutions of the GN equations for two-dimensional traveling waves over a non-uniform, non-stationary bottom surface and stationary waves over a non-uniform bottom surface. 2. Review and discussion of the S-GN equations The development in the remainder of this work limits attention to twodimensional flow. Specifically, with reference to fixed Cartesian base vectors ei (i=1,2,3), consider a two-dimensional flow in the e1 − e3 plane with the force of gravity g per unit mass acting in the negative e3 -direction. The notation used here is the same as that used in [6]. A material fluid particle has location x in the horizontal e1 direction, z in the vertical e3 direction, and the bottom, middle and top surfaces of the fluid are located by z = (α, ψ, β) respectively, such that 1 ψ = α + φ, β = α + φ, (2.1) 2 where φ is the current depth of the fluid and all functions, including α, depend on (x, t).
NEW EXACT SOLUTIONS
3
The horizontal velocity u, the vertical velocity λ of the middle surface and the rate of change w of the vertical depth are defined by u = x, ˙
˙ λ = ψ,
˙ w = φ,
(2.2)
where a superposed (·) denotes material time differentiation φ˙ = φt + φx u.
(2.3)
Also, subscripts are used to denote partial differentiation with respect to t, x or later ξ. The fluid motion in the GN model, developed in [6], is characterized by the equations w + φux = 0,
(2.4a)
ρ∗ φu˙ = pˆβx − pαx − px ,
(2.4b)
ρ∗ φλ˙ = −ρ∗ gφ + p − pˆ,
1 ∗ 1 p ρ φw˙ = − (p + pˆ) + , 12 2 φ
(2.4c) (2.4d)
with (2.4a) being the conservation of mass, (2.4b) being the horizontal component of linear momentum, (2.4c) being the vertical component of linear momentum and (2.4d) being the balance of director momentum. In these equations, ρ∗ is the constant mass density of the fluid, (p, pˆ) are the pressures acting on the bottom and top surfaces of the fluid, respectively, and p is an integrated average pressure through the depth of the fluid, which is an arbitrary function of (x, t) due to the incompressibility assumption. The S equations, cf. [11], can be written as (2.4a) and an equation for u of the form 1 1 1 1 2 2 2 φu+ ˙ φ g + w˙ + αx u˙ + u αxx +φ g + αx u˙ + αxx u + w˙ αx = 0, 2 3 2 2 x (2.5) where the variables (h, ξ) in the S equations in [11] have been replaced by (φ, α), respectively, to unify the notations. Also, use has been made of the expressions u˙ = ut + ux u,
w = −φux ,
w˙ = φ(u2x − uuxx − uxt ).
(2.6)
The equations (2.4b)-(2.4d) have an advantage over (2.5) in that the values and influence of the pressures (p, p, pˆ) are explicit. This was used in [9] to determine the pressure acting on the top surface of the fluid due to the
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bottom surface of a planing boat and in [8] to determine an equation for the bottom free surface in the double free region of the solution for a waterfall. To compare the GN and the S equations, it is convenient to solve equations (2.4c) and (2.4d) for the pressures (p, p) to obtain 2 1 1 ∗ 2 ∗ ¨ + w˙ , p = pˆ + ρ φ g + α ¨ + w˙ , p = pˆφ + ρ φ g + α (2.7) 2 2 3 ˙ Then, using (2.1) where use has been made of (2.1) and (2.2) to eliminate λ. and substituting (2.7) into (2.4b) yields 1 1 1 1 1 2 g + w˙ + α ¨ + ∗ pˆx φ + φ g + α φu˙ + φ ¨ + w˙ αx = 0. (2.8) 2 3 2 ρ 2 x Next, use is made of the expression α ¨ = (αtt + 2αtx u) + (αx u˙ + αxx u2 ),
(2.9)
to rewrite (2.8) in the form 1 1 1 1 2 2 2 φu˙ + φ g + w˙ + (αx u˙ + αxx u ) + φ g + αx u˙ + αxx u + w˙ αx 2 3 2 2 x 1 1 = − ∗ pˆx φ − φ2 (αtt + 2αtx u) − φ(αtt + 2αtx u)αx . ρ 2 x (2.10) It therefore follows that the GN equation (2.10) is more general than the S equation (2.5), since the GN equations do not need to limit attention to a uniform top pressure or a stationary bottom surface. The generality of a non-stationary bottom is essential for the new solutions developed in the next section. 3. Traveling wave solutions with a non-stationary bottom surface For the new class of solutions developed in this section, the vertical location of the bottom surface is specified by α = a(φ − H),
(3.1)
where a is a constant to be determined and H is a constant defining a critical value of the depth φ [see (3.15)]. It then follows from (2.1) that β = (1 + a)φ − aH,
1 ψ = [(1 + 2a)φ − 2aH], 2
1 λ = (1 + 2a)w. 2
(3.2)
NEW EXACT SOLUTIONS
5
Using (3.1), the pressures in (2.7) are given by 2 1 1 ∗ 2 ∗ p = pˆ + ρ φ g + + a w˙ , p = pˆφ + ρ φ g + + a w˙ , (3.3) 2 2 3 and equation (2.8) requires 1 1 1 1 1 2 φu˙ + φ g+ + a w˙ + ∗ pˆx φ + aφ g + + a w˙ φx = 0. 2 3 2 ρ 2 x (3.4) In the remainder of this paper, attention is limit to the case of a constant pressure p0 applied to the top surface pˆ = p0 .
(3.5)
Next, for a traveling wave with constant speed C in the positive x direction, all functions except x depend only on the parameter ξ defined by ξ = x − Ct,
f = f (ξ),
f˙ = ft + fx u = (u − C)fξ ,
(3.6)
and the conservation of mass (2.4a) requires (u − C)φ = k1 , where k1 is a constant of integration. Using (3.5) and the facts that 2 k12 φξ 1 k12 k1 2 φξ , w˙ = , , φu˙ = u˙ = −k1 3 = φ 2 φ2 ξ φ ξ φ φ ξ
(3.7)
(3.8)
equation (3.4) can be rewritten in the form " # 1 2 2 φξ φξ 2 (1 + a) 2 3 φ + + +a φ = −a(1+2a)φξ , (3.9) F H φ 3 φ ξ φ ξ ξ
where use has been made of the Froude number F defined by F2 =
k12 . gH 3
(3.10)
It will be shown that when a vanishes in (3.9) and the bottom surface is flat, the solution yields standard solitary and cnoidal waves, as discussed in subsection 3.1. From (3.9) it can also be seen that when (a = − 12 ) a new nontrivial class of solutions can be obtained which is discussed in subsection
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3.2. To discover a second new class of solutions it is convenient to expand (3.9) and divide the result by 2φ to deduce that " # 2 1 1 φ 1 1 φ 1 1 ξ ξξ (1 + a) 2 3 φ2 + 2 + a2 − + + a = 0. F H 2φ 2 3 φ 3 2 φ ξ
(3.11) in (3.11) is discussed in The class of solutions associated with (a = subsection 3.3. Also, using (3.5), the pressures in (3.3) are given by " # 1 φξ 1 1 +a , p = p0 + ρ∗ gF 2 H 3 φ + 2 3 F H 2 φ φ ξ " (3.12) # 1 ∗ 2 3 2 1 1 φξ 2 p = p0 φ + ρ gF H φ +a . + 2 F 2H 3 3 φ φ ξ − 23 )
3.1. Solutions for a flat bottom (a=0). For a flat bottom a = 0 and (3.9) integrates to deduce that φξ 2 1 H 1 φ2 2 2 φ [k2 −(|F |−1) ]+ = 1− + 2 1 − 2 , (3.13) 3 φ ξ F 2H H φ F H H where k2 is a constant of integration. Then, multiplying (3.13) by φξ /φ2 and integrating yields 1 φ φ φ 1 2 2 φ φ = k2 − (|F | − 1) − 1− 2 −1 , 1− 3 ξ F2 H F H H H (3.14) where the constant of integration has been specified so that φ = H is a critical value of φ, i. e., φξ = 0,
for φ = H.
(3.15)
Moreover, substituting (3.14) into (3.12) yields 2 1 ∗ 1 3 H 3 H2 2 2H p = p0 + ρ gφ + (k2 − (|F | − 1) ) + 3F 2 1 − + , 2 2 2 φ φ 2 φ2 2 1 ∗ 2 H 2 φ 2 p = p0 φ + ρ gH (k2 − (|F | − 1) ) 2 + 1 + 2F 1 − . 2 H φ (3.16) Furthermore, it can be shown that (3.14) can be written in the form 1 2 1 φ − φ1 φ2 − φ φ φ = 2 −1 , (3.17) 3 ξ F H H H
NEW EXACT SOLUTIONS
where the roots (φ1 , φ2 ) are real and non-negative p 4k2 |F | + k22 k2 φ1 = |F | + − ≥ 0, H 2 2 p 4k2 |F | + k22 φ2 k2 = |F | + + ≥ 0, H 2 2
7
(3.18)
for k2 ≥ 0, so the solution of (3.17) corresponds to cnoidal waves. As a special case, (3.17) yields 2 1 2 φ φ φξ = 1 − 2 −1 for k2 = (|F | − 1)2 , (3.19) 3 F H H which is satisfied by the well-known solitary wave, e.g. [10], ! r 2 − 1) ξ 3(F φ(ξ) = H + H(F 2 − 1)sech2 for F 2 ≥ 1. 2 F 2H
(3.20)
3.2. A new class of solutions for (a = − 12 ). For a non-uniform, nonstationary bottom surface with a = − 21 , equation (3.9) integrates to obtain " 2 # 1 φξ 1 1 2 H 1 φ2 φ = k3 − |F | − √ + 1− + 2 1− 2 , 6 φ ξ F 2H H φ 2F H H 2 (3.21) where k3 is a constant of integration. Then, multiplying (3.21) by φξ /φ2 and integrating yields " # 2 ! 1 1 φ φ φ 1 2 φ φ = k3 − |F | − √ + 1− −1 −1 , 12 ξ F2 H 2F 2 H H H 2 (3.22) where the constant of integration has been specified using the condition (3.15). Moreover, substituting (3.21) into (3.12) yields p = p0 + ρ∗ gφ,
2 ! 2 1 H 1 H2 1 ∗ 2 p = p0 φ + ρ gφ k3 − |F | − √ + +1 2 φ2 2 φ2 2 2 H 2H + 2F 2 1 − . φ φ
(3.23)
Furthermore, it can be shown that (3.22) can be written in the form 1 2 1 φ − φ1 φ2 − φ φ φ = 2 −1 , (3.24) 6 ξ F H H H
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where the roots (φ1 , φ2 ) are real and non-negative, q √ φ1 √ = 2|F | + k3 − 2 2k3 |F | + k32 ≥ 0, H q √ φ2 √ = 2|F | + k3 + 2 2k3 |F | + k32 ≥ 0, H
(3.25)
for k3 ≥ 0, so the solution of (3.24) corresponds to cnoidal waves. As a special case, (3.22) yields 1 2 φ = 12 ξ
1−
φ 2F 2 H
2 φ −1 , H
2 1 , for k3 = |F | − √ 2
(3.26)
which is satisfied by the solitary wave 1 ψ = H, 2
! 6(2F 2 − 1) ξ φ = H + H(2F − 1)sech , F2 2H ! r 2 − 1) ξ 1 6(2F 2 α = − H(2F 2 − 1)sech , 2 F2 2H ! r 2 − 1) ξ 1 6(2F β = H + H(2F 2 − 1)sech2 2 F2 2H 2
2
r
(3.27)
for 2F 2 ≥ 1.
It is interesting to note that for this solution the height of the middle surface of the fluid remains constant, ψ = H2 , so there is no vertical acceleration of this surface. Also, the result (3.23) that the pressure p is purely hydrostatic indicates that the vertical accelerations of the fluid above and below the middle surface compensate for each other. 3.3. A new class of solutions (a = − 32 ). Taking a = − 23 and integrating (3.11) yields 1 2 3 φ 1 φ2 φ φ = 1+ − 2 2 −1 , (3.28) 6 ξ 2 H F H H where the constant of integration has been determined by the condition (3.15). Differentiating this equation, dividing the result by φ2 φξ and using the expression 1 φξ 1 = 3 (φφξξ − φ2ξ ), (3.29) φ φ ξ φ
NEW EXACT SOLUTIONS
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it can be shown that 1 1 φξ 1 3 1 φ 1 φ2 1 φ3 = − + . − 6 φ φ ξ φ3 2 F 2 H F 2 H 2 2F 2 H 3 Then, substituting (3.29) into (3.12) yields 3 ∗ φ F 2H 2 2 H p = p0 + ρ gH − −1 , + 2 H φ2 3 φ 1 p = p0 φ + ρ∗ gφ2 . 2
(3.30)
(3.31)
Furthermore, it can be shown that (3.28) can be written in the form 1 2 φ2 − φ φ 1 φ − φ1 φ = 2 −1 , (3.32) 6 ξ F H H H where the roots (φ1 , φ2 ) are real, with one the other being non-negative, " r 3F 2 φ1 = 1− 1+ H 4 " r 3F 2 φ2 = 1+ 1+ H 4
of them being non-positive and 8 3F 2 8 3F 2
# #
≤ 0,
(3.33)
≥ 0,
so the solution of (3.32) corresponds to cnoidal waves with either φ2 ≤ φ < H for 3F 2 < 1
or
H ≤ φ ≤ φ2 for 3F 2 ≥ 1.
(3.34)
3.4. Physical restrictions. For these solutions to be physical, the constants of integration must be restricted so that φ2ξ and the pressures (p, p) remain non-negative. To prove existence of solutions defined in subsection 3.1 which are physical take k2 = k(|F | − 1)2 ,
k ≥ 1 and F 2 ≥ 1.
(3.35)
These conditions ensure that φ2 ≥ H ≥ φ1 in (3.18), with cnoidal waves oscillating between the two largest roots, H ≤ φ ≤ φ2 .
(3.36)
The specifications (3.35) and the restriction (3.36) ensure that all terms in the expressions (3.16) are non-negative for p0 ≥ 0 so the pressures (p, p) are non-negative.
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Similarly, to prove the existence of a class of solutions defined in subsection 3.2 which are physical, take 2 1 , k ≥ 1 and 2F 2 ≥ 1. (3.37) k3 = k |F | − √ 2 These conditions ensure that φ2 ≥ H ≥ φ1 in (3.25) so the solution of (3.24) requires (3.36). The specifications (3.37) and the restriction (3.36) ensure that all terms in the expressions in (3.23) are non-negative for p0 ≥ 0 so the pressures (p, p) are non-negative. Finally, to develop restrictions which ensure that the solution in subsection 3.3 is physical, it is noted that p in (3.31) can be rewritten in the form 3 H2 2 2 φ . p = p0 + ρ∗ gH( 2 )f (x) , f (x) = x3 − x2 + x − F 2 , x = 2 φ 3 3 H
(3.38)
The discriminant of the cubic equation f (x) = 0 is negative for all values of course F 2 , which indicates that f (x) has only one real root. Furthermore, it can be shown that p(φ2 ) ≥ 0 for p0 ≥ 0 and all values of F 2 . Moreover, the value of p(H) is given by 3 p(H) = p0 + ρ∗ gH(1 − F 2 ) . 2
(3.39)
Consequently, the pressures (p, p) in (3.31) will be non-negative for p0 ≥ 0 only when F 2 is bounded above by F2 ≤ 1 +
2p0 . 3ρ∗ gH
(3.40)
It is emphasized that these restrictions on the fluid motion without explicit equations for the pressures (p, p), which are not known in the S equations for motion of shallow water waves. 4. Examples To compare the solitary wave solutions discussed in subsections 3.1 and 3.2, it is noted that for the flat bottom solution, the vertical location of the bottom and top surfaces of the waves are given by s ! 2 3(F − 1) ξ F , αF = 0, βF = φF (ξ) = H + H(FF2 − 1)sech2 FF2 2H (4.1) for FF2 ≥ 1,
NEW EXACT SOLUTIONS
11
where FF is the associated Froude number defined in (3.10). Similarly, the non-uniform, non-stationary, variable bottom solution in subsection 3.2 can be written in the form s ! 2 6(2F − 1) ξ 1 V , αV = − H(2FV2 − 1)sech2 2 FV2 2H (4.2) s ! 2 1 ξ 6(2F − 1) V βV = H + H(2FV2 − 1)sech2 , for 2FV2 ≥ 1, 2 FV2 2H where FV is the associated Froude number. Since the GN equations are invariant under superposed rigid body motions and allow bidirectional wave propagation, the Froude numbers (FF , FV ) can be determined by traveling waves using (3.7), (3.10) and the boundary conditions φ = H, φ = H,
u = 0, u = 0,
CF2 , gH C2 C = CV for ξ → ±∞ =⇒ FV2 = V , gH C = CF for ξ → ±∞ =⇒ FF2 =
(4.3)
or they can be determined by the velocity of the fluid past a stationary depression using (3.7), (3.10) and the boundary conditions φ = H, φ = H,
u = uF , u = uV ,
u2F , gH u2 C = 0 for ξ → ±∞ =⇒ FV2 = V . gH
C = 0 for ξ → ±∞ =⇒ FF2 =
(4.4)
In particular, for the traveling waves (4.3), it can be seen from (4.1) and (4.2) that the bottom of the fluid far from the peak of the traveling wave has the same vertical location for both solutions αF = αV = 0 for ξ → ±∞, and the peaks of the waves (at ξ = 0) are given by 1 2 2 + FV . βF = HFF , βV = H 2
(4.5)
(4.6)
Figure 1 plots the flat bottom solution (F ) and the variable bottom solution (V ) when the waves travel with the same speed FF2 = FV2 = 1.2,
(4.7)
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F KOGELBAUER AND MB RUBIN
and Figure 2 plots these solutions when the waves travel with different speeds, 1 2 2 FF = + FV = 1.7, FV2 = 1.2, (4.8) 2 but have the same peak height (βF = βV ). From these figures, it can be seen that the variable bottom traveling wave has a narrower peak than the flat bottom traveling wave. 5. Conclusions The main conclusions of this paper are summarized as: • The general GN equations describe three-dimensional fluid motion over a non-uniform, non-stationary bottom surface with non-uniform pressure applied to the top surface. In contrast, the generalized S equations are valid only for a stationary bottom surface and a uniform pressure applied to the top surface. • The GN equations include explicit expressions for the pressures (p, pˆ) on the bottom and top surfaces, respectively, and for the average pressure p through the depth of the fluid. Physical restrictions on the solutions have been proposed which require these pressures to remain non-negative to prevent cavitation. These pressures are not determined by the S equations. • Two new classes of exact nonlinear solutions for two-dimensional motion have been obtained for traveling waves over a non-uniform, nonstationary bottom surface. One class admits both solitary and cnoidal waves while the other class admits only cnoidal waves. • Since the equations are invariant under translational velocity, these solutions also describe steady motion over a non-uniform, stationary bottom surface. Acknowledgments. This research was partially supported by MB Rubin’s Gerald Swope Chair in Mechanics. References [1] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche, and M. Tissier. Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes. European Journal of Mechanics - B/Fluids, 30(6):589 – 597, 2011. Special Issue: Nearshore Hydrodynamics.
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Figure 1. Comparison of the solitary wave solutions for a flat bottom (F ) and a variable bottom (V ): Traveling at the same speed (FF2 = FV2 = 1.2)
Figure 2. Comparison of the solitary wave solutions for a flat bottom (F ) and a variable bottom (V ): Traveling at different speeds (FF2 = 1.7 , FV2 = 1.2) but having the same peak height.
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[2] H. Demiray. Weakly nonlinear waves in water of variable depth: variablecoefficient Korteweg–de Vries equation. Computers & Mathematics with Applications, 60(6):1747–1755, 2010. [3] F. Dias and P. Milewski. On the fully-nonlinear shallow-water generalized Serre equations. Physics Letters A, 374(8):1049 – 1053, 2010. [4] D. Dutykh and D. Clamond. Modified shallow water equations for significantly varying seabeds. Applied Mathematical Modelling, 40(23-24):9767–9787, 2016. [5] A. E. Green and P. Naghdi. Directed fluid sheets. Proceedings of the Royal Society of London A, 347(1651):447–473, 1976. [6] A. E. Green and P. M. Naghdi. A derivation of equations for wave propagation in water of variable depth. Journal of Fluid Mechanics, 78(2):237–246, 1976. [7] R. Grimshaw. The solitary wave in water of variable depth. Journal of Fluid Mechanics, 42(3):639–656, 1970. [8] P. Naghdi and M. Rubin. On inviscid flow in a waterfall. Journal of Fluid Mechanics, 103:375–387, 1981. [9] P. Naghdi and M. Rubin. On the transition to planing of a boat. Journal of Fluid Mechanics, 103:345–374, 1981. [10] D. H. Peregrine. Long waves on a beach. Journal of Fluid Mechanics, 27(4):815–827, 1967. [11] F. J. Seabra-Santos, D. P. Renouard, and A. M. Temperville. Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle. Journal of Fluid Mechanics, 176:117–134, 1987. [12] F. Serre. Contribution `a l’´etude des ´ecoulements permanents et variables dans les canaux. La Houille Blanche, (6):830–872, 1953. [13] F. Serre. Contribution `a l’´etude des ´ecoulements permanents et variables dans les canaux. La Houille Blanche, (3):374–388, 1953. ¨ rich, Leonhardstrasse 21, Institute for Mechanical Systems, ETH Zu ¨ 8092 Zurich, Switzerland E-mail address: [email protected] Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel E-mail address: [email protected]