J Eng Math (2011) 71:253–277 DOI 10.1007/s10665-011-9451-y
On capillary gravity-wave motion in two-layer fluids S. C. Mohapatra · D. Karmakar · T. Sahoo
Received: 6 March 2010 / Accepted: 31 December 2010 / Published online: 15 January 2011 © Springer Science+Business Media B.V. 2011
Abstract Generalized expansion formulae for the velocity potentials associated with plane gravity-wave problems in the presence of surface tension and interfacial tension are derived in both the cases of finite and infinite water depths in two-layer fluids. As a part of the expansion formulae, orthogonal mode-coupling relations, associated with the eigenfunctions of the velocity potential, are derived. The dispersion relations are analyzed to determine the characteristics of the two propagating modes in the presence of surface and interfacial tension in both the cases of deep-water and shallow-water waves. The expansion formulae are then generalized to deal with boundary-value problems satisfying higher-order boundary conditions at the free surface and interface. As applications of the expansion formulae, the solutions associated with the source potential, forced oscillation and reflection of capillary–gravity waves in the presence of interfacial tension are derived. Keywords
Expansion formulae · Interfacial tension · Surface tension · Two-layer fluid · Wave oscillation
1 Introduction Havelock [1] was the first to study the classical plane wave-maker problem under the assumption of linearized theory of water waves in a single-layer fluid system using an eigenfunction expansion method. Rhodes-Robinson [2] generalized Havelock’s [1] expansion formulae in the presence of surface tension using Green’s identity in both the cases of finite and infinite depths. Hocking and Mahdmina [3] analyzed the capillary–gravity waves produced by a wave-maker for fluid of finite depth with the wave-maker extending from the free surface till the bottom of the fluid. Henry [4] analyzed the particle trajectories in linear periodic capillary and capillary–gravity water waves propagating over a flat sea bed. One of the major applications of capillary–gravity waves is that the dynamics of these waves strongly impact remote sensing; see [5]. Unlike the case of plane gravity waves, in the presence of surface tension, the free-surface boundary condition is of third order associated with the boundary-value problem in which the governing equation is Laplace’s equation S. C. Mohapatra · T. Sahoo (B) Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur 721 302, India e-mail:
[email protected];
[email protected] D. Karmakar Centre for Marine Technology and Engineering, Instituto Superior Tecnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
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which is not of the standard Sturm–Liouville type. Lawrie and Abrahams [6] established an orthogonal relation for a general class of BVP of Helmholtz-type a partial differential equation with higher-order boundary conditions in a semi-infinite strip type domain, and utilized these relations to solve problems of acoustics. Later on, Manam et al. [7] derived the general expansion formulae and related mode-coupling relations based on the application of Fourier analysis in both the cases of a semi-infinite strip as well as a quarter plane to tackle a general class of boundary-value problem. Karmakar et al. [8] obtained an alternate derivation of the expansion formulae and the corresponding mode-coupling relations are obtained for the boundary-value problem associated with Laplace’s equation satisfying a more general type of higher-order boundary condition on one of the boundaries in a quarter plane, whilst Bhattacharjee et al. [9] derived the corresponding formulae for a semi-infinite strip. The above class of problems are associated with boundary-value problem in a single homogeneous domain. Apart from wave motion at a free surface, there is significant interest in the generation of gravity waves at the interface boundary of two fluids by different densities in which the top and bottom fluid are covered by rigid lids; see [10,11]. Kalisch [12] analyzed the interfacial capillary–gravity waves in deep water using matched asymptotic expansions. Parau et al. [13] analyzed three-dimensional capillary–gravity interfacial flows due to an immersed disturbance that propagates to a constant velocity along the interface between two semi-infinite fluids. Lu and Ng [14] studied interfacial capillary–gravity waves due to a fundamental singularity in a system of two semi-infinite fluids in both two- and three-dimensional cases. In the above two class of problems, there exists only one propagating wave. On the other hand, in the case of gravity-wave propagation in a two layer fluid having a free surface and an interface, two progressive waves exist which are generated at the free surface and the interface. For example; (i) Barthelemy et al. [15] analyzed the scattering of surface waves by a step bottom in a two-layer WKBJ technique and (ii) Manam et al. [16] developed the expansion formulae for a two-layer fluid and analyzed the wave scattering by a porous barrier for both surface and interfacial modes in a two-layer fluid. Unlike the case of surface waves which exist in open water surface, internal waves occur when there is a sharp density change between two layers of different fluids. These internal waves are most commonly observed in shallow, coastal areas like straits, marginal seas, continental shelfs and fjords, which have also been observed in deep oceans. These waves in the ocean influence marine activities such as oil drilling, and are thought to redistribute zooplankton, nutrients and other particulate substances and consequently influence the movement and feeding habits of marine animals; see [17]. A more general class of problems one comes across is the flexural gravity-wave motion in a two-layer fluid having a common interface in which a fifth-order boundary condition is satisfied in addition to the interfacial condition being satisfied at the mean interface for a boundary-value problem associated with Laplace’s equation. Schulkes et al. [18] studied the effect of stratification in a two-layer model by analyzing the flexural gravity-wave propagation in an ice-covered sea surface having an interface. Bhattacharjee and Sahoo [19] derived the expansion formulae for flexural gravity-wave motion in two-layer fluids having an ice-covered surface and an interface in both the cases of water of finite and infinite depths. However, there is little progress on wave motion in two-layer fluids in the presence of surface and interfacial tension. In this case, two propagating waves are developed at the surface and the interface which may be referred to as capillary–gravity waves in surface and internal modes. The wavelength of these waves are much smaller compared to the wavelength of the corresponding gravity waves. Further, in these problems, third-order boundary conditions are satisfied both at the mean free surface and at the mean interface which makes the problem more complex. Rhodes-Robinson [20] used the fundamental set of wave-source potentials for the two layers to construct the set of slope potentials that produce discontinuous free surface and interface slopes. In this paper, a technique using Green’s theorem was developed in two layers using the wave-source potential, which obeys a generalized reciprocity principle. Amaouche and Meziani [21] analyzed the interaction of two superimposed inviscid liquids with a flexible side wall of a rectangular container by using normal mode-decomposition in the presence of surface and interfacial tension. In the present study, to understand the progressive wave characteristics in the presence of surface and interfacial tensions, the dispersion relation is analyzed separately in the cases of deep water, shallow water and waves in intermediate depth by analyzing the behavior of wavelength, phase and group velocities and amplitude of surface to interface elevations. The shallow-water equations are derived based on the linearized theory of long waves in
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a direct manner and the results are compared with those obtained from the general case. The expansion formulae for the velocity potentials in the presence of surface and interfacial tension are derived in both the cases of finite and infinite water depths. The utility of the expansion formulae is demonstrated by deriving the solution associated with the line-source potentials, forced oscillation in rectangular semi-infinite channels and wave reflection by a wall in both the cases of water of finite and infinite depths in the presence of surface and interfacial tension. The general form of the expansion formulae and corresponding orthogonal mode-coupling relations are also developed for boundary-value problems associated with the Laplace equation satisfying higher-order boundary condition at the free surface in the presence of a condition similar to interfacial tension as discussed in Sect. 2 and is added as an Appendix in the paper.
2 Mathematical formulation In the present section, we formulate the physical problem as a boundary-value problem associated with the Laplace equation satisfying a free-surface boundary condition along with an interfacial tension at the interface in the twolayer fluid in both the cases of fluid of finite and infinite depths. Under the assumption of the linearized theory of water waves, the problem is considered in a two-dimensional Cartesian co-ordinate system with the x-axis being in the horizontal direction and the y-axis in the vertically downward positive direction as in Fig. 1. The upper fluid layer is of constant density ρ1 ; in the presence of surface tension T1 it occupies the region 0 < y < h, −∞ < x < ∞ with y = 0 being the mean free surface. The lower fluid is assumed to be of constant density ρ2 in the presence of the interfacial tension T2 and occupies the region h < y < H , −∞ < x < ∞ in the case of finite depth whilst, h < y < ∞, −∞ < x < ∞ in the case of infinite depth, with the mean interface being at y = h. Assuming that the fluid is inviscid and incompressible and the motion is irrotational, the fluid motion is described by the two velocity potentials j (x, y, t), j = 1, 2 and the displacements at the surface and interface as η(x, t) and ζ (x, t), respectively. The velocity potential j (x, y, t), j = 1, 2 satisfies the partial differential equation ∇ 2 j = 0 in the fluid region.
(1)
The linearized kinematic conditions at the mean free surface and the mean interface are given by ∂η ∂1 = on y = 0, ∂t ∂y ∂ζ ∂1 ∂2 = = at y = h. ∂t ∂y ∂y
(2a) (2b)
In the presence of surface tension, the general relation connecting surface tension and pressure gradient is given by T1 in the case of free surface, R T2 in the case of interface, P2 − P1 = R
P1 − P0 =
(3a) (3b)
Fig. 1 Schematic diagram for two layer fluid with interfacial tension
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where 1/R is the mean curvature, P0 is the constant atmospheric pressure and P j is the hydrodynamic pressure in the fluid region j. The mean curvature 1/R in the Cartesian co-ordinate is given by ⎧ ηx x ⎪ in the case of free surface, ⎨ 1 (1 + η2x )3/2 = (4) ζx x ⎪ R ⎩ in the case of interface. (1 + ζx2 )3/2 The hydrodynamic pressure P j in the fluid region in the linearised theory of water waves is given by ∂ j + ρ j gy, (5) ∂t where ρ j is the fluid density. Hence, from Eqs. 4 and 5, the linearized dynamic free-surface boundary condition in the presence of surface tension T1 at the mean free surface y = 0 is given by ∂1 ∂ 2η ρ1 gη − = T1 2 . (6) ∂t ∂x Further, from Eqs. 4 and 5, the linearized dynamic condition at the mean interface y = h in the presence of interfacial tension T2 is given by ∂ 2ζ ∂2 ∂1 (7) ρ2 gζ − − ρ1 gζ − = T2 2 . ∂t ∂t ∂x From combining the kinematic and dynamic boundary conditions (2a), (2b), (6) and (7), the linearized boundary conditions at the mean free surface and mean interface are given by ∂1 ∂ 3 1 ∂ 2 1 = T , at y = 0, (8a) ρ1 g − 1 ∂y ∂t 2 ∂ y∂ x 2 ∂ 3 2 ∂2 ∂1 ∂ 2 2 ∂ 2 1 ρ2 g , at y = h. (8b) − ρ g = T2 − − 1 2 2 ∂y ∂t ∂y ∂t ∂ y∂ x 2 The rigid-bottom boundary conditions are given by P j = −ρ j
2 , |∇2 | → 0 as y → ∞ in case of infinite depth,
(9a)
and ∂2 = 0 at y = H in case of finite depth. (9b) ∂y Further, for a unique determination of the solution, edge conditions must be specified near the contact line where the free surface or the interface interact with a boundary. In the context of the present study, appropriate edge conditions are of the forms [3] ∂1 ∂η = λ± at (x, y) = (0±, 0), (10a) s ∂t ∂x and ∂ζ ∂2 = λi± at (x, y) = (0±, h), (10b) ∂t ∂x ± where λ± s and λi are constants assumed to be known which will depend on the nature of the physical problem. Recently, Harter et al. [22] discussed in detail the range of relevant edge conditions near the contact line for analyzing capillary–gravity wave problems. When it is assumed that the fluid motion is simple harmonic in time with angular frequency ω, the velocity potential, free surface and interface elevations can be written in the form j (x, y, t) = Re{φ j (x, y)e−iωt }, η j (x, t) = Re{η¯j (x)e−iωt } and ζ j (x, t) = Re{ζ¯j (x)e−iωt } respectively. Thus, the spatial velocity potentials φ j for j = 1, 2 satisfy (1). The linearized free-surface boundary condition in the presence of surface tension T1 is given by − M1
∂φ1 ∂ 3 φ1 + + K φ1 = 0 on y = 0, ∂ x 2∂ y ∂y
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where M1 = T1 /ρ1 g, K = ω2 /g, with g being the acceleration due to gravity. In addition, the linearized boundary condition at the mean interface y = h in the presence of interfacial tension T2 is given by ∂φ1 ∂φ2 = on y = h, (12a) ∂y ∂y ∂ 3 φ2 ∂φ1 ∂φ2 = K φ2 + − M2 2 on y = h. (12b) s K φ1 + ∂y ∂y ∂x ∂y An equivalent form of the interface condition (12b) is given by ∂ 3 φ1 ∂ 3 φ2 ∂φ1 ∂φ2 s K φ1 + − M˜ 2 2 = K φ2 + − M˜ 2 2 on y = h, (12c) ∂y ∂x ∂y ∂y ∂x ∂y where M2 = T2 /ρ2 g, M˜ 2 = T2 /(ρ2 − ρ1 )g, s = ρ1 /ρ2 with 0 < s < 1, K > 0. Finally, the far-field radiation condition is prescribed in the form ⎧ II ⎪ ⎪ ⎪ An Fn (kn , y)eikn |x| as |x| → ∞ in case of infinite depth, ⎪ ⎨ (13) φ(x, y) ∼ n=I II ⎪ ⎪ iμ |x| ⎪ ⎪ Rn Yn (μn , y)e n as |x| → ∞ in case of finite depth, ⎩ n=I
where kn and μn are the wave numbers of the propagating modes associated with the surface and internal waves. Further, Fn (kn , y) and Yn (μn , y) are the vertical eigenfunctions associated with kn and μn , respectively, whose details will be discussed in the subsequent sections. The rigid bottom boundary condition is given by (14a) φ2 , |∇φ2 | → 0 as y → ∞ in case of infinite depth, ∂φ2 = 0 at y = H in case of finite depth. (14b) ∂y The edge conditions are ∂φ1 ∂ η¯ = λ± at (x, y) = (0±, 0), (15a) − iω s ∂t ∂x and ∂φ2 ∂ ζ¯ = λi± at (x, y) = (0±, h). (15b) − iω ∂t ∂x In the next section, the behavior of the progressive-wave solutions in the presence of surface tension and interfacial tension will be discussed before deriving the expansion formulae.
3 Dispersion relation, phase and group velocities We will analyze the behavior of the progressive-wave solution by analyzing the dispersion relation associated with the physical problem in specific cases. Assuming sinusoidal waves η = Re{η0 ei(kx−ωt) } and ζ = Re{ζ0 ei(kx−ωt) }, we obtain the velocity potentials 1 and 2 satisfying the boundary conditions (2a), (2b) and (9b) in the form: 1 = (A cosh ky − iωη0 k −1 sinh ky)ei(kx−ωt) ,
(16a)
2 = B cosh k(H − y)e
(16b)
i(kx−ωt)
,
where the constants A, B are arbitrary. Using the boundary conditions (2b) and (7), from (16a) and (16b), we can obtain the unknown constants A, B and ζ0 by solving the matrix equation as given by ⎤ ⎡ ⎤⎡ ⎤ ⎡ −(sη0 ω/ik) sinh kh A s cosh kh cosh k(H − h) −(1 − s)g(1 + M˜ 2 k 2 )/iω ⎣ sinh kh sinh k(H − h) ⎦ ⎣ B ⎦ = ⎣ (iωη0 /k) cosh kh ⎦ , 0 0 ζ0 0 − sinh k(H − h) iω/k
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where M˜ 2 = T2 /(ρ2 − ρ1 )g. Solving this system of equations, we find in particular that iω (17) 1 = − (μ cosh ky + sinh ky)η, k where s tanh kh + coth k(H − h) − k(1 − s)g(1 + M˜ 2 k 2 )/ω2 μ=− coth kh, (18) s coth kh + coth k(H − h) − k(1 − s)g(1 + M˜ 2 k 2 )/ω2 and the ratio of the amplitudes at the free surface and the interface η0 /ζ0 is of the form
η0 1 (19) = sinh kh s coth kh + coth k(H − h) − (1 − s)g(1 + M˜ 2 k 2 )k/ω2 . ζ0 s It may be noted that, if the value of η0 /ζ0 is real and positive, then the surface and interfacial waves are said to be in phase and, if negative, then the surface and interfacial waves are said to be 180◦ out of phase. The dispersion relation for uniform waves in the presence of surface tension T1 and interfacial tension T2 is given by ω2 = −
(M1 k 2 + 1)gk , μ
(20)
where μ is given by (18). The expression designated by μ contains an ω2 term, so the dispersion relation is quadratic in ω2 . Solving for ω2 , we obtain the explicit quadratic roots 2 = ω±
Q ± (Q 2 − 4P R)1/2 , 2P
(21)
where P = s + coth kh coth k(H − h), Q = (1 − s)gk( M˜ 2 k 2 + 1) coth kh + gk(M1 k 2 + 1){s coth kh + coth k(H − h)}, R = k 2 g 2 (M1 k 2 + 1)( M˜ 2 k 2 + 1)(1 − s).
(22a) (22b) (22c)
In Eq. 21, the subscript with the + sign refers to waves in surface mode and subscript with the − sign refers to waves in the interfacial mode and this terminology will be used throughout the paper. A similar approach for deriving the wave speed in the case of solitary waves was used by Michallet and Dias [23]. Next, we will analyze the roots of the dispersion relation in particular cases. 3.1 Deep water In the case of deep water, kh >> 1, k(H − h) >> 1 which yields tanh kh → 1 and tanh k(H − h) → 1. So, the expressions for P, Q and R are obtained as P = (1 + s),
(23a)
Q = (1 − s)gk( M˜ 2 k + 1) + gk(M1 k + 1)(1 + s), R = g 2 k 2 (M1 k 2 + 1)( M˜ 2 k 2 + 1)(1 − s). 2
2
(23b) (23c)
In particular, when M1 = M˜ 2 = M (say), from Eq. 21 it can be easily derived that (1 − s) 2 2 gk(1 + Mk 2 ). ω+ = gk(1 + Mk 2 ) and ω− = (24) (1 + s) The values of ω+ and ω− ensure that the frequency of the interfacial wave is smaller than that of the surface wave. From the theory of equations it is evident that the equation for the surface mode has one real root k0 (say) and two complex roots k I and k I I which are given by i 4M + 3M 2 k02 i 4M + 3M 2 k02 −k0 −k0 + and k I I = − , (25) kI = 2 2M 2 2M
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and similar roots can be obtained for the interfacial mode. The phase velocity for both the surface and the interfacial mode are found to be 1/2 (1 − s) 1/2 and c− = , (26) g(1/k + Mk) c+ = {g(1/k + Mk)} (1 + s) which suggest that the phase speed of the surface mode is higher than that of the interfacial mode. Again, the group velocity for both the surface and the interfacial modes are cg+
1 = 2
g k(1 + Mk 2 )
1/2 (1 + 3Mk ) and cg− 2
1 = 2
g(1 − s) (1 + s)k(1 + Mk 2 )
1/2 (1 + 3Mk 2 ).
(27)
In this case the ratio of the amplitudes of the waves at the free surface to that of the interface from Eq. 19 is given by η0 ekh 2 g 2 (1 + Mk c = (1 + s) − (1 − s) ) . ± 2 ζ0 k 2sc±
(28)
Considering the phase speed c+ , we have η0 /ζ0 = ekh and for the phase speed c− , we have η0 /ζ0 = 0. This shows that the amplitude of the surface wave is higher than that of the amplitude of interfacial wave in case of phase speed c+ and the waves are also in phase. On the other hand, in case of phase speed c− , the amplitude of the interfacial wave is much higher than the amplitude of the surface wave for deep water. Again, it is observed that the phase velocity for both the surface and the interfacial waves attains a minimum for k = kmin =
1 M
1/2 .
(29)
In such a situation, the phase velocity at the free surface and at the interface is given by cmin+ =
2g √ M
1/2
and cmin− =
(1 − s) 2g √ (1 + s) M
1/2 .
(30)
Remark It may be noted that in case of deep water for M1 = M˜ 2 = M, the dispersion relation, phase and group velocities in surface mode are the same as in the case of capillary–gravity waves in homogeneous fluid of infinite depth. As noted by Crapper [24, Sect. 2.2], when a wind begins to blow over a flat surface, no waves are produced until it reaches a speed of cmin+ in case of a single-layer fluid. The present observation suggests that a similar situation can exist in case of a two-layer fluid for this particular value of M. We will give a brief graphical presentation of the wave motion in a two-layer fluid in case of surface and interface mode keeping the values of fluid densities fixed as ρ1 = 1,000 kg m−3 and ρ2 = 1,015 kg m−3 unless explicity mentioned. In Fig. 2a, the variation of phase velocity c+ and group velocity cg+ in a surface-mode wave are plotted versus wave number k for various values of the surface tension T1 with M1 = M˜ 2 . As has been discussed, the phase and group velocities attain their minimum for an intermediate value of the wave number. These observations are similar to that of capillary–gravity wave motion in a homogeneous fluid domain. In Fig. 2b, the variation of phase velocity c− and group velocity cg− versus wave number k in internal mode are plotted for various values of surface tension T1 , with M1 = M˜ 2 . The observations are similar to that of Fig. 2a. However, the phase and group velocities in surface mode are higher than that of the internal mode. In Fig. 3, the surface elevation η and interface elevation ζ for various values of water depth h with T = 4 s, T1 = 0.074 Nm−1 , ωt = 0 and M1 = M˜ 2 are plotted. It is observed that the amplitude of the waves in the surface mode is much larger compared to the waves in the internal mode. Further, it is observed that, with increasing depth of the interface, the free-surface elevation increases without much change in the interface elevation, which is obvious from relation (28).
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(a)
(b) 10
0
0.1
10
Cg+,C+ (ms−1)
Cg , C (ms−1) − −
c+
−0.1
10
T1=0.05N/m T =0.074N/m 1 T1=0.09N/m T =0.1N/m
T =0.05N/m 1 T1=0.074N/m T =0.09N/m 1 T1=0.1N/m
0
10
−0.2
10
−0.3
10
−0.4
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−0.5
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−1
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c−
−0.6
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c
g+
c
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g−
−2
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3
2
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10
−1
10
−1
k (m )
k (m )
Fig. 2 Variation of a Phase velocity c+ and group velocity cg+ of surface wave and b Phase velocity c− and group velocity cg− of internal wave versus wave number k for various values of surface tension T1 40
surface and interface elevation
Fig. 3 Variation of surface elevation η and interface elevation ζ for various values of water depth h with T1 = 0.074 Nm−1 in the case of deep water
h=10m h=12m h=14m
30 20 η
10
ζ
0 −10 −20 −30 −40
0
10
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30
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x
3.2 Intermediate depth In the case of intermediate depth it is assumed that coth k(H − h) ≈ 1. So the expressions for P, Q and R are given by P = s + coth kh, Q = (1 − s)gk( M˜ 2 k 2 + 1) coth kh + gk(M1 k 2 + 1)(s coth kh + 1),
(31b)
R = g k (1 − s)(M1 k + 1)( M˜ 2 k 2 + 1).
(31c)
2 2
2
Considering M1 = M˜ 2 = M we obtain
1/2 gk(1 + Mk 2 ) 2 . 1 ± s 2 + (1 − s)(1 − coth kh) = ω± (s + coth kh)
(31a)
(32)
If we consider s → 1 then, 2 ω+ =
gk(1 + Mk 2 ) 2 (1 + s) and ω− = 0. (s + coth kh)
This shows that only the waves at the free surface exist.
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3.3 Shallow water depth In the case of waves in shallow water depth, it is assumed that tanh kh ≈ kh and tanh k(H − h) ≈ k(H − h). So the expressions for P, Q and R are given by P = 1,
(34a)
Q = gk (1 − s)( M˜ 2 k + 1)(H − h) + gk (M1 k + 1) {s H + h(1 − s)}, R = k 4 g 2 (1 − s)(M1 k 2 + 1)( M˜ 2 k 2 + 1)h(H − h). 2
2
2
2
(34b) (34c)
Considering M = M1 = M˜ 2 we obtain, 2 2 ω+ = gk 2 H (1 + Mk 2 )A+ and ω− = gk 2 H (1 + Mk 2 )A− , (35)
1/2 −h)g and g = g(1 − s). From the theory of equations, it is evident that the where A± = 21 1 ± 1 − 4h(H H2g
equation for the surface mode has two real roots of opposite signs ±k0 (say) and two imaginary roots k I and k I I which are complex conjugates and are given by i 1 + Mk02 i 1 + Mk02 and k I I = − , (36) kI = √ √ M M and similar roots can be obtained for the interfacial mode. In this case, it is easily derived that neither c± nor cg± attains a minimum value. Further, it may be noted that in this case cg+ A+ c+ = = . (37) cg− c− A− 4 Shallow-water approximation In this section, we will discuss the motion of capillary–gravity waves in the presence of interfacial tension under the assumption of shallow-water approximation in a direct manner. Integrating equation of continuity over water depth, we obtain the equation of continuity for linearized long waves as ∂ 2 φ1 ∂η ∂ζ − =h 2 , (38) ∂t ∂t ∂x Taking the double derivative of Eq. 6 with respect to x and using Eq. 38, the linearized long-wave equation at y = 0 in the presence of surface tension T1 is obtained as ∂ 2 (η − ζ ) ∂ 2η ∂ 4η − ρ gh + T h = 0. (39) 1 1 ∂t 2 ∂x2 ∂x4 Proceeding in a similar manner, the equation of continuity for the linearized long-wave equation at the interface y = h is given by
ρ1
∂ζ ∂ 2 φ2 = (H − h) 2 . (40) ∂t ∂x From the linearized dynamic condition as in (7) and the continuity equations as in (38) and (40), after eliminating φ1 and φ2 , the equation of continuity in shallow water in the presence of surface and interfacial tension is obtained as ∂ 4ζ ∂ 2ζ ∂ 2ζ ρ1 ∂ 2 η − {ρ /(H − h) + ρ / h} + = T . (41) 2 1 2 ∂x2 ∂t 2 h ∂t 2 ∂x4 Assuming sinusoidal waves η = Re{η0 ei(kx−ωt) } and ζ = Re{ζ0 ei(kx−ωt) }, from Eqs. 39 and 41, two homogeneous system of equations are obtained as
ω2 − ghk 2 (1 + M1 k 4 ) η0 − ω2 ζ0 = 0, (42) sω2 (H − h)η0 − {h(1 − s) + s H }ω2 − (1 − s)gh(H − h)k 2 (1 + M˜ 2 k 2 ) ζ0 = 0. (43)
(ρ2 − ρ1 )g
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Eliminating η0 and ζ0 , we obtain the dispersion relation in shallow water in the presence of surface and interfacial tension as: ω4 P − ω2 Q − R = 0,
(44)
where P, Q and R are the same as obtained for shallow water depth as in (34a)–(34c). The ratio of the amplitudes of the waves at the free surface to that of interface from (42) with M1 = M is given by η0 c2 = 2 . ζ0 c − gh(1 + Mk 2 )
(45)
It is observed that the equation in ω has two roots for ω2 which indicates that a two-layer fluid supports two types of wave motion generated due to the presence of free surface (refers to wave in surface mode) and interface (refers to wave in internal mode). The two modes supported by the two-layer fluid are very different and are best studied by assuming that the densities of the two fluids differ slightly.
4.1 Structure of wave modes The phase speed of waves in shallow-water gravity waves with surface and interfacial tension in a two-layered fluid is given by 1/2 1 4h(H − h)g 2 = g H (1 + Mk 2 ) 1 ± 1 − . (46) c± 2 H 2g This equation has two roots for c. Assuming g = (1 − s) 1, g we obtain 4g h(H − h)
1. gH2 Using Taylor-series expansion, it can be shown that, for x 1, in the form 1 2h(H − h)g 2 2 . c± = g H (1 + Mk ) 1 ± 1 − 2 H 2g
√ 1−x ∼ = 1 − x2 . So, the phase speed is rewritten (47)
4.1.1 Structure of wave mode with phase speed c+ For the wave mode with a positive root, the phase speed is given by 2 c+ = g H (1 + Mk 2 ),
(48)
which is identical to that of an external gravity wave on the surface with surface tension M of a homogeneous fluid of depth H . The ratio of the amplitudes is given by H η0 . = ζ0 (H − h)
(49)
In Fig. 4a, the variation of the surface elevation η and internal elevation ζ for various values of time period T with h/H = 0.4, T1 = 0.074 Nm−1 , ωt = 0, and M1 = M˜ 2 are plotted in the case of shallow water for waves in surface mode. It is clear that the disturbance on the free surface is larger in amplitude than the disturbance at the interface. However, a comparison with Fig. 4 suggests that the ratio of the amplitude of surface to interface elevation is comparatively smaller compared to that of deep water.
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(b) η
1.5
ζ
T=0.8s T=1s
surface and interface elevation
surface and interface elevation
(a) 1 0.5 0 −0.5 −1 −1.5 0
2
4
6
8
x
10
1
ζ
T=0.8s T=1s
0.8 0.6 0.4 0.2 0
η
−0.2 −0.4 −0.6 −0.8 0
2
4
6
8
10
x
Fig. 4 Variation of surface elevation η and internal elevation ζ for various values of time period T with a T1 = 0.074 Nm−1 and h/H = 0.4 for waves in surface mode and b T1 = 0.074 Nm−1 and ωt = 0 for waves in internal mode in the case of shallow water
4.1.2 Structure of wave mode with phase speed c− For the wave mode with the negative root, the phase speed is h(H − h) 2 . (50) = g (1 + Mk 2 ) c− H Differentiating the expression with respect to h, we conclude that the phase and group speeds of the internal wave are maximum in shallow water when h = H/2 which is not so in the case of the mode with negative root. The ratio of the amplitudes is given by η0 g (H − h) . (51) = ζ0 g (H − h) − g H Since g (H −h) 1, and for small x, x ∼ = −x, so we have gH
1−x
η0 g (H − h) . (52) =− ζ0 gH This shows that the disturbance on the free surface is much smaller in amplitude than the disturbance on the interface, and of opposite sign, so the two interfaces are 180◦ out of phase. We will illustrate the same in Fig. 4b, in which the variations of surface elevation η and interface elevation ζ for various values of time period T with h/H = 0.4, T1 = 0.074 Nm−1 , ωt = 0, and M1 = M˜ 2 are plotted for shallow water for waves in the internal mode. It may be noted that the two-layer results in the absence of surface tension can be obtained by using M1 , M˜ 2 → 0 (as [25, Chap. 14]) from the various results derived in this section. 5 Expansion formulae for capillary–gravity waves In this section, we will derive the expansion formulae for gravity-wave problems in the presence of surface and interfacial tensions in both the cases of finite and infinite depths. 5.1 Case of infinite depth The velocity potential φ(x, y) satisfying (1) along with the boundary conditions (11), (12a), (12b), (13) and (14a) in the case of infinite depth can be expanded as ∞ II ikn x φ(x, y) = An Fn (kn , y)e + A(ξ )L(ξ, y)e−ξ x dξ for x > 0, (53) n=I
0
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where the An are unknown constants and A(ξ ) is the unknown function to be determined with L 1 (ξ, y) for 0 < y < h, L(ξ, y) = L 2 (ξ, y) for h < y < ∞,
(54)
2 where L 1 (ξ, y) = K {(1 − M12ξ )ξ cos ξ y − K sin ξ y}, L 2 (ξ, y) = L1 (ξ, y) − W (ξ ) cos ξ(y − h), W (ξ ) = (1 − s)L 1 (ξ, h) + ξ (1 − M2 ξ ) − s ξ(1 − M1 ξ 2 ) sin ξ h − K cos ξ h . Using the boundary condition (11), we obtain the eigenfunctions Fn (kn , y) as follows: f (kn , y) for 0 < y < h, n = I, I I ; (55) Fn (kn , y) = ksn (h−y) e for h < y < ∞,
where f s (kn , y) =
K {(1 +
iL 1 (ikn , y) . sinh kn h − K cosh kn h}
M1 kn2 )kn
The eigenvalues kn satisfy the dispersion relation which is derived from the boundary condition (12b) and Eq. 55 and is given by kn2 (1 + M1 kn2 ) (1 + M2 kn2 ) − s − K kn (1 + M1 kn2 )(1 + s coth kn h)
+ (1 + M2 kn2 ) − s coth kn h + K 2 [s + coth kn h] = 0. (56) It may be noted that the dispersion relation (56) has two distinct real positive roots k I and k I I , four complex roots k I I I , k I V , k V and k V I of the form −a ± ib and −c ± id; see [20]. A brief discussion about root behavior in specific cases is provided in Sect. 3. It may be noted that L 1 (ikn , h) = 0 for all 0 < h < ∞. Using the boundedness criteria of the velocity potential φ(x, y) at the far field, we have considered the two positive real roots and neglected the contribution of the complex roots in the expansion formulae (53). The set of eigenfunctions Fn and L(ξ, y) are orthogonal with respect to the mode-coupling relation as given by Fm , Fn = s Fm , Fn 1 + Fm , Fn 2 ,
(57)
with h Fm , Fn 1 =
Fm (y)Fn (y)dy +
M1 F (0)Fn (0), K m
0
∞ Fm , Fn 2 = lim
→0
e− y Fm (y)Fn (y)dy +
M2 F (h)Fn (h). K m
h
It can be easily derived that 0 for m = n; m = n = I, I I, Fm , Fn = E n for m = n = I, I I. Using the eigenfunction (55), we have the value of E n as follows:
s 1 (1 + M1 kn2 )2 kn2 (sinh 2kn h + 2kn h) En = 4kn K 2 {kn (1 + M1 kn2 ) sinh kn h − K cosh kn h}2 2M2 3 2 2 2 . +K (sinh 2kn h − 2kn h) − 2K kn {(1 + M1 kn ) cosh 2kn h − (1 + 3M1 kn )} + 2 1 + k K n
(58)
(59)
Again, it can be derived that Fn , L(ξ, y) = 0 for all ξ > 0 and n = I, I I.
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Relations (58) and (60) ensure that the eigenfunctions Fn (kn , y), n = I, I I and L(ξ, y) are orthogonal with respect to the mode-coupling relation defined in (57). Using the above-mentioned orthogonal mode-coupling relation, we obtain the coefficients associated with the expansion formulae as An eikn x =
φ(x, y), Fn (kn , y) 2 , n = I, I I ; A(ξ )e−ξ x = φ(x, y), L(ξ, y) Fn , Fn π (ξ )
(61)
with (ξ ) = W 2 (ξ ) − 2W (ξ )L 1 (ξ, h) + {ξ 2 (1 − M1 ξ 2 )2 + K 2 }K 2 . Thus, the expansion formula for the velocity potential in the case of infinite depth is derived completely. Corollary 1 The equivalent form of the eigenfunctions Fn (kn , y) as in (55) is given by for 0 < y < h, f (kn , y) n = I, I I, Fn (kn , y) = −K sekn (2h−y) for h < y < ∞, where
L 1 (ikn , y) f (kn , y) = (1 + M2 kn2 )kn − (skn + K ) ekn h . L 1 (ikn , h) This form of Fn (kn , y) is derived from (55) by using the alternate form of the dispersion relation as given by Ks K {(1 + M2 kn2 )kn − (skn + K )} = . 2 iL 1 (ikn , h) {kn (1 + M1 kn ) sinh kn h − K cosh kn h} Using the orthogonal mode-coupling relation as in (57) and eigenfunctions as in Corollary 1, an equivalent form of E n as in (59) can be obtained as s K e2kn h K {(1 + M2 kn2 )kn − (skn + K )}2 En = − (1 + M1 kn2 )2 kn2 (sinh 2kn h + 2kn h) 4kn L 21 (ikn , h) 2M2 3 kn . +K 2 (sinh 2kn h − 2kn h) − 2K kn {(1 + M1 kn2 ) cosh 2kn h − (1 + 3M1 kn2 )} + 2s K 1 + K Remark It may be noted that in the case of M1 = M˜ 2 = M, the solution of the dispersion relation k(1 + Mk 2 ) = K associated with the single-layer capillary–gravity waves is one of the solutions of the generalized dispersion relation as in (56). In such a case, the terms associated with A I , i.e, FI (k I , y)eik I x in the expression for velocity potential in (53) is replaced by a0 e−k0 y+ik0 x , where k0 is the wave number associated with the free-surface gravity-wave dispersion relation and a0 = ek0 h is a constant. It can be easily derived that k I I satisfies the dispersion relation
(1 − s)kn (1 + Mkn2 ) − s K tanh kn h − K = 0, with Fn (kn , y) =
i(1−s)L 1 (ikn ,y) , 0 < y < h K for n = I I ; −K sekn (2h−y) , h < y < ∞
Further, in this particular case, in the expansion formulae for the velocity potential in (53), W (ξ ) reduces to
W (ξ ) = (1 − s) ξ 2 (1 − Mξ 2 )2 + K 2 sin ξ h. A similar observation was made by Rhodes-Robinson [20] on the wave numbers associated with the propagating and non-propagating modes.
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5.2 Case of finite depth The velocity potential φ(x, y) satisfying (1) along with the boundary conditions (11), (12a), (12b), (13) and (14b) in the case of finite depth can be expanded as φ(x, y) =
VI
Rn Yn (μn , y)eiμn x +
∞
An Yn ( pn , y)e− pn x for x > 0,
(62)
n=1
n=I
where Rn and An are unknown constants to be determined. Using the boundary condition (12a), we obtain the eigenfunction Yn as ⎧ i tanh μn (H − h)L 1 (iμn , y) ⎨ for 0 < y < h, Yn (μn , y) = K {(1 + M1 μ2n )μn sinh μn h − K cosh μn h} (63) ⎩ cosh μn (H − y)/ cosh μn (H − h) for h < y < H, with the eigenvalues μn , n = I, I I, I I I, I V satisfying the dispersion relation
D(μn ) ≡ μ2n (1 + M1 μ2n ) (1 + M2 μ2n ) − s − K μn (1 + M1 μ2n ) {coth μn (H − h) + s coth μn h}
+ (1 + M2 μ2n ) − s coth μn h + K 2 {coth μn h coth μn (H − h) + s} = 0
(64)
and μn = i pn for n = 1, 2, . . .. Further, it can be easily seen that the dispersion relations in (20) and (64) are equivalent. Keeping in mind that such boundary-value problems are of physical interest and our objective is to derive expansion formulae which can be used for a wide class of problems in fluid–structure interaction, it is assumed that the known coefficients are such that the dispersion relation (64) has two distinct positive real roots μn , n = I, I I , four complex roots μn , n = I I I, . . . , V I with −a ± ib and −c ± id and an infinite number of purely imaginary roots pn , n = 1, 2, . . .. (The behavior of the two positive roots has already been analyzed in particular cases in Sect. 3.) From the boundedness property of the velocity potential, it is seen that μV and μV I will vanish which corresponds to the roots with negative imaginary parts in the expansion formula (62). As discussed in the case of infinite depth, for finite depth also, the orthogonal mode-coupling relation can be defined in a similar manner by 0 for m = n, m, n = I, I I, I I I, I V, 1, 2, . . . , Ym , Yn = s Ym , Yn 3 + Ym , Yn 4 = (65) E n for m = n = I, I I, I I I, I V, 1, 2, . . . , where h Ym , Yn 3 =
Ym (y)Yn (y)dy +
M1 Y (0)Yn (0), K m
Ym (y)Yn (y)dy +
M2 Y (h)Yn (h). K m
0
H Ym , Yn 4 = h
Using the eigenfunction (63), we may obtain the value of E n as
s tanh2 μn (H − h) 1 2 2 2 En = 2 μn (1 + M1 μn ) (sinh 2μn h + 2μn h) 4μn μn (1 + M1 μ2n ) sinh μn h − K cosh μn h +K 2 (sinh 2μn h − 2μn h) − 2K μn {(1 + M1 μ2n ) cosh 2μn h − (1 + 3M1 μ2n )} 4M2 3 1 {sinh 2μn (H − h) + 2μn (H − h)} + μ sinh2 μn (H − h) + . K n cosh2 μn (H − h)
(66)
It is assumed that the set of eigenfunctions Yn , n = I, I I, I I I, I V, 1, 2, . . . are complete in view of the fact that the corresponding eigenvalues are discrete and hence there are no branch-cut distributions (a similar argument was
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On capillary gravity-wave motion in two-layer fluids
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made by Lawrie [26]). When dealing with a specific physical problem having a free surface and an interface, we must incorporate the edge conditions of the type as in (15a) and (15b) to obtain a unique solution and this must be used in conjunction with (65). Using the orthogonal mode-coupling relation (65), we obtain the unknown coefficients as φ(x, y), Yn (μn , y) , n = I, I I, I I I, I V, Rn eiμn x = Yn (μn , y), Yn (μn , y) φ(x, y), Yn ( pn , y) , n = 1, 2, . . . . (67) An e− pn x = Yn ( pn , y), Yn ( pn , y) This completes the derivation of the expansion formula and the associated mode-coupling relation in the case of finite water depth. Corollary 2 An equivalent form of the eigenfunctions Yn (μn , y) as in (63) is given by ⎧ L 1 (iμn , y) ⎨ μn h for 0 < y < h, e {(1 + M2 μ2n )μn − sμn } tanh μn (H − h) − K Yn (μn , y) = L 1 (iμn , h) ⎩ −K seμn h cosh μ (H − y)/ cosh μ (H − h) for h < y < H. n
n
This form of Yn (μn , y) is derived from (63) by using the alternate form of the dispersion relation as given by K [{(1 + M2 μ2n )μn − sμn } tanh μn (H − h) − K ] K s tanh μn (H − h) = . iL 1 (ikn , h) {(1 + M1 μ2n )μn sinh μn h − K cosh μn h} Using the relation (65) and the eigenfunctions as in Corollary 2, we obtain an equivalent form of E n as in (66) as s K e2μn h −K {{(1 + M2 μ2n )μn − sμn } tanh μn (H − h) − K }2 2 μn (1 + M1 μ2n )2 (sinh 2μn h + 2μn h) En = 4μn L 2 (iμn , h) +K 2 (sinh 2μn h − 2μn h) − 2K μn {(1 + M1 μ2n ) cosh 2μn h − (1 + 3M1 μ2n )} 4M2 3 sK 2 {sinh 2μn (H − h) + 2μn (H − h)} + + μ sinh μn (H − h) . K n cosh2 μn (H − h) It may be noted that for finite water depth for deriving the dispersion relation from the eigenfunctions as in (63), one has to use the interface boundary condition (12a), whilst the form of the eigenfunction in Corollary 2 can be used along with the boundary condition (12b) to derive the same dispersion relation. On the other hand, in the case of infinite depth, the dispersion relation (56) can be obtained from the eigenfunction (55) using the boundary condition. Corollary 3 The derivative of D(μn ) given in (64) is related to the expression for E n given in (66) by the following relation 2K 3 μ2n E n , (68) D (μn ) = [Yn (0)]2 sinh μn h[μn (1 + M1 μ2n ) sinh μn h − K cosh μn h] which can be written alternately in terms of Yn (h) as D (μn ) =
2K μ2n E n [μn (1 + M1 μ2n ) sinh μn h − K cosh μn h] . [Yn (h)]2 sinh μn h
(69)
Lemma 1 The eigenfunctions Yn (y) satisfy the following two identities s M1 K
∞
[Yn (0)]2 =1 En
(70)
[Yn (h)]2 = 1. En
(71)
n=I,I I,I I I,I V,1
and M2 K
∞ n=I,I I,I I I,I V,1
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Proof To prove the identity in (70), we proceed with the integral 1 (s M1 /K )K 3 α 2 1 − dα, I1 (α) = 2π i D(α) sinh αh[α(1 + M1 α 2 ) sinh αh − K cosh αh] α
(72)
c
where the contour c is a circle with centre at the origin of radius R. Using Jordan’s lemma and proceeding in a similar manner as in [27], from (72), it can be easily derived that ∞
I1 = 2 =
(s M1 /K )K 3 μ2n −1 M1 μ2n ) sinh μn h − K cosh μn h]
D (μn ) sinh μn h[μn (1 + n=I,I I,I I I,I V,1 ∞ [Yn (0)]2 s M1 K
n=I,I I,I I I,I V,1
En
− 1 = 0,
which yields identity (70). Proceeding in a similar manner by considering the integral (M2 /K )K α 2 [α(1 + M1 α 2 ) sinh αh − K cosh αh] 1 1 dα, I2 (α) = − 2π i D(α) sinh αh α
(73)
c
We may derive ∞
I2 = 2 =
n=I,I I,I I I,I V,1 ∞ M2
K
(M2 /K )K μ2n [μn (1 + M1 μ2n ) sinh μn h − K cosh μn h] −1 D (μn ) sinh μn h
n=I,I I,I I I,I V,1
[Yn (h)]2 − 1 = 0, En
which proves identity (71).
6 Applications of expansion formulae As applications of the expansion formulae, the solutions associated with the line source potential, the problem of forced oscillation in a rectangular semi-infinite channel and wave reflection by a wall are derived in both the cases of water of finite and infinite depths. 6.1 Derivation of line-source potential The generation of water waves with surface and interfacial tension involves the consideration of different types of singularities in the fluid. When these waves are generated by a body present in the fluid, the resulting motion can be described by a series of line singularities placed within the body. The symmetric wave-source potential in the presence of surface and interfacial tension is essentially the function φ(x, y|x0 , y0 ) which satisfies Eq. 1 in the fluid region, except at the source point (x0 , y0 ), where 0 < y0 < ∞ in case of infinite depth and 0 < y0 < H in case of finite depth along with the appropriate boundary conditions in (11)–(14a,14b). At the source point, i.e., as (x, y) → (x0 , y0 ), 1 log(r ) as r = (x − x0 )2 + (y − y0 )2 → 0. φ∼ (74) 2π Due to the symmetry property of the fundamental wave-source potential, we discuss the case when x > x0 and the result will be same in the case of x < x0 . The condition (74) can be recast as ∂φ = δ(y − y0 )/2 on x = x0 , in both the cases of finite and infinite depth. (75) ∂x
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Further, the velocity potential φ satisfies the edge conditions (Similar for capillary–gravity waves as in [2]) ∂ 2φ ∂ 2φ (0+, 0, z) = 0 and (0+, h, z) = 0. ∂ x∂ y ∂ x∂ y Using the generalized identity ∞ F(y0 ) if y0 > 0, δ(y − y0 )F(y)dy = F(y0 )/2 if y0 = 0,
(76)
0
and the expansion formula as discussed in Sect. 5, we will derive the source potentials next. 6.1.1 Case of infinite depth The source potential φ(x, y|x0 , y0 ) for infinite depth is represented as ∞ II ikn (x−x0 ) An Fn (kn , y)e + A(ξ )L(ξ, y)dξ for x > x0 , φ(x, y|x0 , y0 ) = n=I
(77)
0
where An , n = I, I I are unknown constants and A(ξ ) is an unknown function to be determined. Utilizing (75), the property of the delta function as in (76), we obtain the unknowns An and A(ξ ) in (77) as ⎧ −isδ1 f s (kn , y0 ) ⎪ ⎪ for 0 ≤ y0 < h, ⎪ ⎪ {2k s f (k , y), f s (kn , y) 1 + 1 + 2M2 kn3 /K } ⎪ n s n ⎪ ⎨ −iδ1 ekn (h−y0 ) (78) An = for h < y0 ≤ ∞, ⎪ {2kn s f s (kn , y), f s (kn , y) 1 + 1 + 2M2 kn3 /K } ⎪ ⎪ ⎪ i{s f s (kn , h) + 1} ⎪ ⎪ for y0 = h, ⎩ 2{2kn s f s (kn , y), f s (kn , y) 1 + 1 + 2M2 kn3 /K } and ⎧ −sδ1 K {ξ(1 − M1 ξ 2 ) cos ξ y0 − K sin ξ y0 } ⎪ ⎪ for 0 ≤ y0 < h, ⎪ ⎪ ⎪ π ξ (ξ ) ⎪ ⎨ −[K {ξ(1 − M1 ξ 2 ) cos ξ y0 − K sin ξ y0 } − W (ξ ) cos ξ(y0 − h)] A(ξ ) = (79) for h < y0 ≤ ∞, ⎪ π ξ (ξ ) ⎪ ⎪ ⎪ −[(1 + s)K {ξ(1 − M1 ξ 2 ) cos ξ h − K sin ξ h} − W (ξ )] ⎪ ⎪ ⎩ for y0 = h, 2π ξ (ξ ) where kn , Fn (kn , y), L(ξ, y), f s (kn , y), W (ξ ) and (ξ ) are the same as given in Sect. 5.1 with δ1 = 1 for 0 < y0 < h and δ1 = 1/2 for y0 = 0. 6.1.2 Case of finite depth The source potential φ(x, y|x0 , y0 ) for finite depth is represented as φ(x, y|x0 , y0 ) =
IV n=I
An Yn (μn , y)eiμn (x−x0 ) +
∞
Bn Yn ( pn , y)e− pn (x−x0 ) for x > x0 ,
(80)
n=1
Utilizing the condition (75), we obtain the unknowns An and Bn in the relation (80) as ⎧ iδ2 s sinh μn (H − h)V1 (μn ; y0 ) ⎪ ⎪ for 0 ≤ y0 < h, ⎪ ⎪ ⎪ n En ⎨ −iδ cosh μ 2μ 2 n (H − y0 ) for h < y0 ≤ H, An = ⎪ 2μn E n ⎪ ⎪ ⎪ is sinh μn (H − h)V1 (μn ; h) i cosh μn (H − h) ⎪ ⎩ − for y0 = h, 4μn E n 4μn E n and
(81)
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⎧ iδ2 s sin pn (H − h)V1 ( pn ; y0 ) ⎪ ⎪ for 0 ≤ y0 < h, ⎪ ⎪ pn E n ⎪ ⎨ −iδ cos p 2(H − y0 ) 2 n for h < y0 ≤ H, Bn = ⎪ 2 pn E n ⎪ ⎪ ⎪ is sin pn (H − h)V1 ( pn ; h) i cos pn (H − h) ⎪ ⎩ − for y0 = h, 4 pn E n 4 pn E n
(82)
where δ2 = 1 for y0 ∈ (0, h) ∪ (h, H ) and δ2 = 1/2 for y0 = 0, H and μn , pn , Yn (μn , y), E n are the same as in Sect. 5.2 with V1 (μn ; y0 ) = {μn (1 + M1 μ2n ) cosh μn y0 − K sinh μn y0 }/{μn (1 + M1 μ2n ) cosh μn h − K sinh μn h} and pn = iμn for n = 1, 2, . . ..
6.2 Forced oscillation in a rectangular semi-infinite channel Suppose that an anti-symmetric three-dimensional wave motion is set up in a semi-infinite canal by the small motion of a vertical board in a two-layer fluid in the presence of surface and interfacial tension considering only the first term of the Fourier-cosine series in z. The solution of the physical problem is obtained for the fundamental mode of frequency, i.e, n = 1. With the assumption that the fluid is inviscid and incompressible, the motion is irrotational and simple harmonic in time with angular frequency ω, the velocity potential is of the form j (x, y, z, t) = Re{φ j (x, y, z)e−iωt } for j = 1, 2 where Re denotes the real part. The velocity potentials φ j , j = 1, 2 in the region 0 < x < ∞, 0 ≤ z ≤ b and y ∈ (0, h) ∪ (h, H ) in case of finite water depth and y ∈ (0, h) ∪ (h, ∞) in case of infinite depth satisfy the three-dimensional Laplace equation given by ∇x2yz φ j = 0.
(83)
The velocity potential satisfies the free-surface condition (11), interface condition (12a), (12b) and the bottom boundary condition as in (14a) and (14b). At the vertical boundary x = 0, the velocity potential satisfies πz ∂φ j . (84) = U (y) cos ∂x b Again at the two sides the velocity potential satisfies ∂φ j = 0, for z = 0 and z = b. ∂z The edge conditions (as in [20]) are given by
(85)
∂ 2 φ1 ∂ 2 φ2 πz πz (0+, 0, z) = π λ1 cos and (0+, h, z) = π λ2 cos , ∂ x∂ y b ∂ x∂ y b
(86)
where λ1 and λ2 are assumed to be known (Similar to the edge conditions as in [3] for a single-layer fluid). We will next analyze this problem in both the cases of finite and infinite water depths separately. 6.2.1 Case of infinite depth The velocity potential satisfying the governing equation (83) and the boundary conditions (11), (12a), (14a) and (85) is of the form ⎫ ⎧ ∞ ⎬ πz ⎨ ¯ (87) φ(x, y, z) = cos A I eim I x FI (k I , y) + A I I eim I I x FI I (k I I , y) + A(ξ )L(ξ, y)e−βx dξ , ⎭ b ⎩ 0
where the An for n = I, I I are the unknown far-field wave amplitudes in surface and internal modes to be deter!2 !2 mined, with m 2n = kn2 − πb for n = I, I I , β¯ 2 = ξ 2 + πb and Fn , n = I, I I are same as defined in the case of infinite depth in Sect. 5.1. Substituting for the velocity potential φ in the vertical boundary condition at x = 0
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(a)
(b) s=0.85 s=0.9 s=0.95
0.7
0.12
0.5
0.1
⏐A ⏐
0.4
II
I
s=0.85 s=0.9 s=0.95
0.14
0.6
⏐A ⏐
0.16
0.08
0.3
0.06
0.2
0.04
0.1
0.02
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0.5
1
1.5
Time period
2
2.5
3
3.5
4
4.5
Time period
Fig. 5 Variation of (a) the amplitude |A I | in surface mode and (b) the amplitude |A I I | in internal mode versus time period T for various values of s with h = 1 m, λ1 = 1 and λ2 = 1
and applying the orthogonal mode-coupling relation as defined in (57), we obtain the unknown constants An for n = I, I I and the unknown function A(ξ ) given by An =
2 U (y), L(ξ, y) n , n = I, I I, A(ξ ) = , ¯ im n Yn , Yn π β (ξ )
(88)
where h n = s
∞ U (y)Fn (kn , y)dy +
0
U (y)Fn (kn , y)dy h
M 2 π λ2 k n s M 1 π λ1 k n + , − 2 [kn (1 + M1 kn ) sinh kn h − K cosh kn h] K h U (y), L(ξ, y) = s
∞ U (y)L 1 (ξ, y)dy +
0
U (y)L 2 (ξ, y)dy h
− M1 π λ1 ξ − M2 π λ2 ξ (1 − M1 ξ 2 )ξ sin ξ h + K cos ξ h , with (ξ ) being the same as defined in (61). It may be noted that the edge conditions are incorporated into the solution directly while applying the orthogonal mode-coupling relation. In Fig. 5a and b, variations of amplitude of capillary–gravity waves in surface and internal modes in case of a wavemaker in water of infinite depth are plotted for different values of density ratios s. It is observed that with an increase of the density ratio s, the wave amplitudes in the far field in surface mode for different values of s are very similar in nature. However, there is a significant decrease in wave amplitude in internal mode with an increase in the density ratio s. In surface mode, with an increase in time period, the wave amplitude attains optimum values for different time periods alternately. However, in internal mode, the occurrence of a maximum is observed for certain values of the time period. However, in general, the wave amplitude in surface mode is much higher than that in internal mode. In Fig. 6a and b, variations of wave amplitudes of capillary–gravity waves in surface and internal modes are plotted for different values of surface tension T1 in case of a wavemaker in infinite water depth. In this case, the changes in wave amplitude due to a change in surface tension in surface mode is negligible for different values of the surface tension T1 . However, the wave amplitude in internal mode decreases with an increase in the values of the surface tension T1 , as expected. However, the pattern is similar to that in Fig. 5a and b.
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(a)
(b) 0.16
T1=0.004N/m
0.7
1
1
0.6
T =0.09N/m 1
II
⏐A ⏐
I
⏐A ⏐
0.4
0.08 0.06
0.2
0.04
0.1
0.02 1.5
2
2.5
3
1
T1=0.09N/m
0.1
0.3
1
T =0.074N/m
0.12
0.5
0.5
T =0.004N/m
0.14
T =0.074N/m
3.5
4
4.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time period
Time period
Fig. 6 Variation of (a) the amplitude |A I | in surface mode and (b) the amplitude |A I I | in internal mode versus time period T for various values of surface tension T1 with h = 1 m, λ1 = 1 and λ2 = 1
6.2.2 Case of finite depth The velocity potential satisfying the governing equation (83) and the boundary conditions (11), (12a), (14b) and (85) is of the form " IV ∞ πz im n x −m n x φ(x, y, z) = cos An e Yn (μn , y) + An e Yn ( pn , y) , (89) b n=1
n=I
!2 where the An for n = I, I I, I I I, I V, 1, 2, . . . are unknown constants to be determined with m 2n = μ2n − πb for n = !2 I, I I, I I I, I V, m 2n = pn2 + πb for n = 1, 2, . . . and Yn , n = I, I I, I I I, I V, 1, 2, . . . are same as defined in the case of finite depth of Sect. 5.2. Using the vertical boundary condition with the application of the orthogonal modecoupling relation and the edge condition, we obtain the unknown constants An , n = I, I I, I I I, I V, 1, 2, . . . by An =
n , im n Yn , Yn
(90)
where h n = s
H U (y)Yn (μn , y)dy +
0
U (y)Yn (μn , y)dy h
M2 π λ2 μn s M1 π λ1 μn tanh μn (H − h) − tanh μn (H − h), + [μn (1 + M1 μ2n ) sinh μn h − K cosh μn h] K with m n = im n and μn = i pn , for n = 1, 2, . . .. It may be noted that An becomes unbounded for μn → π/b for n = I, I I, I I I, I V . Thus, there are two cut-off frequencies in this case. One is associated with the wave propagating in the surface mode and the other with the wave propagating in the internal mode. Further, the amplitudes of oscillation depend fully on the edge behavior at the free surface and the interface.
6.3 Wave reflection by a wall A sinusoidal wave is obliquely incident from the positive x-direction making an angle θ with the x-axis in a twolayer fluid with a vertical wall at x = 0. With the assumption that the fluid is inviscid and incompressible, the motion is irrotational and simple harmonic in time with angular frequency ω, the velocity potential is of the form j (x, y, z, t) = Re{φ j (x, y)eilz−iωt }, j = 1, 2 where Re denotes the real part, φ j (x, y) denotes the spatial parts of the complex velocity potential with l as the component of the wave number along the z-direction. Thus, the
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spatial velocity potential φ j , j = 1, 2 in the region 0 < x < ∞, ∞ < z < ∞, y ∈ (0, h) ∪ (h, H ) for finite depth and y ∈ (0, h) ∪ (h, ∞) for infinite depth satisfies the Helmholtz equation given by ∇x2y − l 2 φ j = 0. (91) At the vertical boundary x = 0, the velocity potential satisfies the same condition as in (84) with U (y) = 0 and also satisfies the edge condition as given by ∂ 2 φ1 ∂ 2 φ2 (0+, 0) = π λ1 and (0+, h) = π λ2 , ∂ x∂ y ∂ x∂ y where λ1 and λ2 are assumed to be known (Similar to that defined in (86)).
(92)
6.3.1 Case of infinite depth The velocity potential satisfying the governing equation (91) and the boundary conditions (11), (12a), (12b) and (14a) is of the form
φ(x, y) = A I eim I x + R I e−im I x FI (k I , y) + A I I eim I I x + R I I e−im I I x FI I (k I I , y) ∞ +
¯
R(ξ )L(ξ, y)e−βx dξ,
(93)
0
where the An are known constants and the Rn for n = I, I I are the unknown constants to be determined with l = kn sin θ, m n = kn cos θ, m 2n = kn2 − l 2 for n = I, I I , β¯ 2 = ξ 2 + l 2 and Fn , n = I, I I and L(ξ, y) are same as defined in the case of infinite depth. Using the vertical boundary condition with the application of the orthogonal mode-coupling relation and edge condition the unknown constants An , n = I, I I and the unknown functions R(ξ ) are given by n Rn = A n − , n = I, I I, im n Yn , Yn # $ 2 −s M1 π λ1 ξ − M2 π λ2 ξ (1 − M1 ξ 2 )ξ sin ξ h + K cos ξ h R(ξ ) = , (94) ¯ π β (ξ ) where n = −
M 2 π λ2 k n s M 1 π λ1 k n + , [kn (1 + M1 kn2 ) sinh kn h − K cosh kn h] K
with (ξ ) being the same as defined in the case of infinite depth as in Sect. 5.1. 6.3.2 Case of finite depth The velocity potential satisfying the governing equation (91) and the boundary conditions (11), (12a) and (14b) is of the form IV ∞ φ(x, y, z) = An eim n x + Rn e−im n x Yn (μn , y) + Rn e−m n x Yn ( pn , y), (95) n=I
n=1
where A I , A I I , A I I I , A I V are known constants and the Rn for n = I, I I, I I I, I V, 1, 2, . . . are unknown constants to be determined with l = μn sin θ, m n = μn cos θ, m 2n = μ2n − l 2 for n = I, I I, I I I, I V, m 2n = pn2 + l 2 for n = 1, 2, . . . and Yn , n = I, I I, I I I, I V, 1, 2, .. are same as defined in Sect. 5.2. Using the vertical boundary condition with the application of the orthogonal mode-coupling relation and edge condition, the unknown constants An , n = I, I I, I I I, I V, 1, 2, . . . are given by n Rn = A n − , n = I, I I, I I I, I V, (96) im n Yn , Yn
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where
s M1 π λ1 μn tanh μn (H − h) M2 π λ2 μn − tanh μn (H − h), 2 [μn (1 + M1 μn ) sinh μn h − K cosh μn h] K with An = 0, m n = im n and μn = i pn , for n = 1, 2, . . ..
n =
7 Conclusion In the present paper, the capillary–gravity wave motion in the presence of surface and interfacial tension has been investigated in a two-layer fluid in both the cases of finite and infinite water depths. In order to analyze the existence of the surface and internal modes, the dispersion relation was analyzed in case of various water depths in the presence of surface and interfacial tension. The wave characteristics in surface and internal modes were analyzed in both the cases of deep and shallow-water waves by analyzing the phase and group velocities. It was found that for all the water depths, the amplitude of the surface wave is higher than the amplitude of the interfacial wave. Further, it has been observed that for certain values of the surface-tension parameter in case of deep water, the phase velocities in surface and internal modes attend minimum values, which are similar to that of the wave motion in case of capillary–gravity waves in a single-layer fluid. An interesting aspect of the corresponding boundary-value problem is the occurrence of higher-order boundary conditions on the free surface and interface boundaries associated with Laplace’s equation. As a result, the corresponding boundary-value problem gives rise to non-Sturm–Liouville eigen sub-system and the corresponding eigenfunctions are not orthogonal in the usual sense. In the present study, expansion formulae for two-layer fluid in the presence of surface tension and interfacial tension have been developed along with the orthogonal mode-coupling relations satisfied by the eigenfunctions in both the cases of finite and infinite water depths in two dimensions. Further, various characteristics of the eigenfunctions were derived which are expected to simplify the computation of several physical problems. As applications of the expansion formulae, (i) the source potentials associated with the two-layer capillary–gravity wave motion and (ii) problem of forced oscillation of capillary–gravity waves in the presence of surface and interfacial tension in a semi-infinitely extended channel were derived in both the cases of finite and infinite water depth. The results of forced oscillation were used to analyze the wave reflection by a rigid wall in a channel of finite width. The expansion formulae associated with the capillary gravity wave problems in the two layer fluid were further generalized to deal with boundary-value problems satisfying higher-order boundary conditions at the free surface and interface. The developed expansion formulae can be generalized to deal with similar problems in three dimensions and various characteristics of the corresponding eigen-system can be studied. Further, study can be made of the convergence of several expansion formulae and associated series/integrals as discussed in the present paper. The explicit formulae and various results derived on phase and group velocities can be used as a benchmark solution for more complex problems. The present study on the propagation of waves in a two-layer fluid in the presence of surface and interfacial tension will facilitate the analysis of various physical problems in Ocean Engineering including oil spilling, wave oscillation in tanks and other branches of mathematical physics, where higher-order boundary conditions arise in a natural way. Acknowledgement DK acknowledges the support of CSIR, Govt. of India, for the financial support received as a Senior Research Fellow at IIT Kharagpur to pursue this research work. The partial support of NRB, New Delhi, to pursue this research work is gratefully acknowledged.
Appendix Under the assumption of the linearized theory of water waves, the spatial velocity potential φ(x, y) satisfies the Laplace equation (1). On the free surface at y = 0, we assume that the velocity potential φ satisfies a general type of boundary condition of the form
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& % & ∂ ∂ ∂φ +M φ = 0, ∂x ∂y ∂x where L and M are the linear differential operators of the form % & % & l0 m0 ∂ ∂ ∂ 2n ∂ 2n L cn 2n , M dn 2n , = = ∂x ∂x ∂x ∂x %
L
n=0
(A1)
(A2)
n=0
with cn and dn are known constants (as in [6,7,26]). Keeping in mind various physical problems of concern, only the even derivatives in x are included. In the present work, it is assumed that l0 ≥ m 0 so that l0 determines the highest-order derivative in the boundary condition at y = 0. The linearized condition at the interface y = h, bottom boundary condition and the far-field radiation condition are the same as defined in Sect. 2. In case of infinite depth the kn are assumed to be real and positive and satisfy the following relation in k: k 2 P(k; l0 ){(1 − Q 2 k 2 ) − s} − k [K P(k; l0 ){1 + s coth kh} +{(1 − Q 2 k 2 ) − s}Q(k; m 0 ) coth kh + K Q(k; m 0 ){s + coth kh} = 0,
(A3)
and in case of finite depth, the μn are assumed to be real and positive and satisfy the following relation in k: k 2 P(k; l0 ){(1 − Q 2 k 2 ) − s} − k [K P(k; l0 ){coth k(H − h) + s coth kh} +{(1 − Q 2 k 2 ) − s}Q(k; m 0 ) coth kh + K Q(k; m 0 ){s + coth k(H − h) coth kh} = 0,
(A4)
where P(k; l0 ) =
l0
(−1)n cn k 2n ;
Q(k; m 0 ) =
n=0
m0 (−1)n dn k 2n
(A5)
n=0
! ! are the characteristic polynomials associated with the linear differential operator L ∂∂x and M ∂∂x , respectively. Equations A3 and A4 are important for analyzing the wave characteristics in wave-structure interaction problems and represent the dispersion relations in case of finite and infinite depths, respectively. The generalized orthogonal mode-coupling relation in the presence of interfacial tension and any higher-order boundary condition at the free surface of (2l0 + 1)th order for both the cases of finite and infinite depth is given by Fm , Fn = s Fm , Fn 1 + Fm , Fn 2 ,
(A6)
where h Fm , Fn 1 =
Fm (y)Fn (y)dy +
j=1
0
+
l0 (−1) j
m0 (−1) j+1 j=1
j
k=1
dj 2 j−2k Fm2k−2 (0)Fn (0), P(kn ; l0 ) j
k=1
H or ∞
Fm , Fn 2 =
cj 2 j−(2k−1) Fm2k−1 (0)Fn (0) Q(kn ; m 0 )
Fm (y)Fn (y)dy −
Q2 F (h)Fn (h). K m
h
A.1 Infinite depth The general form of velocity potential φ(x, y) satisfying (1) along with the boundary conditions (A1), (11), (12a), (12b) and (14a) in the case of infinite depth can be expanded as follows ∞ 2l0 ik0 x ikn x φ(x, y) = A0 F0 (k0 , y)e + An Fn (kn , y)e + A(ξ )L(ξ, y)e−ξ x dξ for x > 0, (A7) n=I
0
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where the An are unknown constants and A(ξ ) is an unknown function to be determined with L 1 (ξ, y) for 0 < y < h, L(ξ, y) = L 2 (ξ, y) for h < y < ∞,
(A8)
where L 1 (ξ, y) = K {P(iξ ; l0 ) cos ξ y − Q(iξ ; m 0 ) sin ξ y}, L 2 (ξ, y) = L 1 (ξ, y) − W (ξ ) cos ξ(y − h),
W (ξ ) = ξ (1 − s) {K P(iξ ; l0 ) − Q(iξ ; m 0 )} − K (1 + Q 2 ξ 2 ) − K cos ξ h
− K (1 − s)Q(iξ ; m 0 ) + ξ 2 P(iξ ; l0 ) (1 + Q 2 ξ 2 ) − s sin ξ h. By use of (A6) the unknowns An and A(ξ ) are obtained as An =
φ(x, y), Fn (kn , y) 2 φ(x, y), L(ξ, y) , n = 0, I, . . . , 2l0 and A(ξ ) = , Fn (kn , y), Fn (kn , y) π
(ξ )
(A9)
where
(ξ ) = W 2 (ξ ) − 2K W (ξ ) {ξ P(iξ ; l0 ) cos ξ h − Q(iξ ; m 0 ) sin ξ h} + K 2 ξ 2 P 2 (iξ ; l0 ) + Q 2 (iξ ; m 0 ) .
A.2 Finite depth The general form of the velocity potential φ(x, y) satisfying Eq. 1 along with the boundary conditions (A1), (11), (12a), (12b) and (14b) in the case of finite depth can be expanded as follows: φ(x, y) = R0 Y0 (μ0 , y)e
iμ0 x
+
2l0
Rn Yn (μn , y)e
iμn x
n=I
+
∞
An Yn ( pn , y)e− pn x for x > 0.
(A10)
n=1
Using the orthogonal mode-coupling relation as in (A6) the Rn are obtained as Rn =
φ(x, y), Yn (μn , y) , n = 0, I, . . . , 2l0 ; Yn (μn , y), Yn (μn , y)
An =
φ(x, y), Yn ( pn , y) , n = 1, 2, . . . . Yn ( pn , y), Yn ( pn , y)
(A11)
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