Generation Of Unsteady Waves By Concentrated Disturbances In An Inviscid Fluid With An Inertial Surface

Acta Mech Sin (2008) 24:267–275 DOI 10.1007/s10409-008-0155-0

RESEARCH PAPER

Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface D. Q. Lu · S. Q. Dai

Received: 17 May 2007 / Revised: 5 February 2008 / Accepted: 3 March 2008 / Published online: 8 May 2008 © Springer-Verlag 2008

Abstract The surface waves generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface are analytically investigated for two- and three-dimensional cases. The fluid is assumed to be inviscid, incompressible and homogenous. The inertial surface represents the effect of a thin uniform distribution of non-interacting floating matter. Four types of unsteady concentrated disturbances and two kinds of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an instantaneous/oscillating impulse on the surface, an initial impulse on the surface of the fluid, and an initial displacement of the surface. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the surface elevation are obtained by means of a joint Laplace–Fourier transform. The asymptotic representations of the wave motion for large time with a fixed distanceto-time ratio are derived by using the method of stationary phase. The effect of the presence of an inertial surface on the wave motion is analyzed. It is found that the wavelengths of the transient dispersive waves increase while those of the steady-state progressive waves decrease. All the wave amplitudes decrease in comparison with those of conventional The project supported by the National Natural Science Foundation of China (10602032), the Shanghai Rising-Star Program (07QA14022), and the Shanghai Leading Academic Discipline Project (Y0103). D. Q. Lu (B) Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 200072 Shanghai, China e-mail: [email protected]; [email protected] S. Q. Dai Shanghai Key Laboratory of Mechanics in Energy and Environment Engineering, 200072 Shanghai, China e-mail: [email protected]

free-surface waves. The explicit expressions for the freesurface gravity waves can readily be recovered by the present results as the inertial surface disappears. Keywords Waves · Inertial surface · Concentrated disturbances · Asymptotic representation

1 Introduction As is well known, the oceans in polar regions are generally covered by ice. According to the physical nature of the problem considered, two kinds of mathematical models for the ice-covered ocean surface are customarily employed [1]. When an ocean area is covered by a thin uniform distribution of non-interacting floating matter, for example, broken ice, the ocean surface is mathematically modeled as an inertial surface [2–6]. When an ocean area is covered by an ice sheet of small thickness, the ocean surface is mathematically idealized as a thin homogenous elastic plate [1,7–14]. As the flexural rigidity of the plate tends to zero, the inertial surface model will be useful. The generation of the gravity waves due to the initial elevation and impulse at the surface of an inviscid fluid with an inertial surface was first considered by Mandal and Mukherjee [4,5] for two- and three-dimensional cases. However, it seems that the general characteristics of the wave responses to disturbances acting on the fluid with an inertial surface have not been fully elucidated. In this paper, dynamic responses of a fluid with an inertial surface to line- and point-concentrated disturbances will be considered. The fluid is initially quiescent, infinitely deep and is assumed to be inviscid, incompressible and homogenous. Four types of unsteady concentrated disturbances and two kind of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an

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instantaneous/oscillating load on the surface, an initial impulse on the surface, and an initial displacement of the surface. In Sect. 2, the general mathematical model is formulated as an initial-boundary-value problem for two- and threedimensional cases. The fluid is assumed to be covered by an inertial surface of area density ερ with 0 ≤ ε < 1, where ρ is the density of the fluid and ε is a parameter having the dimension of length. The special case with ε = 0 corresponds to a fluid with a clean surface. All the types of disturbances are simultaneously included in the linear mathematical formulation. In Sect. 3, exact solutions in closed integral forms for the transient waves due to impulsive line disturbances and initial concentrated disturbances are given by use of a joint Laplace–Fourier transform. In order to obtain the principal physical features of the wave motion, it is necessary to adopt the asymptotic analysis for the integral-form solution. The method of stationary phase shall be employed to obtain the asymptotic representation of the wave motion for large time with a fixed distance-to-time ratio. This approximation has been successfully used to study the classical Cauchy–Poisson problems [15, Sect. 6.4],[16, Sect. 3.7] and their generalized cases [1,4,5,11–14,17–23]. It is found that the transient waves are dispersive. In Sect. 4, the asymptotic representations for the unsteady waves due to oscillating line disturbances are derived. It is found that the unsteady waves consist of steady-state progressive waves and transient dispersive waves. In Sect. 5, the solutions for the three-dimensional cases are provided. Finally, discussion and conclusions are given in Sect. 6.

2 General mathematical formulation Without loss of generality, a Cartesian system is used, in which the z axis points vertically upward while z = 0 represents the undisturbed inertial surface. Therefore, the governing equation is ∇ 2  = Mδ(x − x 0 ),

(1)

where (x, t; x 0 ) is the velocity potential for the perturbed flow, M(t) the time-dependent strength of the simple source, δ(·) the Dirac delta function, x an observation point, t the time, and x 0 the source point. For two-dimensional cases, x = (x, z), x 0 = (0, z 0 ) and ∂/∂ y = 0 while for threedimensional cases, x = (x, y, z) and x 0 = (0, 0, z 0 ). It is assumed that the wave amplitude is very small in comparison with the wavelength. Thus, the linearized boundary conditions will be applied on the undisturbed inertial surface. According to Rhodes-Robinson [2,3] the kinematic and dynamic conditions at z = 0 are given by ∂η ∂ − = 0, ∂t ∂z

123

(2)

∂ 2η P ∂ + gη + ε 2 = − δ(z), ∂t ∂t ρ

(3)

where η is the elevation of the inertial surface, g the acceleration of gravity, P(t) the time-dependent strength of the applied load, and z the field point at the undisturbed inertial surface. For the two-dimensional cases, z = (x, 0) while for the three-dimensional cases, z = (x, y, 0). We impose the initial conditions on , η and the first time derivative of η in order to meet the requirement for Eqs. (2) and (3) to be a well-posed initial-boundary-value problem. Therefore, the initial conditions at z = 0 are I0 δ(z), ρ = E 0 δ(z),

|t=0 = −

(4)

η|t=0
∂η

= 0. ∂t
t=0

(5) (6)

Equations (4) and (5) imply that the fluid is initially subject to a surface impulse and elevation concentrated at the origin, where I0 and E 0 are the corresponding constant magnitudes. Equation (6) shows that the initial velocity of the inertial surface is zero, which is consistent with the assumption that the entire fluid is at rest for t < 0. Moreover, since the finite disturbance must die out at infinity, it is required that ∇ → 0 as z → ∞,

(7)

which imposes a uniqueness on the problem considered. Thus, Eqs. (1) to (7) constitute a well-posed problem for  and η, mathematically as well as physically. The assumption of linearity allows us to envisage the perturbed flow as the superposition of a singular flow and a regular flow. The former represents the effect of the moving singularity in an unbounded domain while the latter the influence of the boundary. Thus, we write  = S (x, t; x 0 ) + R (x, t),

(8)

S

where is the potential due to the singularity while R is a continuous function everywhere in the corresponding domain and ∇ 2 R = 0. Thus, the relation between the singular and regular components can be established through the boundary conditions at z = 0: ∂S ∂η ∂R − = , ∂t ∂z ∂z ∂R ∂ 2η ∂S P + gη + ε 2 = − δ(z) − . ∂t ∂t ρ ∂t

(9) (10)

3 Transient waves due to instantaneous line disturbances In this section, the instantaneous line disturbances are considered. The time-dependent strengths of the simple source

Generation of unsteady waves by concentrated disturbances

269

and applied load are mathematically represented as M(t) = M0 δ(t) and P(t) = P0 δ(t), respectively. Accordingly, the singular component in Eq. (8) is the fundamental solution of two-dimensional Laplace equation in an unbounded domain,   1 M0 S ln δ(t), (11)  =− 2π r where r = ||x − x 0 ||. By taking the Laplace–Fourier transform over Eq. (1) and applying the Jordan lemma and the Cauchy residue theorem, an alternative representation for Eq. (11) can be given by S = −

M0 8π 2 i

Equation (16) is known as the dispersion relation, as shown in Fig. 1. A special case of Eq. (16) with ε = 0 corresponds to the dispersion relation for the classical Cauchy–Poisson wave problem for an inviscid fluid of infinite depth. The dispersion relation for gravity waves on the inertial surface is herein derived by using the Laplace–Fourier integral transform. In fact, one can also obtain the dispersion equation by means of the method of separation of variables. It can be seen from Eq. (16) that ωlim (ε) = lim ω(k, ε) = k→∞

c+i∞ 

+∞ 1 ds dα k

g/ε.

(18)

Equation (16) can be re-written as

−∞

c−i∞

× exp(−k|z − z 0 | + iαx + st),

(12)

where k = |α|, c is the Laplace convergence abscissa. In order to obtain the formal solution of this initialboundary-value problem, it is convenient to introduce a combination of the Laplace transform with respect to t and the Fourier transform with respect to spatial variables. For the two-dimensional cases, we have 1 { , η} = 4π 2 i



c+i∞ 

+∞ ds dα

R

−∞

c−i∞ R

˜ exp(kz), η} ˜ exp(iαx + st). ×{

ω2 . g − εω2

(19)

Since k > 0, it follows from Eq. (19) that ω < ωlim . Once ε is given, progressive waves with ω ≥ ωlim will not be able to appear since “the inertial surface is too heavy” [2]. However, the propagation of progressive waves with any frequency is possible if the surface tension is included in the mathematical formulation [2]. The ratio of group velocity to phase velocity for the gravity waves on the inertial surface is given by

(13)

By substituting Eqs. (12) and (13) into the Laplace–Fourier transforms of boundary conditions (9) and (10), two simultaneous algebraic equations are set up for the unknown func˜ R and η, ˜ which can readily be solved. Consequently, tions  the formal integral expression for the surface elevation can be written as η = M0 ηm (z, t; x 0 ) +

k=

σ =

kC g 1 = , ω 2(1 + εk)

(20)

and is shown in Fig. 2.

P0 I0 ηp (z, t) + ηi (z, t) + E 0 ηe (z, t), ρ ρ (14)

where +∞ exp(iαx) 1 {ηm , ηp , ηi , ηe } = dα 2π 1 + εk −∞  k × exp(kz 0 ) cos(ωt), − sin(ωt), ω  k (15) − sin(ωt), (1 + εk) cos(ωt) , ω ω(k, ε) =

ω0 , (1 + εk)1/2

ω0 = ω(k, 0) =



gk.

(16)

(17)

Fig. 1 Wave frequencies versus wave-number. (i) ε = 0 m; (ii) ε = 0.005 m; (iii) ε = 0.01 m; (iv) ε = 0.05 m; (v) ε = 0.1 m

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Fig. 2 The ratio of group velocities to phase speeds. (i) ε = 0 m; (ii) ε = 0.005 m; (iii) ε = 0.01 m; (iv) ε = 0.05 m; (v) ε = 0.1 m

It is noted that ηi in Eq. (15) is identical with ηp and will not be shown next. We re-write Eq. (15) as +∞ 2 2  1  exp(itmn ) {ηm , ηp , ηe } = dk 4π 1 + εk m=1 n=1 0   n+1 ik × exp(kz 0 ), (−1) , 1 + εk , ω

(21)

where mn (k) = (−1)m+1 k

x + (−1)n+1 ω. t

(22)

According to the stationary-phase approximation, the dominant contribution to the integral in Eq. (21) stems from the stationary points of mn . It is easily seen that for x > 0, 12 and 21 have stationary points while for x < 0, 22 and 11 have stationary points, and the stationary points for both x > 0 and x < 0 are the same. The solutions for the stationary points are determined by ∂mn = 0. ∂k

(23)

A straightforward derivation for Eq. (23) yields |x| − Cg Q(k, k0 , ε) = t

√ g 1 1 1 = 0, = √ −√ · 2 k0 k (1 + εk)3/2

where  2 2 1/3   2ε k0 εk0 b 1/3 a=4 − , 3b 18 b = −9 + (81 + 768εk0 )1/2 .

(27) (28)

For a fixed x/t, kε decreases from k0 with increasing ε, as shown in Fig. 4. By a straightforward application of the Stokes method of stationary phase, the asymptotic representation of Eq. (21) can be given as 1 {ηm , ηp , ηe }   1/2 (2π |ωε |t) (1 + εkε )   kε sin ψ, (1 + εkε ) cos ψ , × exp(kε z 0 ) cos ψ, ωε

(29)

(24)

ψ = kε |x| − ωε t + π/4,

(30) (31)

(25)

ωε = ω(kε , ε), √ g(1 + 4εkε ) ∂ 2 ω(kε , ε)  ωε = = − 3/2 . 2 ∂k 4kε (1 + εkε )5/2

is the root of Q(k, k0 , 0) = 0. Equation (24) is shown in Fig. 3. It is noted that Eq. (24) can be transformed into a

123

biquadratic equation, for which the single real positive root with respect to k, denoted by kε , can be exactly given as   1/2 3 1 1 − + −a k = kε = 2ε 2 4  −1/2 1/2  1 1 1 , (26) +a+ −a + 2 4 4

where

where C g (k, ε) = ∂ω/∂k is the group velocity and k0 = gt 2 /4x 2

Fig. 3 Group velocities versus wave-number. (i) ε = 0 m; (ii) ε = 0.005 m; (iii) ε = 0.01 m; (iv) ε = 0.05 m; (v) ε = 0.1 m

(32)

It is noted that ωε < 0 holds for all kε > 0. Thus, Eq. (29) holds for |x|/t > 0.

Generation of unsteady waves by concentrated disturbances

271

where +∞ 2  1  exp[(−1)n+1 ikx] = dk 2π 1 + εk n=1 0   iµ exp(kz 0 + iµt) k exp(iνt) , × ,− 2 ω2 − µ2 ω − ν2 +∞ 2  2 1  exp(itmn ) {ζmT , ζpT } = dk 4π 1 + εk

{ζmS , ζpS }



(36)

m=1 n=1 0

exp(kz 0 )[(−1)n+1 iω + iµ] , ω2 − µ2  k[1 + (−1)n+1 ω−1 ν] . ω2 − ν 2

× −

Fig. 4 Wave-numbers versus x at t = 15 s. (i) ε = 0 m; (ii) ε = 0.005 m; (iii) ε = 0.01 m; (iv) ε = 0.05 m; (v) ε = 0.1 m

4 Unsteady waves due to oscillating line disturbances In this section, the oscillating line disturbances are considered. Accordingly, the time-dependent strengths of the simple source and applied load are mathematically represented as M(t) = M0 exp(iµt)H(t) and P(t) = P0 exp(iνt)H(t), respectively, where H(t) is the Heaviside step function, µ and ν are constant. Thus, the governing equation and the boundary conditions are Eqs. (1) to (3) while the initial conditions are Eqs. (4) to (6) with I0 = E 0 = 0. Upon the integral transform and other mathematical manipulation, the formal integral expression for the surface elevation due to the oscillating line disturbances can be written as η = M0 ζm (x, t; x 0 ) +

P0 ζp (z, t), ρ

(33)

where 1 {ζm , ζp } = 4π 2 i ×

c+i∞  +∞ c−i∞ −∞

exp(kz 0 )s k ,− s − iµ s − iν

exp(iαx + st) dαds. (1 + εk)(s 2 + ω2 )



(34)

By taking a contour integration in the complex s plane, Eq. (34) can be re-written as {ζm , ζp } = {ζmS , ζpS } + {ζmT , ζpT },

(35)

(37)

Two approaches available for the asymptotic representation of Eq. (36) with large |x| are given by Debnath [24, Sect. 2.5] and will not be reproduced here. Following Debnath [24], we have  µ exp(kµ z 0 − ikµ x + iµt) S S {ζm , ζp }  , 1 + εkµ  ikν exp(−ikν x + iνt) , (38) 1 + εkν where {kµ , kν } =



 µ2 ν2 , . g − εµ2 g − εν 2

(39)

ζmS and ζpS represent the progressive waves with frequencies µ and ν, traveling with the phase speeds (g − εµ2 )/µ and (g − εν 2 )/ν and the group velocities (g − εµ)2 /2gµ and (g − εν)2 /2gν, respectively. It can be seen from Eqs. (38) and (39) that no outgoing progressive waves are possible for √ √ µ ≥ g/ε or ν ≥ g/ε, as was remarked by RhodesRobinson [2,3]. In this case, the wave motion consists of the transient component (37) only. Obviously, the analytic form of Eq. (37) is similar to that of Eq. (21). ζmT and ζpT represent the transient dispersive waves. For the k integration in Eq. (37), the method of stationary phase is used for large t with |x|/t held fixed. The mathematical procedure for the asymptotic representation of Eq. (37) is similar to that in Sect. 3 for Eq. (21). Finally, we have 1 {ζmT , ζpT }   1/2 (2π |ωε |t) (1 + εkε )  exp(kε z 0 )(ωε sin ψ − iµ cos ψ) , × − ωε2 − µ2  kε (cos ψ + iωε−1 ν sin ψ) , (40) ωε2 − ν 2 where kε , ωε , ωε and ψ are given in Sect. 3. It should be noted that the asymptotic solution (38) is derived for large |x| while the asymptotic solution (40) is

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D. Q. Lu, S. Q. Dai

obtained for large t with |x|/t fixed. ζmT and ζpT break down at the critical points ωε = µ2 and ωε = ν 2 , respectively, where the poles for Eq. (36) and the stationary points for Eq. (37) are merged. Thus, a special asymptotic device should be invoked [24]. In combining these two solutions, two possible cases for the region of validity, (i) kε < kµ and kε < kν ; and (ii) kε > kµ and kε > kν , should be considered. It is noted that case (i) is inadmissible for the steady response in view of the group velocity. However, case (ii) is always possible for sufficiently large time [25]. Therefore, the formal solutions for the wave profiles due to an oscillating disturbances are {ζm , ζp } = {ζmS H(t − 2gµ|x|/(g − εµ)2 ), ζpS H(t − 2gν|x|/(g − εν)2 )} + {ζmT , ζpT }.

(41)

Equation (43) can be re-written as +∞ K J0 (K R) 1 dK {ηm , ηp , ηe } = 2π 1 + εK 0  K × exp(K z 0 ) cos(t), − sin(t),   (1 + εK ) cos(t) ,

where J0 (K R) is the zeroth-order Bessel function of the first kind. For the sake of consistency, we replace K and  with k and ω for Eq. (44) without change in its exactness, respectively. Furthermore, we may replace J0 (k R) by its asymptotic formula for large k R, 

5 Waves due to point disturbances

J0 (k R) 

Firstly, the instantaneous point disturbances are considered. The time-dependent strengths of the simple source and applied load are mathematically represented as M(t) = M0 δ(t) and P(t) = P0 δ(t), respectively. Accordingly, the singular component in Eq. (8) is the fundamental solution of threedimensional Laplace equation in an unbounded domain, M δ(t) 4πr c+i∞  +∞ +∞ M ds dαdβ =− 16π 3 i

S = −

c−i∞

×

where (K , ε) = ω(K , ε). With a change of variables {x, y} = R{cos θ, sin θ }, {α, β} = K {cos φ, sin φ}.

123

1/2

  π . cos k R − 4

(45)

Thus, we have an approximation for Eq. (44) as follows +∞  2  2 k 1/2 exp(itmn ) 1  {ηm , ηp , ηe }  dk 4π 2π R 1 + εk m=1 n=1 0   ik × exp(kz 0 ), (−1)n+1 , 1 + εk , (46) ω

mn =

  π (−1)m+1 kR − + (−1)n+1 ω. t 4

(47)

(42)

 where K = α 2 + β 2 , F = αx + βy. By taking a Laplace–Fourier transform similar to Eq. (13), we have the solutions for the transient waves due to the instantaneous point disturbances: +∞ +∞ exp(iF) 1 dαdβ {ηm , ηp , ηe } = 2 4π 1 + εK −∞ −∞  K × exp(K z 0 ) cos(t), − sin(t),   (1 + εK ) cos(t) ,

2 πk R

where

−∞ −∞

1 exp[−K |z − z 0 | + iF + st], K

(44)

(43)

The mathematical procedure to obtain the asymptotic representation of Eq. (46) follows that in Sect. 3 for Eq. (21). Thus, the asymptotic representation of Eq. (46) can be given as  1/2 kε exp(kε z 0 ) cos ϕ, 2π(|ωε |Rt)1/2 (1 + εkε )  kε sin ϕ, (1 + εkε ) cos ϕ , (48) ωε ϕ = kε R − ωε t. (49) {ηm , ηp , ηe } 

Next, the oscillating point disturbances are considered. Upon a similar mathematical manipulation as above, the formal integral expression for the surface elevation due to the oscillating point disturbances can be written as {ζm , ζp } = {ζmS H(t − 2gµR/(g − εµ)2 ), ζpS H(t − 2gν R/(g − εν)2 )} + {ζmT , ζpT },

(50)

Generation of unsteady waves by concentrated disturbances

273

where +∞ kJ0 (k R) 1 dk 2π 1 + εk 0   iµ exp(kz 0 + iµt) k exp(iνt) × , , − ω2 − µ2 ω2 − ν 2 +∞ 2  1  exp[(−1)n+1 iωt] T T {ζm , ζp } = kJ0 (k R)dk 2π 1 + εk {ζmS , ζpS } =



(51)

n=1 0

exp(kz 0 )[(−1)n+1 iω + iµ] , ω2 − µ2  k[1 + (−1)n+1 ω−1 ν] . ω2 − ν 2

× −

solutions for the wave numbers (i.e. kε , kµ and kν ) and the asymptotic representations for the wave profiles (i.e. ηm , ηp , ηe , ζm and ζp ) due to the instantaneous and oscillating concentrated disturbances have been analytically obtained. It can be seen from Figs. 1, 2 and 3 that for a fixed k, the wave frequency, group velocity, and the ratio of group velocity to phase velocity for the dispersive waves decrease as the ε increase from zero. Figure 4 shows that for fixed x and t, the wavelength increases with increasing ε. The effect of the presence of an inertial surface on the transient and steadystate wave profiles due to the line disturbances is shown in Figs. 5, 6, 7, 8, and 9, respectively. The wave profiles in

(52)

The asymptotic representations of Eqs. (51) and (52) are {ζmS , ζpS }

 1/2 µkµ exp[kµ z 0 +i(µt −kµ R +π/4)] 1 ,  1/2 (2π R) 1+εkµ  3/2 ikν exp[i(νt − kν R + π/4)] , (53) 1 + εkν 1/2

{ζmT , ζpT } 

kε 2π(|ωε |Rt)1/2 (1 + εkε )  exp(kε z 0 )(ωε sin ϕ − iµ cos ϕ) , × − ωε2 − µ2  kε (cos ϕ + iωε−1 ν sin ϕ) . ωε2 − ν 2

(54) Fig. 5 Two-dimensional transient waves ηm versus x at t = 15 s, z 0 = −1 m

6 Discussion and conclusions We might check the present results by taking a corresponding limit to recover the previous results for the pure gravity waves. When ε = 0, the asymptotic solutions given by Stoker [15, Eqs. (6.5.13) and (6.5.12)] can be recovered from ηp and ηe in Eq. (29) by replacing kε , ωε and ωε with k0 , ω0 and ω0 , respectively, where ω0 = ∂ 2 ω0 (k0 )/∂k 2 ; the solution obtained by Miles [26, Eq. (3.12)] can be recovered from ζp in Eq. (41); the solution given by Stoker [15, Eq. (6.5.15)] can be recovered from ηp in Eq. (48); the result obtained by Debnath [24, Eq. (3.5.7)] can be recovered from ζp in Eqs. (53) and (54). The procedure presented here for obtaining the wave elevations with an inertial surface provides an alternative method to obtain the free-surface gravity waves formulated in the Cauchy–Poisson problem. In addition, the wave profiles due to a submerged source are derived. The principal interest herein is to study the effect of the presence of an inertial surface on the dispersion relation and the generation and propagation of the gravity waves. The exact

Fig. 6 Two-dimensional transient waves ηp versus x at t = 15 s

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Fig. 7 Two-dimensional transient waves ηe versus x at t = 15 s

D. Q. Lu, S. Q. Dai

Fig. 9 Two-dimensional steady wave profile ζmS versus x at t = 15 s, z 0 = −1 m and ω = 4 s−1

factors in two- and three dimensions do not differ essentially from each other, while the wave amplitude factors are different.

References

Fig. 8 Two-dimensional steady wave profile ζmS versus x at t = 15 s, z 0 = −1 m and ω = 2 s−1

Figs. 5, 6, 7, 8, and 9 are obtained by summing up the views of many observers moving with speed x/t for the same instant. Figures 5, 6 and 7 indicate that the amplitudes of the transient dispersive waves decease while the wavelengths increase due to the presence of an inertial surface. Figures 8 and 9 show that both the amplitudes and the wavelengths of the steady-state progressive waves decease due to the presence of an inertial surface. As the frequency of the applied source or external load increase, the effect of inertial surface become more significant. It is noted that the dispersive characteristics and wave systems for the case of three-dimensional problem are of the same nature as those for the two dimensions. The oscillatory

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