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CHAPTER 7 Envelope Solitary Waves CHAPTER 7 Envelope solitary waves R.H.J. Grimshaw Department of Mathematical Sciences, Loughborough University, Loughborough, UK. Abstract In this chapter, we discuss envelope solitary waves that arise when a plane peri- odic wave is amplitude modulated. In general, the modulation leads to a rapidly oscillating carrier wave with a slowly varying envelope. In the weakly nonlinear regime, envelope waves are described by the nonlinear Schrödinger equation when attention is confined to just one spatial dimension. We shall focus our attention on capillary–gravity waves, although envelope waves can arise in many other fluid flow situations. When the nonlinear Schrödinger equation is of the focusing type, it supports solitary waves (solitons). However, these solitary waves do not lead to a steady solution of the full physical system, as the carrier’s phase speed and the envelope’s group velocity are not usually equal. But in the special circumstances when the phase and group velocities do coincide, there is a bifurcation to a steady envelope solitary wave. Our main interest is in capillary–gravity waves, and for these we give many details. Extensions to waves in more than one space dimension is based on the Benney–Roskes system, and leads to ‘lump’ solitary waves. 1 Linear waves and group velocity As noted in Chapter 1, there are two classes of solitary waves of interest, each of which can be regarded as a bifurcation from those points in the linear spec- trum where the group velocity and phase velocity are equal. When this bifurcation occurs in the long-wave limit, it leads typically to the Korteweg–de Vries equation and its solitary wave solutions. These have been described in detail in the preceding chapters. In this chapter, we will describe the alternative situation when the bifur- cation occurs at a finite non-zero wavenumber. In this case, the outcome for weakly WIT Transactions on State of the Art in Science and Engineering, Vol 9, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-84564-157-3/07 160 Solitary Waves in Fluids nonlinear waves is related to the nonlinear Schrödinger (NLS) equation and leads to envelope solitary waves; these waves have an underlying rapidly varying oscil- lation, contained with a smoothly varying envelope; consequently, their decaying tails are also oscillatory. However, before coming to the specific case of envelope solitary waves, it is instructive to consider wave envelopes in a more general setting. Let us consider a fluid flow supporting waves and move to the linearized x = (x, y). Then a sinusoidal wave can be described by η(x, t) = a exp (ik · x − iωt) + c.c. (1) Here, η is one of the relevant physical flow variables, k is the wavenumber vector , ω is the wave frequency, while a is the constant amplitude and φ is a constant phase. This sinusoidal wave has a wavelength λ = 2π/κ, where κ =|k| is the magnitude of the√ wavenumber vector (for instance, in two spatial dimensions, k = (k, l) and so κ = k2 + l2) and the wave period T = 2π/ω; a is the (complex) amplitude and c.c. denotes the complex conjugate. The phase velocity has a magnitude c = ω/κ and is in the direction of k. Equation (1) is a kinematic expression, valid for all physical systems that support waves. The dynamics of the fluid flow system in question will lead to the dispersion relation, ω = ω(k), (2) defining the frequency as a function of wavenumber. The phase velocity c = ω/κ is likewise a function of wavenumber. Here we consider only stable waves for which ω is real-valued for all real wavenumbers k. For instance, for water waves, the dispersion relation is (see, for instance, Lamb 1932; Mei 1983), ω2 = gκ(1 + Bq2) tanh q, q = κh, (3) where the Bond number B = /gh2 (ρ is the coefficient of surface tension and ρ is the water density, which has a value of 74 dynes/cm at 20◦C). The concept of group velocity arises when one considers a linear superposition of waves of several wavenumbers and corresponding frequencies. For simplicity, we consider the one-dimensional case when k = (k, 0) and η depends only on x and t. Thus, in general, we can replace (1) with ∞ η(x, t) = F(k) exp (ikx − iω(k)t)dk,(4) −∞ where F(k) is the Fourier transform of η(x, 0). Thus, the initial conditions deter- mine the Fourier transform, each component of which evolves independently with frequency ω related to the wavenumber k through the dispersion relation ω = ω(k) (the one-dimensional version of (2)). To obtain a wave packet, we suppose that the initial conditions are such that F(k) has a dominant component centred at k = k0. The dispersion relation is then approximated by 2 ω = ω0 + cg(k − k0) + δ(k − k0) , (5) WIT Transactions on State of the Art in Science and Engineering, Vol 9, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Solitary Waves in Fluids 161 where dω 1 d2ω c = and δ = . (6) g dk 2 dk2 Here, both cg and δ are evaluated at k = k0. The expression (4) then becomes η(x, t) ≈ A(x, t) exp (i(k0x − ω0t)) + c.c., (7) where ∞ 2 A(x, t) = F(k0 + κ) exp (i(κ(x − cgt)) − iδκ t)dκ, (8) −∞ and the variable of integration has been changed from k to κ = k − k0. Here, the sinusoidal factor exp (i(k0x − ω0t)) is a carrier wave with a phase velocity ω0/k0, while the (complex) amplitude A(x, t) describes the wave packet. Since the term proportional to κ2 in the exponent in (8) is a small correction term, it can be seen that to leading order, the amplitude A propagates with the group velocity cg, while the aforementioned small correction term gives a dispersive term proportional to t−1/2. It is important to note here that within this narrow-band approximation, the amplitude A satisfies the linear Schrödinger equation i(At + cgAx) + δAxx = 0. (9) Further, it can be shown that as t →∞, π 1/2 iπ A(x, t)|x=c t ∼ F(k ) exp sign δ ,ast →∞. (10) g 0 |δ|t 4 This same result can be obtained directly from (4) by using the method of stationary phase, valid here in the limit when t →∞ (see, for instance, Whitham 1974; Lighthill 1978). In this case, it is not necessary to assume also that F(k) is centred at k0, and instead k0 is defined as a function of x/t by the relation x = cg(k0)t. Thus, in this linearized approximation, wave packets are the generic longtime outcome of the initial-value problem. The natural extension of the group velocity to higher spatial dimensions is cg =∇k ω(k), (11) obtained by differentiation of the dispersion relation (2). This result can be readily obtained by extending the above argument to several space dimensions, or perhaps more naturally by using the kinematic theory of waves (seeWhitham 1974; Lighthill 1978). Thus, let the wave field be defined asymptotically by η(x, t) ∼ A(x, t) exp (i (x, t)) + c.c., (12) WIT Transactions on State of the Art in Science and Engineering, Vol 9, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 162 Solitary Waves in Fluids where A(x, t) is the (complex) wave amplitude, and (x, t) is the phase, which is assumed to be rapidly varying compared to the amplitude. Then it is natural to define the local wave frequency and wavenumber by ω =− t, k =∇x . (13) Note that expression (10) has the required form (12). Then cross-differentiation leads to the kinematic equation for the conservation of waves, kt +∇xω = 0. (14) But now, if we suppose that the dispersion relation (2) holds for the frequency and wavenumber defined by (13), then we readily obtain kt + cg ·∇k = 0, (15) with a similar equation for the frequency. Thus, both the wavenumber and frequency propagate with the group velocity, a fact which can also be seen in (10). Equation (15) is itself a simple wave equation, which can be readily integrated by the method of characteristics, or rays. It is important to note that the group velocity cg is a function of k, so that (15) is a nonlinear equation. Because the group velocity is the velocity of the wave packet as a whole, it is no surprise to find that it can also be identified with energy propagation. Indeed, it can be shown that in most linearized physical systems, an equation of the following form can be derived, ∂E +∇·(c E) = 0, (16) ∂t g where E is the wave action density and is proportional to the square of the wave amplitude |A|2, with the factor being a function of the wavenumber k. Typically, the wave action is just the wave energy density divided by the frequency, at least in inertial frames of reference. For water waves, when η in (1) represents the elevation of the free surface, E = 2ρg|A|2. For water waves, the dispersion relation (3) is isotropic, and so the group velocity is in the direction of the wavenumber k and has a magnitude cg =|cg| given by c 1 + 3Bq2 2q c = ∂ω/∂κ = + . (17) g 2 1 + Bq2 sinh 2q Here, we recall that q = κh. Assuming that the Bond number√ B < 1/3, it can be shown that the phase speed c = ω/κ decreases from c0 = gh when κ = 0toa minimum of cm at κ = κm and then increases to infinity as κ increases from κm to infinity.
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