Linear dynamics of a wave packet

Stability of open flows: basic ideas

3.1 Introduction

In this chapter we begin our study of the stability of flows. We start by making our discussion of linear stability introduced in Chapter 2 more precise (Section 3.1). The approach for deciding whether a flow is stable or unstable on the basis of temporal normal modes is justified by studying the spatio-temporal evolution of an arbitrary perturbation (Section 3.2). In the case of open flows, where a perturba-tion never passes the same point more than once (in contrast to closed flow such as Couette flow between two cylinders), we distinguish two types of instability: con-vective and absolute (Section 3.3). The ideas introduced in a rather abstract way in this chapter will be illustratedmore concretely in the following chapters.

3.1.1 Linear dynamics of a wave packet

We have seen that in many cases, normal mode analysis admits the possibility of stable or unstable propagating waves. Here we shall derive some classical results on the propagation of a packet of dispersive waves for the special case of a Gaus-sian packet. A more general discussion can be found in, for example, Lighthill (1978, §3.7). We consider a wave represented by the integral of its spatial Fourier components

u(x,t) =1 2

 _+∞

−∞ ˆu(k)e^{i(kx−ω(k)t)}dk, (3.1) where the wave number k and frequencyω(k) are real and ˆu(k) is the spectrum of the wave packet.¹ Assuming that the waves propagate at positive speed c= ω/k, we choose the corresponding branch of the dispersion relation, namely, the one for

1This spectrum does not correspond exactly to the Fourier transform of u(x) owing to the factor of 1/π which for convenience we have omitted from(3.1).

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which ω(k) has the same sign as k. Then, using the relation ˆu(−k) = ˆu^∗(k), the above integral can be rewritten as

u(x,t) =1 2

 _+∞

0 ˆu(k)e^{i(kx−ω(k)t)}dk+c.c. (3.2) For x large and x/t fixed, this integral can be evaluated using the method of sta-tionary phase.²It can also be evaluated directly in the case of a Gaussian spectrum of widthσ⁻¹centered on a wave number k0(Figure3.1a), i.e.

ˆu(k) = ˆu0e^{−σ2(k−k0)2}, k> 0. (3.3) A Gaussian spectrum is typical of many experimental situations and leads to sim-ple calculations of the wave evolution: in particular, the Fourier transform of a Gaussian is also a Gaussian. The wave (3.2) can also be written as a traveling wave of wave number k0and frequencyω0= ω(k0):

u(x,t) =1

2A(x,t)e^{i(k0x−ω0t)}+c.c., (3.4) where A(x,t) is the envelope of the wave packet, defined as

A(x,t) =

 _∞

0 ˆu(k)e^{i(k−k0)x−i(ω−ω0)t}dk. (3.5) This envelope depends on space and time, and so it differs from the amplitude of a Fourier mode, which can depend on time but is spatially uniform.

The wave envelope at the initial time

Setting v=σ(k −k0−ix/2σ²), the argument of the exponential inside the integral (3.5) can be written at the initial time t= 0 as

−σ²(k −k0)²+i(k −k0)x = −v²− x²

4σ². (3.6)

2 We consider the integral I(x)=k₂

k₁ f(k)e^ixψ(k)dk. When there exists a number k₀inside the integration range such thatψ^(p)(k0)= 0 but ψ^(k0)= ...= ψ^(p−1)(k0)= 0 with f (k0)= 0, then for x large (Bender and Orszag,

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Figure 3.1 (a) Gaussian spectrum of widthσ⁻¹= 0.1k0; (b) the corresponding wave.

We then find the expression for the envelope A(x,t) at t = 0:

A(x,0) = ˆu0 where we have used the classical result

 _∞

−∞e^−v2dv=√ π.

The envelope of a Gaussian wave packet is therefore a Gaussian whose width is larger the narrower the peak of the spectrum, and we recover the classical result for the Fourier transform (Figure3.1b). In particular, a spectrum which is infinitesimally narrow (a Dirac delta-function) corresponds to a sine wave.

Propagation of the wave packet

In the case of a quasi-monochromatic wave packet, i.e., a narrow spectrum cen-tered at k0, the dispersion relation ω = ω(k) can be expanded in a Taylor series about k0, which is then truncated after the first order:

ω −ω0= cg(k −k0), cg=∂ω

∂k(k0), (3.8)

where c_g is the group velocity of the wave packet, which represents the speed of propagation of the wave number k₀or the speed at which the energy of the packet propagates (Lighthill, 1978, §3.6). By a calculation identical to that which gave

(3.7), but with x replaced by x−cgt, we find

Therefore, to the leading order (3.8) of the Taylor expansion of the dispersion relation, the envelope of a quasi-monochromatic wave propagates at the group velocity without distortion.

Dispersion of the wave packet

We can go farther in our description of the evolution of a wave packet by expanding the dispersion relationω = ω(k) through second order:

ω −ω0= cg(k −k0)+ω₀^

2 (k −k0)², (3.10a) cg=∂ω

∂k(k0), ω^0=∂²ω

∂k²(k0). (3.10b)

By a calculation identical to that which gave (3.9), but withσ² replaced byσ²+ iω^₀t/2, we find

This expression for the envelope (which is now complex) shows that at short times, i.e., times t σ²/ω₀^, the wave packet propagates without significant dispersion (i.e., without flattening or spreading out), consistent with the previous result (3.9) obtained at the leading order. Dispersion is manifested after a time of orderσ²/ω₀^, when the wave packet has traveled a distance of orderσ²cg/ω^₀. Then, the width of the packet grows linearly with time, and its amplitude decreases as 1/√

t, as illustrated in Figure 3.2.The wave energy, which is proportional to the product of the squared amplitude and the width of the packet, therefore remains constant, consistent with the fact that we have neglected any dissipative phenomena. The spreading of the packet corresponds to the dispersion of the Fourier components of the packet: the wave numbers with higher speeds outrun the slower ones (see Exercise3.4.1).

The above results remain valid for non-Gaussian spectra, provided the width δk ∼ 1/σ is small, or δk/k0 1. In particular, the characteristic dispersion time and the distance traveled by the wave packet in that time can easily be obtained by considering the propagation of two adjacent wave numbers k₀ and k₀+δk. Since the group velocity corresponds to the propagation speed of the wave number, the

0.2

0

–0.2

0.2

0

–0.2

–50 0 50 100 150 200

–50 0 50 100 150 200

k₀x k₀x

u0/u u0/u

(a)

(b)

Figure 3.2 The wave corresponding to the envelope(3.11) forσ⁻¹k0= 0.1 and ω₀^= 4cg/k0at (a) the initial time t= 0 and (b) cgt= 100/k0.

difference of the propagation speeds of these two wave numbers isδcg=ω^₀(k0)δk.

Defining the dispersion time as that needed for the separation between the two wave numbers to be 1/δk, this time is (1/δk)/δcg=1/ω^₀δk². This dispersion time is of the same order as that of the Gaussian spectrum,σ²cg/ω₀^. This type of analy-sis of the characteristic scales forms the baanaly-sis for nonlinear analyses in Chapters9 and10.