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Chemical Wave Packet Propagation, Reflection, and Spreading 11676 J. Phys. Chem. A 2002, 106, 11676-11682 Chemical Wave Packet Propagation, Reflection, and Spreading Lingfa Yang and Irving R. Epstein* Department of Chemistry and Center for Complex Systems, MS 015, Brandeis UniVersity, Waltham, Massachusetts 02454-9110 ReceiVed: May 8, 2002; In Final Form: July 10, 2002 Chemical waves can travel in well-defined packets. Two types of phase wave packets, distinguished by whether their component waves move toward or away from an initiating perturbation, are found in a reaction-diffusion model with a finite wave instability. Their propagation, reflection, and spreading are studied numerically and analytically. Reflection from a boundary or collision of two identical packets can result in standing waves. When two wave packets collide, they can interact briefly and then pass through each other without modification. The phase velocity, group velocity, and spreading velocity calculated by linear stability analysis with the inclusion of quadratic dispersion agree well with the results of numerical simulations. 1. Introduction stable if it asymptotically decays in any uniformly translating frame. Otherwise, the packet is unstable. An unstable packet is Wave packets in reaction-diffusion systems are rarely seen, 1 absolutely unstable if, at any fixed location, the waves’ even though wave packets have been intensively studied in amplitude grows in time. It is convectively unstable if the atomic, molecular, optical, and solid state physics, and also in amplitude decays at all points in a fixed frame but grows in laser chemistry. In those fields, manipulation or measurement some moving frame.10 of wave packet coherence is an important technique with In this paper, we study wave packets in a homogeneous potential applications to exploring the boundary between reaction-diffusion model. We call these chemical wave packets, classical and quantum mechanics, controlling chemical reaction because the waves consist of concentrations varying in space dynamics with lasers, and storing information in quantum and are generated by a wave instability arising from the interplay computing or communication. of chemical reaction and diffusive transport. In homogeneous - Wave packets in reaction diffusion systems were first reaction-diffusion systems, traveling waves, resulting from 2 reported by Zhabotinsky et al. in a model calculation, where excitability or a Hopf bifurcation, usually occur as wave trains, the wave packet arises from a wave instability. The first as in the case of the familiar spiral waves and target patterns. - experimental observation of wave packets in reaction diffusion In contrast to wave packets, which move in space and occupy - systems was made in the 1,4-cyclohexanedione Belousov a finite region, wave trains continue indefinitely (in principle) 3 Zhabotinsky (CHD-BZ) system, where the behavior was to emerge from a fixed source and to expand. We find that attributed to anomalous dispersion of the medium. Recently, chemical waves can travel in well-defined packets when the wave packets were found in the BZ reaction in a reverse wave instability is present. From an initially localized impulse 4,5 microemulsion (BZ-AOT) system arising from a wave perturbation, the wave instability gives rise to packets that instability. Wave packets also occur in a differential flow always move outward, away from the initial perturbation. The 6 system due to convective instability. component waves, however, can propagate either outward or A wave packet consists of waves whose wavenumbers lie in inward. Outwardly propagating (OP) waves are more common; a small range, the so-called “narrowband”. The packet is inwardly propagating (IP) waves are rarely seen. IP wave characterized by a width ∆x in real space, and a mean packets are also referred to as backward volume waves wavenumber, k0, in momentum space. The propagation is (BVW).11 One special feature we find for these chemical wave V usually described by two velocities, the phase velocity, p,at packets is that two packets can pass through each other like which the component waves (carrier waves) move, and the group solitons, rather than annihilating as in the case of spiral waves. velocity, Vg, at which the peak of the wave packet travels. The concept of group velocity appears to have been first enunciated 2. Phase Velocity, Group Velocity, and Speed of 7 by Hamilton in 1839. A group velocity can be positive or Spreading of a Wave Packet negative. A negative group velocity implies that the individual waves move in a direction opposite to that of the packet as a The wave instability plays an important role in pattern whole. Such a phenomenon was first pointed out in 1904 as a formation in many systems, such as binary fluid convection,12 consequence of anomalous (negative) dispersion8 and has been surface reactions,13 electrochemical reactions,14 and the BZ-AOT of interest recently in connection with superluminal light.9 system.4 When a wave instability emerges in a homogeneous The spreading of the wave packet depends on the medium reaction-diffusion system, the symmetry of the uniform steady through which it propagates. A packet, which represents the state solution is broken, generating oscillatory patterns periodic * Green function of an initially localized impulse perturbation, is both in time (oscillatory frequency ωc 0) and in space 12 (wavenumber kc * 0). * To whom correspondence should be addressed. E-mail: epstein@ Just above the wave bifurcation, a small perturbation can brandeis.edu. cause the system to leave the steady state, producing waves 10.1021/jp0260907 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/30/2002 Wave Packet Propagation, Reflection, and Spreading J. Phys. Chem. A, Vol. 106, No. 47, 2002 11677 with wavenumbers lying within a narrowband around the characteristic wavenumber kc. Superposition of waves in the narrowband results in amplitude modulation and may form wave packets. For simplicity, we take as an example two waves with slightly different wavenumbers k1 ) kc - ∆k and k2 ) kc + ∆k. The corresponding frequencies are ω1 ) ω(kc) - ∆ω and ω2 ) ω(kc) + ∆ω. The superposition gives ) + ) - - A(x, t) A1(x, t) A2(x, t) A0 cos[(kc ∆k)x - + + - + ) (ω ∆ω)t] A0 cos[(kc ∆k)x (ω ∆ω)t] - - 2A0 cos(kcx ωt) cos(∆kcx ∆ωt) (1) Figure 1. Schematic plots showing a linearly convectively unstable Because kc . ∆k, the term cos(∆kcx - ∆ωt)ineq1canbe (growing amplitude) packet with a large group velocity (a) and a linearly absolutely unstable packet with a small group velocity giving a wave regarded as a modulation of the cos(kcx - ωt) (plane wave) term. The phase velocity of the plane waves is then train (b). we include the third term (quadratic dispersion) V ) ω(k)| p k)k (2) +∞ k c ) - i(kx-ωt) Ψ(x, t) ∫-∞ g(k kc) e dk V ) +∞ 2 2 and the velocity of the packet (group velocity) is g ∆ω(k)/ ) 2 1/4 -k σ × ∫-∞ (8πσ ) e ∆k.As∆k f 0 and k f kc, we have + - -R - - - 2 ei[(k kc)x ω0t (k kc)t (1/2)â(k kc) t] dk dω(k) i(kcx-ω0t) 2 1/4 2 1/2 V ) | ) (3) ) e (8πσ ) [π/(σ + iât/2)] × g k kc dk - -R 2 2+ e (x t) /4(σ iât/2) (7) The spreading of a wave packet is due to the dependence of At time t, the packet is centered at x(t) ) x(0) + |V |t, and its the phase velocity on the wavenumber. g rms width, from the real part of the exponential function in eq 7, is dV V )V + p g p k (4) dk ât 2 ∆x(t) ) σ2 + (8) x (2σ) On the basis of the sign of dVp/dk, three types of media are possible as follows: normal dispersive (dVp/dk < 0), nondis- Thus, for sufficiently long times, the rms width of the packet persive (dVp/dk ) 0), and anomalous (dVp/dk > 0). The group increases linearly with time. velocity describes the motion of the point of maximum A wave packet emanating from an initial disturbance may amplitude in the packet. For a quantum particle wave, it equals evolve as an isolated packet, as a wave train, or even as a the speed of the particle,15 with which the energy in the group standing wave, depending upon the magnitude of the group V < V is transmitted. In normal media, d p/dk 0, which implies g velocity, |Vg|.If|Vg| is large relative to the speed of packet e Vp; in nondispersive media, for example, light in a vacuum spreading, â/2σ, given by eq 8, an isolated packet leaves the with constant velocity c, we have Vg )Vp; in anomalous site of the initial pulse perturbation (the source) quickly, and V >V dispersion, g p. Superluminal (faster than c) light has been the initial perturbation dies away (Figure 1a). If |Vg| is slower reported due to anomalous dispersion.9 than the spreading of the packet, the packet tail moves backward, A wave packet centered around the wavenumber kc can be toward the source, resulting in continued local oscillation in written as the inverse Fourier transform the region between the source and the outermost edge of the packet (Figure 1b). In effect, the initial perturbation remains as +∞ - a source of waves, creating a wave train rather than a wave Ψ(x, t) ) ∫ g(k - k ) ei(kx ωt) dk (5) -∞ c packet. If |Vg| ) 0, the center of the wave packet remains at the source, producing waves of equal amplitude in both directions, +∞ -ikx 2 1/4 -k2σ2 which can interact to form standing waves: where g(k) ) ∫-∞ G(x) e dx ) (8πσ ) e is the Fourier transform of the Gaussian function, G(x) ) (2 2)-1/4e-x2/4σ2 πσ ) - + + ) with a root-mean-square (rms) width ∆x ) σ.
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