A: Dispersive and nondispersive waves 1.1 Background theory nondispersive waves 1.1.1 Oscillations Oscillations (e:g:, small amplitude oscillations of a simple pendulum) are often described by an equation of the form Figure 1: Characteristics of a simple oscillation. d 2 dt 2 + 2 = 0 (1) where t is time and is some system variable (angular displacement in the case of the simple pendulum). (1) has solutions of the form (t) = Re Y e i!t = Y 0 cos(!t ) (2) where frequency! =, amplitude Y 0 and phase are real constants, and Y = Y 0 e i is a complex amplitude. Thus, as shown in Fig. 1, an oscillation is characterized by three constants: amplitude, frequency, and phase. 1
1.1.2 Nondispersive waves Unlike such simple oscillations, waves are functions of both time and space. The simplest wave equation is of the form @ 2 c 2 @ 2 @t 2 0 @x = 0 (3) 2 where c 0 is some constant. Thisn equation. could represent the electric or magnetic elds in vacuo, with c 0 the speed of light; or pressure perturbations in a compressible uid, with c 0 the sound speed, or waves on shallow water. We can nd solutions to (3) by separating the variables, writing (x; t) = Re [A(t)B(x)] : More speci cally, if we look for wave-like solutions for which B(x) = e ikx, where k is a real wavenumber (so 2=k is wavelength) 1, then d 2 B=dx 2 = k 2 B, so @ 2 @x = Re A(t) d2 B 2 dx (x) = k 2 Re [A(t)B(x)] ; 2 and (3) becomes This has solutions like d 2 A dt 2 + k2 c 2 0A = 0 : A = + e i!t ; A = e +i!t where + and are constant (complex) amplitudes and the frequency! satis es! 2 = k 2 c 2 0 : (4) The full solution is (x; t) = Re + e i(kx!t) + e i(kx+!t) : (5) Each of the two terms in (5) describes a progressive wave (Fig. 2): at any instant, it is just a sinusoidal wave disturbance, of wavelength 2=k 1 In general, any function of x can be expressed as a Fourier integral of such waves. 2
Figure 2: Characteristics of a progressive wave. at any xed location x = x 0, it is just an oscillation of the form De i!t, where D = e ikx 0 is its complex amplitude, of period T = 2=! it propagates with phase speed c =!=k = c 0. Note from (5) that the rst term of is constant along characteristics with kx!t =constant, i.e., x =! t+constant. The second term is constant k along x =! t+ constant. Thus, (5) represents two solutions, propagating in opposite directions with phase speed c 0 k. Eq. (4) is the dispersion relation for the wave: for a given wavenumber k, it tells us the wave s frequency. This form is particularly simple, as shown in Fig. 3. [Note that this is drawn for positive k only we may de ne k positive, without loss of generality, as long as we do not try to constrain the sign of!.] The phase speed c =!=k = c 0 ; the waves can propagate in either direction. These waves are nondispersive, i:e:, their phase speed is independent of wavenumber. Thus, all waves, of any wavenumber, propagate at the same speed (in either direction), which means that non-sinusoidal disturbances propagate without change of shape. In fact, any function (x; t) = F (x c 0 t) (6) 3
Figure 3: The dispersion relation for 1-D shallow water waves. is a solution to (3) 2. Eq. (6) just describes any shape of disturbance, including a localized one, that propagates at speed c without changing its shape (Fig. 4). 1.1.3 Two-dimensional waves In two dimensions (x; y), (3) is replaced by where r 2 @2 form @x 2 + @2 @ 2 @t 2 c 2 0r 2 = 0 (7) @y 2. We can now look for plane wave solutions of the (x; y; t) = Re A 0 e i(kx+ly!t) : (8) 2 To see this, note that if X = x c 0 t, then F = F (X) and the chain rule gives us So, (3) is satis ed by (6). @ = @F @x @x = df @X dx @x = df dx ; @ 2 @x 2 = @ df = @X d 2 F @x dx @x dx 2 = d2 F dx 2 ; @ = @F @t @t = df @X dx @t = c df 0 dx ; @ 2 @t 2 = @ df @X d 2 F c 0 = c 0 @t dx @t dx 2 = d 2 F c2 0 dx 2 : 4
Figure 4: Nondispersive waves: arbitrary disturbances propagate without change of shape. Then r 2 = 2 Re i(kx+ly A 0 e!t) where = p k 2 + l 2 is the total wavenumber. Therefore, substituting into (7) gives the dispersion relation for this case! 2 = 2 c 2 0 : (9) [Note that the one-dimensional case we discussed above is just a special case of the two-dimensional problem, with l = 0.] Eq. (8) describes a plane wave because is constant along lines of constant phase kx + ly!t = constant, so at any instant in time, kx + ly = constant; see Fig. 1.1.3. The wave pattern moves at right angles to the phase lines, with speed c. 5
Plane waves are a special, and particularly simple, form of 2-D waves. Exactly what shape the wavefronts have will in general depend on the geometry of the system and of the process that generated the wave. If the source is very localized (e:g:, a stone dropped into water), the wavefronts will be circular, as shown in Fig. 5. Note that, far from the source (in the dashed rectangle), the wavefronts will look almost plane. Figure 5: Circular wave fronts radiating from a localized source. 1.2 Background theory dispersive waves 1.2.1 Dispersion The general dispersion relation for the frequency! of 1-D waves of wavenumber k can be written! =!(k) (10) and the phase speed is c =!(k) : (11) k Clearly, c is independent of k for all k only if!(k) =constantk this is the nondispersive case we discussed earlier and, as we saw, it implies that all disturbances, including localized ones, propagate without change of shape. This can be thought of in terms of Fourier components. Any non-sinusoidal disturbance can be described a sum of components of di erent wavenumber; 6
if all these waves propagate at the same speed, so will the disturbance itself, and its shape will not change. For many kinds of wave motion, however (including surface waves on deep water, internal gravity waves, and Rossby waves), c varies with k, in which case the di erent wavenumber components will have di erent speeds, a phenomenon known as wave dispersion. Therefore the way they interfere with one another will change with time so the shape of the disturbance will change. 1.2.2 Group velocity There is one particularly important aspect of dispersion, which concerns the way that modulations propagate on a wave train. A monochromatic plane wave of the form (x; t) = Re A 0 e i[k 0x!(k 0 )t] has a single wavenumber k. This is a special case of a more general form Z 1 (x; t) = Re A(k) e i[kx!(k)t] dk (12) 1 where, in the plane wave case, the wavenumber spectrum has the simple form of a -function, A(k) = A 0 (k k 0 ) [Fig. 6(a)]. The wave behaves in the way we have discussed, propagating with a speed c =!(k 0 )=k 0. Consider Figure 6: Wavenumber spectra for (a) a monochromatic wave, and (b) an almost-monochromatic wave. now an almost monochromatic wave, with a narrow spectrum [Fig. 6(b)]. In 7
this case, the spectrum has nite but small width such that A(k) is nonzero only for wavenumbers in the close vicinity of k 0. Writing k = k 0 + k, we can rewrite (12) as (x; t) = Re where! =!(k 0 + k) where Z 1 1 F (x) = A(k 0 + k) e i[k 0x!(k 0 )t] e i[k x! t] d (k) ;!(k 0 ) ' (@!=@k)k. At t = 0, this is simply (x; t) = Re F (x) e ik 0x Z 1 1 A(k 0 + k) e i k x d (k) is the modulating envelope of the wave train, which has carrier wavenumber k 0, as illustrated in Fig. 7. Now, for t > 0, the wave packet behaves as Figure 7: An almost-monochromatic wave packet, comprising many wavelengths. The phase of the carrier wave propagates at the phase speed, but the modulation envelope propagates at the group velocity. where Z 1 (x; t) = Re A(k 0 + k) e i[k(x 1 cgt)] d (k) e i[k 0(x ct)] = Re F (x c g t) e i[k 0(x ct)] ; (13) c g = @! @k (14) k=k0 8
is the group velocity. Thus, while the carrier wave propagates at the phase speed, the modulation envelope propagates at the group velocity. This is an important concept, as it is the latter velocity that governs the propagation of information, as we shall see. Nondispersive waves have! = c 0 k, with constant phase speed c 0, and so their group velocity is the same as their phase velocity. But the group velocity of dispersive waves di ers from the phase speed, so in a wave packet like that shown in Fig. 7 the wave crests will move at a di erent speed than the envelope. If c > c g (which, as we shall see, is the case for deep water waves), new wave crests appear at the rear of the wave packet, move forward through the packet, and disappear at its leading edge. We shall see some examples of this below. In general, it is easy to get a feel for both phase and group propagation graphically from the dispersion relation, as shown in Fig. 8. For any Figure 8: The disperion relation!(k). The phase velocity at k = k 0 is tan, the group velocity tan. wavenumber k, the phase velocity is just given by the slope of a line joining the point (!; k) to the origin, while the group velocity is given by the slope tan of the tangent to the curve at (!; k). 9
1.2.3 Group velocity in multi-dimensional waves For future reference, we note here that for 2- or 3-dimensional waves, the general dispersion relation is of the form! =!(k) (15) where k = (k; l; m) is the (2-D or 3-D) vector wavenumber. The phase velocity 3 is! c = k ;! l m ;! ; (16) and the group velocity c g = @! @k ; @! @l ; @! @m : (17) As we shall see (and a comparison of (16) and (17) implies), not only may c g and c be of di erent magnitudes, they may also be in di erent directions. 3 The phase velocity is in fact not a vector, even though it has magnitude and direction. It does not transform like a vector under rotation this stems from the fact that phase propagation has no meaning along the phase lines. 10