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Lecture12

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Lecture12

  1. 1. Lecture 12Group Velocity and Phase Velocity Many modern ideas on wave propagation originated in the famous works of LordRaleigh, and the problems we intend to discuss are no exception to this rule. Thedistinction between phase velocity and group velocity appears very early in Rayleigh’spapers. The problem is discussed in particular in connection with measurements of thevelocity of light and this is a place where a curious error was introduced regarding theangle of aberration. Let us recall that the usual velocity of waves is defined as giving the phasedifference between the vibrations observed at two different points in a free plane wave. Awaveψ = A cos ( k <x − ω t ) , ˆ ktravels with a phase velocity v p = ω . kAnother velocity can be defined, if we consider the propagation of a peculiarity (to useRayleigh’s term), that is, of a change of amplitude impressed on a train of waves.This is what we now call a modulation impressed on a carrier. The modulation results inthe building up of some “groups” of large amplitude (Rayleigh) which move along withthe group velocity U. In wave mechanics, Schrodinger called these groups “wavepackets.” A simple combination of groups obtains when we superimpose on one plane 1
  2. 2. wave another plane wave with slightly different phase velocity and frequency such thatwe have cos ( k.x − ω t ) + cos ª( k + δ k ) .x − (ω + δ ω ) t º ¬ ¼ ª ( k.x − ω t ) + {( k + δ k ) .x − (ω + δ ω ) t} º = 2 cos « » « ¬ 2 » ¼ ª ( k.x − ω t ) − {( k + δ k ) .x − (ω + δ ω ) t} º cos « » « ¬ 2 » ¼ ª§ δk· § δω · º §δ k δω · = 2 cos «¨ k + ¸ .x − ¨ ω + ¸ t » cos ¨ .x − t¸ , ¬© 2 ¹ © 2 ¹ ¼ © 2 2 ¹where the first term corresponds to a carrier wave and the second term corresponds to themodulation envelope. Thus the superposition results in a carrier wave propagating with aphase velocity § ˆ δω · k v p = ¨ω + ¸ , © 2 ¹ δk k+ 2where § δk· ¨k + ¸ 2 ¹ k= © ˆ . δk k+ 2and a modulation envelope propagating with a group velocity δω ˆ vg = k" , δkwhere ˆ δk k"= . δkIn three dimensions the group velocity therefore becomes 2
  3. 3. δω δω δω v g = x1 ˆ + x2 ˆ + x1 ˆ . δ k1 δ k2 δ k3This expression is convenient to use if the dispersion relation is given explicitly as b gω = f k1 , k 2 , k 3 .However from Christoffel equation we have seen that the dispersion relation can beobtained in the following implicit form () k 2Γ ij l − ρω 2 = Ω (ω , k1, k2 , k3 ) = 0 . ˆ e jwhere l is the propagation direction l ≡ k . The above equation cannot always betransformed to get an explicit form for ω as a function of wave number. In such casesthe required derivatives are obtained by implicit differentiation. For example, from theabove equation, we have § ∂Ω ∂Ω · ¨ δω + δ k1 ¸ = 0 , © ∂ω ∂k1 ¹ k 2 k3 § ∂ω · ∂Ω ∂k1 ¨ ¸ =− ,...etc. © ∂k1 ¹k2 k3 ∂Ω ∂ω 3
  4. 4. Thus the group velocity can always be evaluated as ∇k Ω vg = − , ∂Ω ∂ωwhere the gradient of Ω is taken with respect to k. In loss less medium it can be shown that v e = v g .This provides us with an alternative and simpler way to compute ray velocity/groupvelocity surfaces, which we describe below. Recall that b g b g b g Ω = Ω k , ω = Ω ωp, ω = F p, ω , F ∂p I 1 ∇ Ω = ∇ F = ∇ FG J = ∇ F . k k H ∂k K ω p pAlso we can write ∂Ω ∂p § k · 1 = ∇ pF. = ∇ p F . ¨ − 2 ¸ = − p.∇ p F . ∂ω ∂ω © ω ¹ ωTherefore, 1 ∇pF ∇ F vg = − ω = p . 1 p.∇ p F − p.∇ p F ωor, in terms of the slowness vector, the group velocity can be written as 4
  5. 5. ∇p F 1 n n vg = = = . p. ∇p F b g p cos p, n p. nwhere n is the unit normal to the slowness surface. Thus the group velocity has the samemeaning as the energy velocity (normal to the slowness surface). The relation between group velocity, phase velocity and the slowness surface canbe shown as follows: Phase Velocity Surface n Phase velocity vector normal to the group velocity surfacep p Slowness Surface Group Velocity Surface (lines of constant phase are tangent to this surface)Note 1 vp = p , p n 5 vg = . p. n

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