1. Lecture 12
Group Velocity and Phase Velocity
Many modern ideas on wave propagation originated in the famous works of Lord
Raleigh, and the problems we intend to discuss are no exception to this rule. The
distinction between phase velocity and group velocity appears very early in Rayleigh’s
papers. The problem is discussed in particular in connection with measurements of the
velocity of light and this is a place where a curious error was introduced regarding the
angle of aberration.
Let us recall that the usual velocity of waves is defined as giving the phase
difference between the vibrations observed at two different points in a free plane wave. A
wave
ψ = A cos ( k <x − ω t ) ,
ˆ
k
travels with a phase velocity v p = ω .
k
Another velocity can be defined, if we consider the propagation of a peculiarity (to use
Rayleigh’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 in
the building up of some “groups” of large amplitude (Rayleigh) which move along with
the group velocity U. In wave mechanics, Schrodinger called these groups “wave
packets.” A simple combination of groups obtains when we superimpose on one plane
1

2. wave another plane wave with slightly different phase velocity and frequency such that
we 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 ¹
where the first term corresponds to a carrier wave and the second term corresponds to the
modulation envelope. Thus the superposition results in a carrier wave propagating with a
phase velocity
§ ˆ
δω · k '
v p = ¨ω + ¸ ,
© 2 ¹ δk
k+
2
where
§ δk·
¨k + ¸
2 ¹
k'= ©
ˆ .
δk
k+
2
and a modulation envelope propagating with a group velocity
δω ˆ
vg = k" ,
δk
where
ˆ δk
k"= .
δk
In three dimensions the group velocity therefore becomes
2

3. δω δω δω
v g = x1
ˆ + x2
ˆ + x1
ˆ .
δ k1 δ k2 δ k3
This expression is convenient to use if the dispersion relation is given explicitly as
b g
ω = f k1 , k 2 , k 3 .
However from Christoffel equation we have seen that the dispersion relation can be
obtained in the following implicit form
()
k 2Γ ij l − ρω 2 = Ω (ω , k1, k2 , k3 ) = 0 .
ˆ
e j
where l is the propagation direction l ≡ k . The above equation cannot always be
transformed to get an explicit form for ω as a function of wave number. In such cases
the required derivatives are obtained by implicit differentiation. For example, from the
above 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. Thus the group velocity can always be evaluated as
∇k Ω
vg = − ,
∂Ω ∂ω
where 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/group
velocity 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 p
Also we can write
∂Ω ∂p § k · 1
= ∇ pF. = ∇ p F . ¨ − 2 ¸ = − p.∇ p F .
∂ω ∂ω © ω ¹ ω
Therefore,
1
∇pF ∇ F
vg = − ω = p .
1 p.∇ p F
− p.∇ p F
ω
or, in terms of the slowness vector, the group velocity can be written as
4

5. ∇p F 1 n n
vg = = = .
p. ∇p F b g
p cos p, n p. n
where n is the unit normal to the slowness surface. Thus the group velocity has the same
meaning as the energy velocity (normal to the slowness surface).
The relation between group velocity, phase velocity and the slowness surface can
be shown as follows:
Phase Velocity Surface
n
Phase velocity vector
normal to the group
velocity surface
p
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