Waves: Phase and group velocities of a wave packet
The velocity of a wave can be defined in many different ways,
partly because there are different kinds of waves, and partly
because we can focus on different aspects or components of any
given wave.
The wave function depends on both time, t, and position, x, i.e.:
A  A(x,t) ,
where A is the amplitude.

Waves: Phase and group velocities of a wave packet
At any fixed location on the x axis the function varies sinusoidally
with time.
The angular frequency, , of a wave is the number of radians (or
cycles) per unit of time at a fixed position.
Waves: Phase and group velocities of a wave packet
Similarly, at any fixed instant of time, the function varies
sinusoidally along the horizontal axis.
The wave number, k, of a wave is the number of radians (or
cycles) per unit of distance at a fixed time.
Waves: Phase and group velocities of a wave packet
A pure traveling wave is a function of w and k as follows:
A(t, x)  A0 sin( t  kx) ,
where A0 is the maximum amplitude.
A wave packet is formed from the superposition of several such
waves,with different A, , and k:
A(t, x)   An sin(  n t  kn x) .
n
Waves: Phase and group velocities of a wave packet
Here is the result of superposing two such waves with
A1  A0
k1  1.2k0

and
(or 1  1.2 0 ) :
Waves: Phase and group velocities of a wave packet
Note that the envelope of the wave packet (dashed line) is also a
wave.
Waves: Phase and group velocities of a wave packet
Here is the result of superposing
two sine waves whose
amplitudes, velocities and
propagation directions are the
same, but their frequencies differ
slightly. We can write:
A(t)  A sin( 1t)  Asin(  2 t) 
1   2   1   2  
2A cos
t sin 
t  .
 2    2  
While the frequency of the sine
term is that of the phase, the
frequency of the cosine term is
that of the “envelope”, i.e. the
group velocity.
Animation courtesy of Dr. Dan Russell,
Kettering University
Waves: Phase and group velocities of a wave packet
The speed at which a given phase propagates does not coincide
with the speed of the envelope.
Note that the phase velocity is
greater than the group velocity.
Waves: Phase and group velocities of a wave packet
The group velocity is the velocity with which the envelope of the
wave packet, propagates through space.
The phase velocity is the velocity at which the phase of any one
frequency component of the wave will propagate. You could pick
one particular phase of the wave (for example the crest) and it
would appear to travel at the phase velocity.
Question: Is the P-wave speed a phase or a group velocity?
Waves: Dispersive waves
A wave packet is said to be dispersive if different frequencies
travel at different speeds.
Surface waves are dispersive!
Waves: Dispersive waves
Calculated dispersion curves for Love and Rayleigh waves (from
Shearer's text book):
Waves: Dispersive waves
Question: Why do the longer periods travel faster than shorter
periods?
Reply: Surface waves are dispersive because waves with longer
wavelengths (lower frequency)are traveling deeper than short
wavelengths (higher frequency). Since in general, seismic speed
increases with depth, longer wavelengths travel faster than
shorter wavelengths.
Waves: Fersnel zone
The Fresnel zone is a circular region surrounding the ray path with
a diameter A-A'.
Figure from www.glossary.oilfield.slb.com
Subsurface features whose dimensions are smaller than the
Fresnel zone cannot be detected by seismic waves.
Waves: Fersnel zone
Consider a direct wave traveling in a straight line from the source,
S, to the receiver, R. Other waves that travel slightly off the direct
path may still arrive to the same receiver if they bump into
“something”.
S
R
If they are out of phase with the direct wave they will have wave
canceling effect.
Waves: Fersnel zone
Each Fresnel zone is an ellipsoidal shape as shown below. In
zone 1 the signal will be 0 to 90 degrees out of phase in zone 2,
90 to 270 degrees in zone 3, 270 to 450 degrees and so on.
S
R
Refraction: reminder on a horizontal interface
The refracted wave traveling along the interface between the
upper and the lower layer is a special case of Snell's law, for
which the refraction angle equals 900. We can write:
sin ic sin 90
V0

 sin ic 
,
V0
V1
V1
where ic is the critical angle. The refracted ray that is returned to
the surface is a head wave.

The travel time of the refracted wave is:
2
2
2h
V

V
2h0
X  2h0 tan ic
X
0
1
0
t



.
V0 cosic
V1
V0V1
V1
So this is an equation of a straight line whose slope is equal to
1/V1, and the intercept is a function of the layer thickness and the
velocities above and below the interface.
Refraction: reminder on a horizontal interface
Refracted waves start arriving after a critical distance Xcrit, but
they overtake the direct waves at a crossover distance Xco.
The critical distance is:
Xcrit  2h0 tanic .
The crossover distance is:
X co X co 2h0 V  V



V0
V1
V1V0
2
1

X co  2h0
V1  V0
V1  V0
2
0
.
Refraction: an inclined interface
For the down-dip we get:
2zd
V02
x
td 
1 2  sin( ic   ) .
V0
V1 V0
The apparent head velocity in
the down-dip direction is thus:
V0
Vd 
.
sin( ic   )
Similarly, for the up-dip we get:
2zu
V02
x
tu 
1 2  sin( ic   ) ,

V0
V1 V0
and the apparent head velocity
in the up-dip direction is:
V0
Vu 
.
sin( ic   )
Refraction: an inclined interface
V0, Vu and Vd are inferred
directly from the travel time
curve.
To solve for V1 and  we write:
and:

1  1 V0
1 V0
ic  sin
 sin

2 
Vd
Vu 

1  1 V0
1 V0
  sin
 sin
 .
2 
Vd
Vu 