waves: phase and group velocities of a wave packet the velocity of a wave can be defined in many...
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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.:
where A is the amplitude.
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A = A(x, t) ,
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.
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
Waves: Phase and group velocities of a wave packet
A pure traveling wave is a function of w and k as follows:
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) = A0 sin(ωt − kx) ,
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A(t,x) = An sin(ωn t − knx)n
∑ .
Waves: Phase and group velocities of a wave packet
Here is the result of superposing two such waves with
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A1 = A0
and
k1 =1.2k0 (or ω1 =1.2ω0) :
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
Animation courtesy of Dr. Dan Russell, Kettering University
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:
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.
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A(t) = Asin(ω1t) + Asin(ω2t) =
2Acosω1 −ω2
2
⎛
⎝ ⎜
⎞
⎠ ⎟t
⎛
⎝ ⎜
⎞
⎠ ⎟sin
ω1 +ω2
2
⎛
⎝ ⎜
⎞
⎠ ⎟t
⎛
⎝ ⎜
⎞
⎠ ⎟ .
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
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.
Question: Why do the longer periods travel faster than shorter periods?
Waves: Fersnel zone
The Fresnel zone is a circular region surrounding the ray path with a diameter A-A'.
Subsurface features whose dimensions are smaller than the Fresnel zone cannot be detected by seismic waves.
Figure from www.glossary.oilfield.slb.com
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”.
If they are out of phase with the direct wave they will have wave canceling effect.
Waves: Fersnel zone
SR
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:
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:
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.
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sinicV0
=sin90
V1
⇒ sinic =V0
V1
,
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t =2h0
V0 cosic+X − 2h0 tanic
V1
=2h0 V1
2 −V02
V0V1
+X
V1
.
Refracted waves start arriving after a critical distance Xcrit, but they overtake the direct waves at a crossover distance Xco.
The critical distance is:
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Xcrit = 2h0 tanic .
The crossover distance is:
€
XcoV0
=XcoV1
+2h0 V1
2 −V02
V1V0
⇒
Xco = 2h0
V1 +V0
V1 −V0
.
Refraction: reminder on a horizontal interface
Refraction: an inclined interface
For the down-dip we get:
The apparent head velocity in the down-dip direction is thus:
Similarly, for the up-dip we get:
and the apparent head velocity in the up-dip direction is:
€
td =2zdV0
1−V0
2
V12
+x
V0
sin(ic +α ) .
€
Vd =V0
sin(ic +α ) .
€
tu =2zuV0
1−V0
2
V12
+x
V0
sin(ic −α ) ,
€
Vu =V0
sin(ic −α ) .
Refraction: an inclined interface
V0, Vu and Vd are inferred directly from the travel time curve.
To solve for V1 and we write:
and:
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ic =1
2sin−1 V0
Vd+ sin−1V0
Vu
⎛
⎝ ⎜
⎞
⎠ ⎟
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=1
2sin−1 V0
Vd− sin−1V0
Vu
⎛
⎝ ⎜
⎞
⎠ ⎟ .