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5 Acoustics in Biology and Medicine
5.3 Phase, Group, and Signal Speed
You may have noticed in a bundle of waves spreading in a still pool of water from a
stone’s impact that the group of waves moves with a slower speed than the individual
‘bumps’ within the group. This is possible because as the bundle moves forward,
new bumps grow at the rear of the bundle while the advanced bumps in front die
out. As we will see, we can account for this behavior from the fact that waves in the
bundle with different wavelengths have varying speeds.
First, in the spirit of Fourier, let us express any bundle of waves moving in one
direction as a superposition of simple (sinusoidal) waves. The wave displacement
for the bundle will then have the form
ξ(x, t) =
dkA(κ) exp i(κ x − ω(κ) t) .
(5.24)
We will take a localized bundle of waves 11 with an amplitude A(κ) dominated by
wave numbers centered around some value κ 0 . Expanding ω(κ) about these wave
numbers gives
ξ(x, t) ∼
near κ 0
dk
A(κ)e
(i((κ(x−dω/dκ t)))
e
(i(κ 0 x−ω o t))
(5.25)
i.e., the bundle has sinusoidal waves exp (i(κ 0 x − ω 0 t)) modulated by an ‘envelope’
with a more slowly varying phase,
A(κ) exp iiκ((x − dω/dκ t))
. From this
expression, one can see that the wave with the faster-changing phase moves at the
speed
v p = ω o /κ o ,
(5.26)
called the ‘phase velocity’, while the envelope of the wave moves at a speed of
v g ≡ dω/dκ ,
(5.27)
called the ‘group velocity’ of the bundle of waves.
For a bundle of waves, there is a third speed one can define, which is the speed
that the front of the wave packet is first detected by a distant receiver. For any
wave packet, this speed is limited, according to Einstein’s relativity theory, to be
not greater than the maximum speed c, taken to be the speed of light. Wave pulses
cannot carry signals faster than c. The speed of the leading edge of a wave pulse is
called the ‘signal velocity’, v s . The signal velocity may differ from both the phase
velocity and the group velocity. In fact, either one of the latter may be greater than c.
11 A bundle of waves is also called a ‘wave packet’. If the wave packet is strongly localized, it is
called a wave pulse.
5 Acoustics in Biology and Medicine
5.3 Phase, Group, and Signal Speed
You may have noticed in a bundle of waves spreading in a still pool of water from a
stone’s impact that the group of waves moves with a slower speed than the individual
‘bumps’ within the group. This is possible because as the bundle moves forward,
new bumps grow at the rear of the bundle while the advanced bumps in front die
out. As we will see, we can account for this behavior from the fact that waves in the
bundle with different wavelengths have varying speeds.
First, in the spirit of Fourier, let us express any bundle of waves moving in one
direction as a superposition of simple (sinusoidal) waves. The wave displacement
for the bundle will then have the form
ξ(x, t) =
dkA(κ) exp i(κ x − ω(κ) t) .
(5.24)
We will take a localized bundle of waves 11 with an amplitude A(κ) dominated by
wave numbers centered around some value κ 0 . Expanding ω(κ) about these wave
numbers gives
ξ(x, t) ∼
near κ 0
dk
A(κ)e
(i((κ(x−dω/dκ t)))
e
(i(κ 0 x−ω o t))
(5.25)
i.e., the bundle has sinusoidal waves exp (i(κ 0 x − ω 0 t)) modulated by an ‘envelope’
with a more slowly varying phase,
A(κ) exp iiκ((x − dω/dκ t))
. From this
expression, one can see that the wave with the faster-changing phase moves at the
speed
v p = ω o /κ o ,
(5.26)
called the ‘phase velocity’, while the envelope of the wave moves at a speed of
v g ≡ dω/dκ ,
(5.27)
called the ‘group velocity’ of the bundle of waves.
For a bundle of waves, there is a third speed one can define, which is the speed
that the front of the wave packet is first detected by a distant receiver. For any
wave packet, this speed is limited, according to Einstein’s relativity theory, to be
not greater than the maximum speed c, taken to be the speed of light. Wave pulses
cannot carry signals faster than c. The speed of the leading edge of a wave pulse is
called the ‘signal velocity’, v s . The signal velocity may differ from both the phase
velocity and the group velocity. In fact, either one of the latter may be greater than c.
11 A bundle of waves is also called a ‘wave packet’. If the wave packet is strongly localized, it is
called a wave pulse.