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Since the coefficients of (4.15) are constant in both space
and time we may look for plain waves and Fourier analyse by
assuming

(4.16) 
where is a constant and
. In (4.16) we have

(4.17) 

(4.18) 
and

(4.19) 
This form for is particularly useful since we note that
Thus, defining the total wavenumber as
, (4.15) becomes

(4.20) 
So either (and we have the trivial solution since
, and so
) or or the
coefficient of in (4.20) must vanish. Hence,

(4.21) 
This is the dispersion relation for sound waves and it relates
the frequency with which the waves oscillates in time to the spatial
length scales of the wave through the wave
vector (and the various wave numbers). The dispersion relation,
, can be used to define two important
quantities, namely the phase speed and the group velocity.
The phase speed, in general, is given by

(4.22) 
and in the case of sound waves this is

(4.23) 
Thus, the phase speed is the sound speed.
The group velocity is

(4.24) 
For sound waves
Differentiating this gives

(4.25) 
on using (4.23).
The phase speed gives the speed of an individual wave and the
phase velocity is
. The group velocity gives the
speed and direction of the transport of information and energy.
Next: Alternative approach to sound
Up: Sound Waves  Basic
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Prof. Alan Hood
20000111