157
4.2 Compressibility and Velocity of Sound
propagation velocity is denoted by c. After passage of the sound wave, pressure,
density and velocity become p + dp, ρ + dρ and dv.
We further consider a relative frame, moving along with the sound wave. The
flow is steady within this frame and the laws formulated in the previous section apply. Before the wave, velocity equals c, after the wave c-dv.
(4.9)
(4.10)
Elimination of dv from (4.9) and (4.10) gives
(4.11)
Isentropy (reversible adiabatic flow) means
(4.12)
The compressibility of matter is characterised by the relative volume change caused
by a pressure change. The ratio of the infinitesimal pressure increase to the corresponding infinitesimal relative volume decrease is called the bulk modulus or the
volumetric elasticity modulus E. For a given mass m applies m= ρV, so that dρ/ρ +
dV/V = 0. The compressibility coefficient β, being the inverse of the bulk modulus,
is
For isentropic compression it follows that
Mass:
.
d
dv 0
c
r
r
−
=
Work:
.
dp
cdv
0
r
−
+
=
or
2
2
d
dp
dp
c
c
.
d
r
r
r
r
=
=
or
dp
d
p ~
.
p
g
r
r
g r
=
Thus :
or
.
2
p
p
c
c
g
g r
r
=
=
1
/
/ .
dV V d
E
dp
dp
r r
b = = −
=
2
dp
E
c .
d
r
r
r
=
=