Page 84 - Modern physical chemistry

4.9 Conditions in a Planar Pressure Pulse 73
Plane in front 1 Plane at given

of wave \ phase of wave

Steady state
pressure wave
u

FIGURE 4.4 Variation of pressure with
distance through an elementary sound
2 wave.

Let PI be the density and u l the speed of the fluid at plane 1, while P2 is the density
and U2 the speed at plane 2. The volume of the fluid passing through unit area of plane
j in unit time is then u j , Multiplying this by the density as in pjU j then gives the corre-
sponding mass. In the steady state, there is no accumulation of mass between planes 1
and 2 and
[4.55]

The momentum of mass PIUI is (PIUI)UI; the momentum of mass P2U2 is (P2U2)U2' The
force acting to cause this change in momentum per unit area per unit time is PI - P2 . So
from Newton's second law
[4.56]
or
(P2U2t ~-(PIUlt ~=Pl -P2· [4.57]
P2 Pl
Eliminating P2U2 with equation (4.55) gives

(PlUlt(~-~)=Pl -P2· [4.58]
P2 Pl

Replacing lIPj with the specific volume ~and solving for U I yields
2
2 _v,2 Pl -P2 _V,2( fJJ)
Ul - 1 --- - 1 --. [4.59]
V 2 -V l ~V
In a typical wave, the process is so fast that negligible heat flows and
q=O. [4.60]
And when the amplitude is small, the ratio of increments may be replaced by the deriv-
ative to yield

[4.61]

or

[4.62]