Page 64 - 48Fundamentals of Compressible Fluid Mechanics

26 CHAPTER 3. SPEED OF SOUND

The people had recognized for sev-
eral hundred years that sound is sound wave
a variation of pressure. The ears dU velocity=dU c
sense the variations by frequency P+dP P
and magnitude to transfer to the r+dr r
brain which translates to voice.
Thus, it raises the question: what is
the speed of the small disturbance Fig. 3.1: A very slow moving piston in a still gas
travel in a “quiet” medium. This ve-
locity is referred to as the speed of sound.
To answer this question consider a piston moving from the right to the
left at a relatively small velocity (see Figure 3.1). The information that the piston is
moving passes thorough a single “pressure pulse.” It is assumed that if the velocity
of the piston is infinitesimally small, the pules will be infinitesimally small. Thus,
the pressure and density can be assumed to be continuous.
In the control vol-
ume it is convenient to look
at a control volume which Control volume around
the sound wave
is attached to a pressure c-dU c
pulse. Applying the mass P+dP P
balance yields r+dr r

Fig. 3.2: stationary sound wave and gas move relative to
(3.1)
the pulse

or when the higher term
is neglected yields

(3.2)

From the energy equation (Bernoulli’s equation), assuming isentropic flow and

neglecting the gravity results
(3.3)

) yield

(3.4)
neglecting second term (

from equation (3.2) into equation (3.4) yields

(3.5)
Substituting the expression for