Cdp pdv

Moving Wove

P ° c

p+dp r+dr dv

¿1_

v-0

dv

o v=C

P+dp v=C-dv r+ or

r

-7

Figure 2.8: Moving pressure disturbance in a motionless fluid and fixed wave in a moving fluid

Applying linear momentum balance to the control volume, we get:

0 = C(pAC) - (C - dv)pAC + pA - (p + dp) A Which further reduces to:

By combining equations 2.20 and 2.21 we get:

This expression gives the velocity of propagation of a compressible wave. Since the propagation of infinitesimal expansion and compression waves is called sound, C is then the speed of sound. the above equation is valid for any continuum, be it a solid, a liquid, or a gas. Because of the high speed of travel of the wave there is very little time for any significant heat transfer to take place, so the process is very nearly adiabatic. It should be noted that C is measured relative to the fluid in which the front is propagating. We also assumed a constant value of C, that is we used an inertial control volume with steady flow relative to this control volume. The wave involve infinitesimal pressure variation. Wave with comparatively large pressure variation over a very narrow front are called shock waves. these waves move relative to fluid at speed in excess of the acoustic speed. Then we may consider acoustic wave as limiting cases of shock waves where the change in pressure across wave become infinitesimal. Therefore what we have developed is valid for any weak spherical and cylindrical waves which moves caused by any small disturbance of pressure.

If a continuum were incompressible, equation 2.22 would give an infinite speed of sound. However, no actual liquid or solid can be perfectly incompressible. All materials have a finite speed of sound. For example the speed of sound in water, air, ice and steel at 15oc and a pressure of 101.325 kPa are 1490,340, 3200 and 5059 m/s, respectively.

For liquids and solids it is customary to define the bulk modulus as a parameter relating the volume (or density) change to the applied pressure change:

Water( at 20oc and atmospheric pressure) has a bulk modulus of about 2.2 x 106 Pa while steel about 200 x 106 Pa. The bulk modulus related to young's modulus of elasticity E by the expression:

0 0

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