Sound waves propagate in compressible fluids at a finite sound speed c = . Consi

Question

Sound waves propagate in compressible fluids at a finite sound speed c = . Consider a compressible fluid where the density rho is only a function of the pressure p (such fluids are known as barotropic fluids). If the sound speed in this fluid is a known constant c, and the density of the fluid at sea level (z = 0) is rho0 , find rho(z). the fluid density as a function of the altitude. Then determine the gage pressure as a function of altitude pg(z) in this fluid if the gage pressure at sea level is 0.

Explanation / Answer

as given pressure is a function of density only so partial differntial becomes total differential

so c=sqrt(dP/d)

c2=dP/d

c2 *d=dP

c2 * = P + K1

now P=P(atm)-*g*z

where z= altitude

putting = o at z=0

we get

K1=c2 * 0 -P(atm)

so (c2 +gz)= c2 *0

= c2 *0/(c2 +gz)

guage pressure= P-P(atm)

= - (c2 *0/(c2 +gz) )*g*z

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