Speed of Sound

When a sudden disturbance is introduced into a fluid, such as suddenly shutting of a valve, that disturbance will propagate through the fluid at a finite velocity.  This means that when you shut that valve the disturbance will not be felt instantaneously upstream.  Instead it will move up stream at a finite velocity or acoustic velocity known as the speed of sound.  The speed of sound is related to changes in the pressure and density of the fluid medium.

(Eq 1) $c=\sqrt{\frac{dP}{dρ}}$

$c$ = Speed of Sound

$dP$ = Pressure Differential

$dρ$ = Density Differential

The speed of sound can also be related to the fluid bulk modulus using the following equation.

(Eq 2) $c=\sqrt{\frac{E_v}{ρ}}$

The disturbance that propagates through a fluid is normally small.  Due to this fact the heat transferred by the disturbance is negligible. Because of this it can be assumed that we will be dealing with an isentropic process.  As a result the speed of sound can be calculated by using the following equation.

(Eq 3) $c=\sqrt{\frac{kp}{ρ}}$

 $k$= Specific Heat Ratio

Finally, the ideal gas law can also be used to determine the speed of sound.

(Eq 4) $c=\sqrt{kRT}$

$R$ = Ideal Gas Constant

$T$ = Temperature

Mach Number

Since there is a relationship between pressure and density a pressure wave builds up as an object approaches the speed of sound. Because of this pressure wave, it is impossible for an object to travel exactly at this speed. The reason why is because as the object moves at this speed the pressure wave will continue to build until it destroys the object. This is why jets have after burners. The after burners are used to get the jet past this building pressure wave and essentially break the sound barrier causing the jet to out run its pressure wave which is referred to as the sonic boom. To relate an objects speed to the speed of sound, the Mach number would be used, which is a unit less number. Refer to equation 2.

(Eq 5) $Mach~Number=\frac{v}{c}$

v = Velocity

 

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