Acoustics of a uniform stagnant fluid

1.3 Wave equation

Looking at small perturbations (p’,p’,v’) of a uniform stagnant state (po, po) and neglecting friction and heat transfer, we find, for linear pertur­bations:

dp’

-p + poV • v = 0 ,

(44)

dv’ „ , po – at + Vp = f

(45)

and

ds’ = Qw dt poTo

The corresponding linearized equation of state is:

(46)

‘ 2 ‘ . (dp ‘

p = cop A ds) p s.

(47)

Taking the time derivative of (44), subtracting the divergence of (45) and using (46) and (47) in order to eliminate p’ and s’, we obtain the wave equation for pressure perturbations:

1 32p’

C0 ~dt2

 

1 f &p dQw Topoc2 V ds) p dt

 

V2p’

 

(48)

 

As can be seen from this equation, the unsteady heat production is a source of sound, which is due to the dilatation of the fluid. This is in line with our common experience that turbulent flames are noisy. Also an unsteady non-uniform force field appears to be a source of sound. This is the sound source when considering the whistling of a cylinder placed with its axis normal to a uniform flow. Due to hydrodynamic instability, the wave be­hind the cylinder breaks down into a vortex street of alternating rotation direction. This periodic vortex shedding induces an unsteady force of the flow on the cylinder. The reaction force from the cylinder on the fluid is the source of sound. The so-called Aelonian tone will be discussed in section 7.2.

The next sections will focus on wave propagation and hence assume that Qw = 0 and f = 0. We therefore consider solutions of the homogeneous wave equation of d’Alembert

1 32p’ C0 ~3t2

 

V2p’

 

0 .

 

As the flow is isentropic the equation of state (16) reduces to p’ = c?0p’.

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