How to Calculate Sound Pressure from Displacement Wave?

AI Thread Summary
To calculate sound pressure from a displacement wave, the relevant equations include ΔPmax = ρvωsmax and ΔP = ΔPmaxsin(kx - ωt). The user is unsure how to incorporate the displacement function into these equations, as velocity is not provided. It is suggested to derive velocity from the displacement equation to proceed with the calculations. The discussion emphasizes the need to connect the displacement wave to sound pressure equations for accurate results.
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Homework Statement


Consider a sound wave in air of density 1.2 kg/m3. The displacement wave has the form s(x,t)=smaxcos(kx-\omegat) where k=8.79rad/m, \omega=3021.6 rad/s and smax=2.51 x 10-7m.
Calculate the sound pressure \DeltaP(x,t) of this wave at x=0.282m and t=0.00137s. Answer in units of Pa.


Homework Equations


\DeltaPmax=\rhov\omegasmax
\DeltaP=\DeltaPmaxsin(kx-\omegat)

The Attempt at a Solution



I have a feeling I'm not using the correct equations because the equations I've found include a velocity, which isn't given, and don't include the displacement function. How do I relate the displacement function to the pressure equations?
 
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The only way I can think to solve this is by using the kinetic theory of an ideal gas to derive pressure based on average velocity. You can review this derivation at http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kinthe.html#c3". The important equation is:

\frac{1}{3} \frac{N}{V} m \bar{v}^2

You can calculate velocity from your displacement equation.
 
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