Pressure and displacement amplitudes

[Kaisean]

I'm having some trouble on this problem.

"The human ear is most sensitive to sounds at about f = 3 kHz. A very loud sound at that frequency would have a displacement amplitude of about 4um. What is the pressure amplitude? (Assume the wave to be sinusoidal. For air at room temperature, B = 1.42 x 10^5 Pa and p = 1.20 kg/m^3.) Compare this to the typical diurnal variation in atmospheric pressure, about 500 Pa = 0.005 atm."

 

[berkeman]

What equations would you use to start working on this problem? You need to show some of your own work in order for us to help you.

 

[Andrew Mason]

I'm having some trouble on this problem.

"The human ear is most sensitive to sounds at about f = 3 kHz. A very loud sound at that frequency would have a displacement amplitude of about 4um. What is the pressure amplitude? (Assume the wave to be sinusoidal. For air at room temperature, B = 1.42 x 10^5 Pa and p = 1.20 kg/m^3.) Compare this to the typical diurnal variation in atmospheric pressure, about 500 Pa = 0.005 atm."

p is $\rho$ = density of the air. I am not sure what pressure B is. Standard air pressure is 1.013x10^5 Pa. What is B?

To do this problem, consider a single wavelength of sound $\lambda$ and the space of some volume of air $A\lambda$ where A is the surface area of the wavefront.

In the compression part of the wave, the volume of air is reduced by A x displacement. Since it happens very quickly, it can be treated as an adiabatic compression (no time for heat to be lost). What is the change in pressure? Use:

$$PV^\gamma = K$$

AM

 

[Kaisean]

I would start with the constitutive relationship

P = P0 - Bds/dx

I'm given a B and p; however, no P0. Since this is displacement though, I only need to measure by how much this pressure varies so therefore I get P = Bds/dx where the "-" disappears since I am calculating only for a magnitude of change. I think that s(x, t) can take on the form of s(x, t)=S0sin kx cos wt and from there plug in for ds/dx assuming maximum change in s.