# Pressure and displacement amplitudes

1. Sep 27, 2006

### 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."

2. Sep 27, 2006

### Staff: Mentor

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.

3. Sep 27, 2006

### Andrew Mason

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

4. Sep 27, 2006

### 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.