Sound is propagating in water, which has a density 1,000 kg/m3 and a bulk modulu

Question

Sound is propagating in water, which has a density 1,000 kg/m3 and a bulk modulus 2.2 x 109 N/m2. The intensity of the sound is found to be 0.04 W/m2. The pressure amplitude of the wave is 2,000 N/m2. What is the density amplitude of the sound wave?

A. 1.3 x 10-5 kg/m3

B. 2.5 x 10-3 kg/m3

C. 4.3 x 10-3 kg/m3

D. 9.1 x 10-4 kg/m3

E. 2.1 x 10-2 kg/m3 The correct answer is *D*. --- I know the following formula's which I'm attempting to use in my solution:

1) We can relate the Bulk Modulus to pressure and volume by the formula p=B(delta V/V)

2) rho (density)=m/v...which for wate=1000 kg/m^3

3) I=P/A where I is intensity and P is POWER... *NOT* pressure. This is equiv to I=(1/2)rho*v*omega^2 *Sm^2

Sm in the displacement amplitude of a transverse wave of form s(x,t)=Sm*Cos(kx-omega*t) of a longitudinal wave.

4) I can relate the pressure amplitude to the wave equation using:

delta p (x,t)=delta (Pm)sin(kx-omega*t)

and delta Pm=(v*rho*omega)sm.

Okay, that *should* be all you need to use to solve this problem...but I'm not sure. I'm getting screwed up somehwere in the combining of equations, and since this is an exam q...its supposed to be something you can figure out quickly (IE this should take me...~2 mins max). So, there probably should be a faster way to do this than what I'm currently trying to do.

Finally, if you wouldn't mind...I'd really appreciate it if you could explain why you're doing what you're doing, rather than just doing it and assuming I'll understand.

Explanation / Answer

use p=B(deltaV/V)

since rho = m/V

p=B(delta rho / rho)

substitute p = 2000N/m^2, B= 2.2x10^9 N/m^2, rho for water = 1000kg/m^3

density amplitude delta rho = 9.1x10^-4 (answer D)

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