Intensity of a sound wave problem

[Chase11]

Homework Statement

Two loudspeakers are placed beside each other and produce sound of the same intensity at the position of a listener. One speaker produces a low note of 40 Hz and the other produces a high note of 2560 Hz. What is the ratio of the maximum displacements of the speakers vibrating cones?

Homework Equations

1) I=P/4piR^2
2) I=1/2BωkA^2
3) I=1/2\sqrt{ρB}ω^2A^2

The Attempt at a Solution

I understand that I am supposed to use equation 3 for both frequencies and set them equal to each other to come up with a ratio. I just don't understand how equation 3 is derived from equation 1, or how equation 2 is derived from equation 1 for that matter. If I could see how to manipulate these equations I would understand this type of problem much better. (The only equation I am given on my equation sheet is the first one).

 

[Simon Bridge]

Well what do each of the terms in the equations mean?

 

[Chase11]

Well I is intensity of the wave, p is the pressure, r is radius, ω is angular frequency, k is 2pi/\lambda, A is amplitude, and ρ is density. Right?

 

[Simon Bridge]

Well I is intensity of the wave, p is the pressure, r is radius, ω is angular frequency, k is 2pi/λ, A is amplitude, and ρ is density. Right?

This is incomplete, and you have not been consistent in your notation.

Taken in order:
- Intensity of the sound wave - good;
- there is no "p" in your equations. Do you mean "P" here?
$\qquad$... sound is a pressure wave, so there are lots of pressures all over the place so which pressure does P refer to? Or is that P for "power"?
- there is no "r" in your equations, do you mean "R"? What is R the radius of?
- $\small{\omega}$ = angular frequency of the wave
$\qquad \small{\omega = vt = 2\pi f}$ where v is the wave-speed and f is the frequency of the wave. $\small{k(x-vt)=kx-\omega t}$
- $\small{k=2\pi/\lambda}$ good, it's called the wave number.
- What is A the amplitude of
$\qquad$- if "the sound wave" then is it a pressure or a displacement or something else?
- there are lots of different kinds of density - what is $\small{\rho}$ the density of?
- what is B? You missed it out.

If you don't know what the terms refer to then you won't be able to understand the equations.
I think you need to check your ideas about what sound intensity means:
http://en.wikipedia.org/wiki/Sound_intensity
... the intensity of the sound is the rate that energy is delivered to the listeners location per unit area.
Energy rate = energy per unit time = Power, so $I=P/A$ i.e. is power per unit area.
Revisit the equation list in post #1 with that in mind.