Sound waves propagation and the doppler effect

[Jalo]

Homework Statement

A lighthouse emits a noise with a frequency of 2000 Hz in the direction of the sea, with a power of 100W. Consider the speed of sound, v_s, to be 341 m/s.

A ship approaches the coast with a speed of 30 km/h and produces noise with an intensity of 40dB. Consider it is possible to distinguish two sounds with equal levels of intensity. Find the maximum distance from the coast where the ship can distinguish the sound of the lighthouse.

Homework Equations

I_DB = 20log₁₀(P/P₀) , with P₀ = 2*10^-5 N m^-2
ψ(x,t) = ψ₀cos(wt-kx)
ρ(x,t) = kρ₀ψ₀cos(wt-kx+pi/2)
P(x,t) = v_sωρ₀ψ₀cos(wt-kx+pi/2)
I = 0.5 * v_sωρ₀²ψ₀² =
= 0.5 * P_s0² / (v_sρ₀) , with P_s0 equal to the amplitude of P(x,t), v_sωρ₀ψ₀

The Attempt at a Solution

It will be impossible to hear the sound of the lighthouse when the intensity of the sound is equal to 0 decibels. Therefore:

I_DB = 20log₁₀(P/P₀) = 0 ⇔ P/P₀ = 1 ⇔ P = P₀

I also know that the intensity is equal to the power divided by the area:
I = Power / Area

Therefore, assuming the lighthouse emits in every direction I have:
I = Power / (4∏r^2) , where r will be the distance from the lighthouse.

Equaling this equation with the expression of intensity that depends on pressure we get:
Power / (4∏r^2) = 0.5 * P_s0² / (v_sρ₀) ⇔
⇔ r^2 = 2*Power*(v_sρ₀) / [4∏*P_s0²]

If I solve this equation I will not get to the correct result tho.
If anyone could point me in the right direction I'd really appreciate.
Thanks!

 

[rude man]

For what it's worth:

The units are confusing to me. I suppose I at the ship due to the lighthouse sound is 100W/4πd² or 100W/2πd² depending on whether the ground sound is absorbed or reflected.

The intensity at the boat is said to be +40dB, but I don't know what the intensity reference level is. In one webpage I found I_ref = 1e-12 W/m^2 so by that +40dB would be 1e-8 W/m^2.

So you can equate that level of intensity with what you get from the lighthouse at a distance d.

Maybe you can fill in some of the blanks.

I don't know how to relate sound pressure to either power or intensity but I don't see why that should be necessary here.