A sheet of 6 cm in thickness causes the original intensity to decrease to 59% of its original value. You are trying to build a soundproof room that reduces sound coming in and out by 95%. That is, the intensity after crossing the walls is only 5% of the initial value. What should be the thickness of the mineral wool used in the walls of your soundproof room?

Question

This is the way I'm approaching this question, can someone confirm if it is the correct way, please? Thanks in advance.

I(i)= 1
I(f)= 1-0.59 = 0.41

ΔI = -βIΔz
0.41 = -β(1)(6)
β = -0.41/6

ΔI = -βIΔz
0.95 = -[-0.41/6](1)Δz
Δz = 0.95/(-[-0.41/6])

Answer 1

You modeled the intensity decrease as linear.
I think a more appropriate model is exponental decay.

I= Io e^(-kL) where k is some constant, L is distance.

.59=1 e^(-k6) then solve for k
ln .59= -6k
or k= (-ln .59)/6

solve for k. Then put it in the original equation

.95=e^(-kd) and solve for d.

Answer 2

Thank you!
That really helped. Now I'm getting an answer of 34 which seems reasonable, according to what the question is asking.

Also, if you don't mind answering, how would you know to use the exponential decay equation or the linear equation? Are there certain instances where it is better to use one over the other, and if so, how can you know this?

Answer 3

Experience. Most oscillations dampen in physical systems exponentially. The reason is because the amount of energy lost in the media is proportional to the amount of energy in the wave. In differential equations (adv calculus), you recognize immediately this leads to exponential decay.