Interference (Constructive and Destructive)

[jumbogala]

Homework Statement

You are standing in front of two side by side loudspeakers playing sounds of the same frequency.

Initially you hear no sound. Then one of the speakers is moved away from you. The sound intensity increases until it reaches a maximum when the speakers are 0.75 m apart.

As the speaker continues to move away, the sound starts to decrease. What is the distance between the speakers when the sound intensity is again zero?

Homework Equations

The Attempt at a Solution

My solution manual gives this:
Find the wavelength of the sound:
2pi(Δx)/ λ + ϕ₀ = m*(2pi) for constructive interference. Δx is the separation between the speakers. Since initially we heard no sound when the speakers were side by side, the speakers are out of phase and ϕ₀ = pi. m= 1 because this is the first separation giving constructive interference. Solving for λ gives λ = 1.5 m.

Next use 2pi(Δx)/ λ + ϕ₀ = (m + 0.5)*(2pi) for destructive interference. Using m = 1 because this is the second time we're seeing destructive interference, solve for Δx to get Δx = 1.5 m.

What I don't understand is why m = 1 for the first separation giving constructive interference. Shouldn't m = 0 in that case? The formula takes values of m = 0, 1, 2, 3...

m = 1 makes sense for the second time seeing destructive interference, because m would have been zero when the speakers were side by side.

 

[kuruman]

In your equation for constructive interference, what does ϕ₀ represent and what value does it have?

 

[jumbogala]

Still pi; initially the speakers were perfectly out of phase and they still are.

 

[kuruman]

Right. Now replace φ₀ with π in your equation for constructive interference. Solve for λ by moving everything else to the other side. What do you get?

 

[jumbogala]

Thanks for the help :) I'm confused - isn't that what I did in my first post? I thought it would be 1.5 m.

Then obviously if the sound was initially zero, to get back to zero I need to move the speaker one full wavelength away. So the distance between them will be 1.5 m.

But if I want to use the formulas to do this, rather than reasoning it out, I don't understand what values of m I'm supposed to be using.

 

[kuruman]

I agree, you should use the formula. Can you find an expression (not numbers) for the wavelength using the constructive interference formula as I suggested in posting #4? Once you find such an expression, see what you get for the wavelength when you set m = 0.