Find the correct frequency so two waves are in phase

[Jalo]

Homework Statement

A man is sitting between two speakers, 5 meters from the first and 6.5 meters away from the second.

Speaker 1~~~~~~(6.5m) ~~~~~~ Man ~~~~ (5m) ~~~~ Speaker 2

They both create sound waves in phase. What's the lowest frequency for which the man hears both sounds in phase?
The velocity of sound is 340m/s.

Homework Equations

The Attempt at a Solution

We have two speakers. Speaker 1 is positioned at x = 0, creating waves with a phase of
ϕ₁(x,t) = kx - wt
propagating in the positive direction.
Speaker 2 is positioned at x = 11.5, emiting waves with a phase
ϕ₂(x,t) = kx + wt
which propagate in the negative direction.

The man is sitting at x=6.5.
We know that both speakers emit in phase, therefore for any given instant t we have:
ϕ₁(0,t) = ϕ₂(11.5,t)

We also want the man to receive both waves in phase, therefore:
ϕ₁(6.5,t) = ϕ₂(6.5,t)

If we solve this equations we'll get:
ϕ₁(0,t) = ϕ₂(11.5,t) ⇔ 0*x - w*t = 11.5*k + wt ⇔ 2*wt = -11.5k
ϕ₁(6.5,t) = ϕ₂(6.5,t) ⇔ 6.5k - wt = 6.5k + wt ⇔ wt = 0

The result doesn't make any sense. I've probably assumed something wrongly, but I can't quite figure out what. If anyone could point my logic's holes I'd appreciate.

Thanks.
Daniel

 

[sankalpmittal]

Homework Statement

A man is sitting between two speakers, 5 meters from the first and 6.5 meters away from the second.

Speaker 1~~~~~~(6.5m) ~~~~~~ Man ~~~~ (5m) ~~~~ Speaker 2

They both create sound waves in phase. What's the lowest frequency for which the man hears both sounds in phase?
The velocity of sound is 340m/s.

Homework Equations

The Attempt at a Solution

Snip..

Jesus !
Such a long solution ?

Why do you not use,

Δψ=2π*Δx/λ

In phase can be when there is constructive interference. In this case, what will be phase difference, Δψ ?

 

[Jalo]

Jesus !
Such a long solution ?

Why do you not use,

Δψ=2π*Δx/λ

In phase can be when there is constructive interference. In this case, what will be phase difference, Δψ ?

I'm afraid I did not understand your answer. If Δψ is the phase difference then it will be zero, since both waves must arrive with the same phase. Therefore Δψ = 0 = 2π*Δx/λ , which can't be true.

 

[haruspex]

I'm afraid I did not understand your answer. If Δψ is the phase difference then it will be zero, since both waves must arrive with the same phase. Therefore Δψ = 0 = 2π*Δx/λ , which can't be true.

"In phase" only requires that the phase difference is a whole number of cycles.