Wave interference, finding amplitude in certain situations

[aliaze1]

Homework Statement

Two loudspeakers emit sound waves of the same frequency along the x-axis. The amplitude of each wave is a. The sound intensity is minimum when speaker 2 is 10 cm behind speaker 1. The intensity increases as speaker 2a is moved forward and first reaches maximum, with amplitude 2 a, when it is 30 cm in front of speaker 1.

What is the amplitude of the sound if the speakers are placed side by side?

Homework Equations

ΔΦ = 2π[(Δx)/λ] = 2mπ , m = 0,1,2,3...

The Attempt at a Solution

I found the following:

λ = 80cm
Phase difference = ¾π (0.75π) =~ 2.36radians

I tried drawing it but I am not getting the correct answer, and I don't know what to do...

 

[learningphysics]

You have the amplitude... so try to write the wave equation for the two waves... Use the equation:

y = A cos (kx) for speaker 1

y = A cos (kx + phi) for speaker 2

you know k = 2pi/wavelength. you know A = 80. Try to use the fact that when speaker two is moved 30m in front of speaker 1, then there is constructive interference... hint a motion of 30m, is a shift in the positive x direction.

Once you know the equations for both waves, add them and find the amplitude for the sum.

 

[Draco]

Excuse me but why would A= 80?

 

[Draco]

wouldnt x =80

 

[learningphysics]

I'm sorry. A is not 80... amplitude is unknown variable a. I should have written wavelength is 80cm.

I'm getting a phase difference of -0.7854 radians

Using these and adding

y = a cos (kx) to

y = a cos (kx + phi)

where phi = -0.7854 and k = 2pi/0.80 = 7.854

you can get the amplitude of the sum in terms of a... you can either use trig identites to get the amplitude of the sum... or use phasors...

 

[Draco]

what about x?

y = a cos ( (2pi/.80)(0)

+

y = a cos ( (2pi/.80)(0.30) -0.7854)

is that correct?

 

[learningphysics]

what about x?

y = a cos ( (2pi/.80)(0)

+

y = a cos ( (2pi/.80)(0.30) -0.7854)

is that correct?

No. don't substitute in an x.

2pi/.80 = 7.854

You need the function:

y = acos(7.854x) + acos(7.854x - 0.7854)

don't substitute in an x... you need to find the amplitude of this function... you can write this sum of two cosines as a single cosine...

try to use this identity. cosA + cosB = 2*(cos[(A+B)/2])*(cos[(A-B)/2]), where A = 7.854x and B = 7.854x - 0.7854

you should get something of the form:
y = (unknownamplitude)*cos(7.854x + unknownphase)

 

[Nitrag]

What is The amplitude of the sound if the speakers are placed side by side?

A = 2*a*cos(\Delta\Phi/ 2)

The amplitude has its max value A=2a if cos(\Delta\Phi/2) = +/- 1.
This is not the case so take a out.

2*cos(\Delta\Phi/2)=.765

Something a friend told me that's a little clearer to me and perhaps others.