Wave interference, finding amplitude in certain situations

In summary: The amplitude of the sound if the speakers are placed side by side is equal to two times the amplitude of each wave, as the two waves are in phase with each other. This can be calculated using the formula A = 2*a*cos(\Delta\Phi/ 2), where A is the amplitude of the combined wave, a is the amplitude of each individual wave, and \Delta\Phi is the phase difference between the two waves. In this scenario, the phase difference is not equal to 0 or \pi, so the maximum amplitude is not equal to 2a. Instead, it is equal to 2a multiplied by the cosine of half the phase difference. In this specific case, the amplitude would be equal to
  • #1
aliaze1
174
1

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...
 
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  • #2
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.
 
  • #3
Excuse me but why would A= 80?
 
  • #4
wouldnt x =80
 
  • #5
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...
 
Last edited:
  • #6
what about x?

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

+

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

is that correct?
 
Last edited:
  • #7
Draco said:
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)
 
  • #8
What is The amplitude of the sound if the speakers are placed side by side?

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

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

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

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

1. What is wave interference?

Wave interference is the phenomenon that occurs when two or more waves meet and their amplitudes add together. This can result in either constructive interference, where the amplitudes add and produce a larger wave, or destructive interference, where the amplitudes cancel out and produce a smaller or zero wave.

2. How does wave interference affect amplitude?

Wave interference can either increase or decrease the amplitude of a wave, depending on whether it is constructive or destructive. When two waves with the same amplitude and frequency meet, they will result in a larger amplitude due to constructive interference. Conversely, when two waves with opposite amplitudes and the same frequency meet, they will result in a smaller or zero amplitude due to destructive interference.

3. Can wave interference occur with different types of waves?

Yes, wave interference can occur with any type of wave, including sound waves, electromagnetic waves, and water waves. As long as two or more waves are present and meet at the same point in space, interference can occur.

4. How can we find the amplitude in certain situations of wave interference?

The amplitude of a wave can be found by measuring the distance from the equilibrium point (the middle of the wave) to the crest or trough of the wave. In situations of wave interference, the amplitude can be determined by adding or subtracting the individual amplitudes of the interfering waves.

5. How can wave interference be useful in real-world applications?

Wave interference is a fundamental concept in many fields, including physics, engineering, and telecommunications. It is used to create and manipulate signals in communication systems, such as in radio and satellite transmissions. It is also used in medical imaging techniques, such as ultrasound, to produce clearer images. In addition, the study of wave interference has led to advancements in technologies such as noise-canceling headphones and anti-vibration devices.

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