How Does Destructive Interference Occur With Different Sound Wavelengths?

[keemosabi]

Homework Statement

Two speakers, one directly behind the other and along the x-axis, emit sound waves along the x-axis. The speaker in the back emits sound with a wavelength of .02 m and the speaker in the front emits waves with a wavelength of .05m. If the two speakers are separated by 3 m, where along the x-axis will destructive interference first occur?

Homework Equations

d sin(theta) = m(wavelength)

The Attempt at a Solution

How do I use the above equation if there are two wavelengths that I need to use?

 

[keemosabi]

Am I using the correct equation?

 

[nlightner]

To find the location of destructive interference, we need to consider the phase difference between the two sound waves. Since the two speakers are emitting waves with different wavelengths, the phase difference between them will change as we move along the x-axis. At some points, the waves will be in phase and at others, they will be out of phase.

To find the first point of destructive interference, we need to find the point where the phase difference between the two waves is exactly 180 degrees. This can be achieved by finding the path difference between the two waves and equating it to half the wavelength of one of the waves.

In this case, the path difference between the two waves can be calculated as d = 3 m. Since the wavelength of the back speaker is smaller, we can use this wavelength to calculate the phase difference.

Therefore, using the equation d sin(theta) = m(wavelength), we get:

3 sin(theta) = 0.5 x 0.02

Solving for theta, we get:

sin(theta) = 0.01

theta = sin^-1(0.01)

theta = 0.57 degrees

This means that the first point of destructive interference will occur at an angle of 0.57 degrees from the x-axis. To find the location on the x-axis, we can use the equation:

x = d tan(theta)

x = 3 tan(0.57)

x = 0.029 m

Therefore, destructive interference will first occur at a distance of 0.029 m from the front speaker along the x-axis.