Two identical speakers are pointed at one another, along the x-axis. The speakers, which are in...

Question:

Two identical speakers are pointed at one another, along the x-axis. The speakers, which are in phase with one another, broadcast identical sound waves at a frequency of 170 Hz. Assume the speed of sound to be 340 m/s. Speaker A is positioned at x = 0,and speaker B is located at x = +14.2 m.

(a) In between the speakers, there are a few locations at which the two waves produce completely destructive interference. What is the x-coordinate of the point closest to speaker B at which completely destructive interference occurs?

Interference of Sound Waves

When two sound waves travell on the same path and if they are in the same phase, then constructive interference will occur between the waves and if they are in the different phases, then thair destructive interference will occur.

In constructive interference, the amplitude of the resultant wave will be more than the parent waves and have more energy in the wave.

But in destructive interference, the resultant wave will have less amplitude than the parent waves or even may become zero amplitude.

For constructive interference of the sound wave, the path difference must be:

{eq}x_2-x_1\ =\ 0+n\lambda {/eq}

For destructive interference of the sound wave, the path difference must be:

{eq}x_2-x_1\ =\ (\dfrac{1}{2}+n)\lambda {/eq}

We are given:

• f = 170 Hz = frequency of the sound waves.
• v = 340 m/s = speed of the sound waves.
• {eq}x_1+x_2\ =\ 14.2\ m {/eq} = distance between the speakers.

Now,

The wavelength of the sound wave is:

{eq}\lambda \ =\ \dfrac{v}{f}\ =\ \dfrac{340}{170}\ =\ 2\ m.\\ {/eq}

For destructive interference of sound waves:

{eq}x_2-x_1\ =\ (\dfrac{1}{2}+1)\lambda \\ {/eq}

For first interference nearest to B:

n = 0

{eq}\Rightarrow x_2-x_1\ =\ (\dfrac{1}{2}+0)\times 2\ =\ 1\\ \therefore x_1 =\ x_2-1\\ {/eq}

But we have given:

{eq}x_1+x_2\ =\ 14.2\ m\\ \Rightarrow x_2-1+x_2\ =\ 14.2\\ \Rightarrow 2x_2\ =\ 15.2\\ \therefore x_2\ =\ 7.6\ m.\\ {/eq}

Now,

{eq}x_1\ =\ x_2-1\ =\ 7.6-1\ =\ 6.6\ m.\\ {/eq}

Hence the x-coordinate of point of destructive interference closest to speaker B is 7.6 m.