Question

Two identical loudspeakers separated by distance dd emit 200 Hz sound waves along the x-axis. As you walk along the axis, away from the speakers, you don't hear anything even though both speakers are on.
What are the three lowest possible values of d? Assume a sound speed of 340 m/s.

Answer 1

Answer:

The first possible value of d is 0.85 m

The second possible value of d is 2.55 m

The third possible value of d is 4.25 m

Explanation:

Given that,

Distance =d

Frequency of sound wave= 200 Hz

We need to calculate the wavelength

Using formula of wavelength

$\lambda=\dfrac{v}{f}$

Put the value into the formula

$\lambda=\dfrac{340}{200}$

$\lambda=1.7\ m$

The separation between the speakers in the destructive interference is

$\Delta x= d$

The equation for destructive interference

$2\pi\times\dfrac{\Delta x}{\lambda}-\Delta\phi_{0}=(m+\dfrac{1}{2})2\pi$

The loudspeakers are in phase

So, $\Delta\phi_{0}=0$

The equation for destructive interference is

$2\pi\times\dfrac{d}{\lambda}=(m+\dfrac{1}{2})2\pi$....(I)

Here, m = 0,1,2,3.....

We need to calculate the first possible value of d

For, m = 0

Put the value in the equation (I)

$2\pi\times\dfrac{d_{1}}{1.7}=(0+\dfrac{1}{2})2\pi$

$d_{1}=\dfrac{1.7}{2}$

$d_{1}=0.85\ m$

We need to calculate the second possible value of d

For, m = 1

Put the value in the equation (I)

$2\pi\times\dfrac{d_{2}}{1.7}=(1+\dfrac{1}{2})2\pi$

$d_{2}=\dfrac{1.7\times3}{2}$

$d_{2}=2.55\ m$

We need to calculate the third possible value of d

For, m = 1

Put the value in the equation (I)

$2\pi\times\dfrac{d_{3}}{1.7}=(2+\dfrac{1}{2})2\pi$

$d_{3}=\dfrac{1.7\times5}{2}$

$d_{3}=4.25\ m$

Hence, The first possible value of d is 0.85 m

The second possible value of d is 2.55 m

The third possible value of d is 4.25 m