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}](https://tex.z-dn.net/?f=%5Clambda%3D%5Cdfrac%7Bv%7D%7Bf%7D)
Put the value into the formula
![\lambda=\dfrac{340}{200}](https://tex.z-dn.net/?f=%5Clambda%3D%5Cdfrac%7B340%7D%7B200%7D)
![\lambda=1.7\ m](https://tex.z-dn.net/?f=%5Clambda%3D1.7%5C%20m)
The separation between the speakers in the destructive interference is
![\Delta x= d](https://tex.z-dn.net/?f=%5CDelta%20x%3D%20d)
The equation for destructive interference
![2\pi\times\dfrac{\Delta x}{\lambda}-\Delta\phi_{0}=(m+\dfrac{1}{2})2\pi](https://tex.z-dn.net/?f=2%5Cpi%5Ctimes%5Cdfrac%7B%5CDelta%20x%7D%7B%5Clambda%7D-%5CDelta%5Cphi_%7B0%7D%3D%28m%2B%5Cdfrac%7B1%7D%7B2%7D%292%5Cpi)
The loudspeakers are in phase
So, ![\Delta\phi_{0}=0](https://tex.z-dn.net/?f=%5CDelta%5Cphi_%7B0%7D%3D0)
The equation for destructive interference is
....(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](https://tex.z-dn.net/?f=2%5Cpi%5Ctimes%5Cdfrac%7Bd_%7B1%7D%7D%7B1.7%7D%3D%280%2B%5Cdfrac%7B1%7D%7B2%7D%292%5Cpi)
![d_{1}=\dfrac{1.7}{2}](https://tex.z-dn.net/?f=d_%7B1%7D%3D%5Cdfrac%7B1.7%7D%7B2%7D)
![d_{1}=0.85\ m](https://tex.z-dn.net/?f=d_%7B1%7D%3D0.85%5C%20m)
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](https://tex.z-dn.net/?f=2%5Cpi%5Ctimes%5Cdfrac%7Bd_%7B2%7D%7D%7B1.7%7D%3D%281%2B%5Cdfrac%7B1%7D%7B2%7D%292%5Cpi)
![d_{2}=\dfrac{1.7\times3}{2}](https://tex.z-dn.net/?f=d_%7B2%7D%3D%5Cdfrac%7B1.7%5Ctimes3%7D%7B2%7D)
![d_{2}=2.55\ m](https://tex.z-dn.net/?f=d_%7B2%7D%3D2.55%5C%20m)
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](https://tex.z-dn.net/?f=2%5Cpi%5Ctimes%5Cdfrac%7Bd_%7B3%7D%7D%7B1.7%7D%3D%282%2B%5Cdfrac%7B1%7D%7B2%7D%292%5Cpi)
![d_{3}=\dfrac{1.7\times5}{2}](https://tex.z-dn.net/?f=d_%7B3%7D%3D%5Cdfrac%7B1.7%5Ctimes5%7D%7B2%7D)
![d_{3}=4.25\ m](https://tex.z-dn.net/?f=d_%7B3%7D%3D4.25%5C%20m)
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