Respuesta :
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
[tex]\lambda=\dfrac{v}{f}[/tex]
Put the value into the formula
[tex]\lambda=\dfrac{340}{200}[/tex]
[tex]\lambda=1.7\ m[/tex]
The separation between the speakers in the destructive interference is
[tex]\Delta x= d[/tex]
The equation for destructive interference
[tex]2\pi\times\dfrac{\Delta x}{\lambda}-\Delta\phi_{0}=(m+\dfrac{1}{2})2\pi[/tex]
The loudspeakers are in phase
So, [tex]\Delta\phi_{0}=0[/tex]
The equation for destructive interference is
[tex]2\pi\times\dfrac{d}{\lambda}=(m+\dfrac{1}{2})2\pi[/tex]....(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)
[tex]2\pi\times\dfrac{d_{1}}{1.7}=(0+\dfrac{1}{2})2\pi[/tex]
[tex]d_{1}=\dfrac{1.7}{2}[/tex]
[tex]d_{1}=0.85\ m[/tex]
We need to calculate the second possible value of d
For, m = 1
Put the value in the equation (I)
[tex]2\pi\times\dfrac{d_{2}}{1.7}=(1+\dfrac{1}{2})2\pi[/tex]
[tex]d_{2}=\dfrac{1.7\times3}{2}[/tex]
[tex]d_{2}=2.55\ m[/tex]
We need to calculate the third possible value of d
For, m = 1
Put the value in the equation (I)
[tex]2\pi\times\dfrac{d_{3}}{1.7}=(2+\dfrac{1}{2})2\pi[/tex]
[tex]d_{3}=\dfrac{1.7\times5}{2}[/tex]
[tex]d_{3}=4.25\ m[/tex]
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
