Answer:
The phase difference is [tex]\Delta \phi = 1.9995 rad[/tex]
Explanation:
From the question we are told that
The distance between the loudspeakers is [tex]d = 2m[/tex]
The distance of the listener from the wall [tex]D = 81.7 \ m[/tex]
The frequency of the loudspeakers is [tex]f = 4450Hz[/tex]
The velocity of sound is [tex]v_s = 343 m/s[/tex]
The path difference of the sound wave that is getting to the listener is mathematically represented as
[tex]\Delta z =\sqrt{d^2 + D^2} -D[/tex]
Substituting values
[tex]\Delta z =\sqrt{2^2 + 81.7^2 } -81.7[/tex]
[tex]\Delta z =0.0245m[/tex]
The phase difference is mathematically represented as
[tex]\Delta \phi[/tex] = [tex]\frac{2 \pi}{\lambda } * \Delta z[/tex]
Where [tex]\lambda[/tex] is the wavelength which is mathematically represented as
[tex]\lambda = \frac{v_s }{f}[/tex]
substituting value
[tex]\lambda = \frac{343 }{4450}[/tex]
[tex]\lambda = 0.0770 m[/tex]
Substituting value into the equation for phase difference
[tex]\Delta \phi[/tex] = [tex]\frac{2 * 3.142 * 0.0245}{0.0770}[/tex]
[tex]\Delta \phi = 1.9995 rad[/tex]
