Sound waves with frequency 2400 Hz and speed 343 m/s diffract through the rectan

Question

Sound waves with frequency 2400 Hz and speed 343 m/s diffract through the rectangular opening of a speaker cabinet and into a large auditorium of length 100 m. The opening, which has a horizontal width of 43.5 cm, faces a wall 100 m away (the figure below).

Along that wall, how far from the central axis of that wall in meters will a listener be at the first diffraction minimum and thus have difficulty hearing the sound? (Neglect reflections.)
tolerance is +/-2%
Sound waves with frequency 2400 Hz and speed 343 m/s diffract through the rectangular opening of a speaker cabinet and into a large auditorium of length 100 m. The opening, which has a horizontal width of 43.5 cm, faces a wall 100 m away (the figure below).

Along that wall, how far from the central axis of that wall in meters will a listener be at the first diffraction minimum and thus have difficulty hearing the sound? (Neglect reflections.)
tolerance is +/-2%


Along that wall, how far from the central axis of that wall in meters will a listener be at the first diffraction minimum and thus have difficulty hearing the sound? (Neglect reflections.)
Along that wall, how far from the central axis of that wall in meters will a listener be at the first diffraction minimum and thus have difficulty hearing the sound? (Neglect reflections.)
tolerance is +/-2%

tolerance is +/-2%

Explanation / Answer

Given,

f = 2400 Hz ; v = 343 m/s ; L = 100 m ; w = 43.5 cm = 0.435 m

we know that the angular separation in single slit diffraction is given by:

sin(theta) = lambda/w

lambda = v/f = 343/2400 = 0.143 m

sin(theta) = 0.143/0.435 = 0.329

theta = sin^-1(0.329) = 19.2 deg

the distance from the central axis will be:

D = R tan(theta) = 100 x tan19.2 = 34.82 m

Hence, D = 34.82 m

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