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Sound with a frequency of 1250 Hz leaves a room through a doorway with a width of 1.05 m.At what minimum angle relative to the centerline perpendicular to the doorway will someone outside the room hear no sound? Use 344 m/s for the speed of sound in air and assume that the source and listener are both far enough from the doorway for Fraunhofer diffraction to apply. You can ignore effects of reflections.Theta = ? degrees, ? degrees, ? degrees (should be three terms)

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Answer:

15.19°, 31.61°, 51.84°

Explanation:

We need to fin the angle for m=1,2,3

We know that the expression for wavelenght is,

[tex]\lambda = \frac{c}{f}[/tex]

Substituting,

[tex]\lambda = \frac{344}{1250}[/tex]

[tex]\lambda = 0.2752m[/tex]

Once we have the wavelenght we can find the angle by the equation of the single slit difraction,

[tex]sin\theta = \frac{m \lambda}{W}[/tex]

Where,

W is the width

m is the integer

[tex]\lambda[/tex] the wavelenght

Re-arrange the expression,

[tex]\theta = sin^{-1} \frac{m\lambda}{W}[/tex]

For m=1,

[tex]\theta = sin^{-1} \frac{1 (0.2752)}{1.05}= 15.19\°[/tex]

For m=2,

[tex]\theta = sin^{-1} \frac{2 (0.2752)}{1.05}= 31.61\°[/tex]

For m=3,

[tex]\theta = sin^{-1} \frac{3 (0.2752)}{1.05}= 51.84\°[/tex]

The angle of diffraction is directly proportional to the size of the wavelength.

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