Sound and other waves undergo diffraction just as light does. Suppose a loudspea

Question

Sound and other waves undergo diffraction just as light does. Suppose a loudspeaker in a 20 degree celsius room is emitting a steady tone of 700 Hz . A 1.2-m-wide doorway in front of the speaker diffracts the sound wave. A person on the other side walks parallel to the wall in which the door is set, staying 12 m from the wall. When he is directly in front of the doorway, he can hear the sound clearly and loudly. As he continues walking, the sound intensity decreases.

How far must he walk from the point where he was directly in front of the door until he reaches the first quiet spot?

two significant figures, appropriate units

Explanation / Answer

At 20 °C the speed of sound is ab:
c = 343 m∙sâ» ¹
So the wave length of the sound waves is,
λ = c / f = 343 m∙sâ» ¹ / 700 sâ» ¹ = 0.49... m

The minima observed in single slit diffraction with slit with d are given by:
sin(θ_n) = n∙λ/d with n = 1,2,3... such that n∙λ < d
θ_n is the angle to a line perpendicular to the wall with the slit.

The first minimum from the center of the slit can be observed at an angle of
θ = arcsin( λ/d )
= arcsin( 0.49 m / 1.2 m )
= 24.1

Let x=12 be the distance of the observer to wall and y the distance to the center of the slit parallel to the wall. x and y are the catheti of a right triangle, which hypotenuse is a line from center of the slit to the point of observation. θ is the angle enclosed between x and the hypotenuse. In a right triangle the angle between a leg and the hypotenuse, equals the ration of opposite leg to adjacent leg.
Hence,
tan( θ ) = y/x
=>
y = x∙tan( θ )
= 12 m ∙ tan( 24.1 ° )
= 5.36 m

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