The Minimum Angle for No Sound Transmission Through a Doorway

[jemstone]

Wave Optics!

Homework Statement

Sound with frequency 1240 Hz leaves a room through a doorway with a width of 1.19 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.

Homework Equations

λf = v
θ = (m+.5)λ / d (because no sound is heard and therefore it's destructive?)

The Attempt at a Solution

First I use the first equation to find λ. v/f = λ and therefore 344 m/s / 1240 Hz = .277419 m
Then I set m = 0 and used the second equation to find the angle:
θ = (m+.5)λ / d
θ = (.5 * .277419) / 1.19 m
I found that the angle was .1165 radians, but I know this is wrong. Help!

 

[Kurdt]

What is the condition for a minimum in Fraunhofer diffraction?

 

[nlightner]

I would approach this problem using principles of wave optics. Sound waves can be treated as a form of wave, and therefore, the phenomenon of diffraction can be applied to explain the minimum angle for no sound transmission through a doorway.

Firstly, we need to understand that when a sound wave passes through a doorway, it will encounter an obstacle (the sides of the doorway) that is comparable in size to the wavelength of the sound wave. In this case, the wavelength is 0.277419 m, which is close to the width of the doorway (1.19 m). This means that diffraction will occur, causing the sound wave to spread out and create a diffraction pattern.

Now, according to the principles of Fraunhofer diffraction, the minimum angle for destructive interference (no sound transmission) is given by the equation:

θ = λ / d

where θ is the minimum angle, λ is the wavelength, and d is the width of the doorway.

Substituting the values given in the problem, we get:

θ = 0.277419 m / 1.19 m = 0.2329 radians

This corresponds to an angle of approximately 13.36 degrees. Therefore, at any angle greater than 13.36 degrees relative to the centerline perpendicular to the doorway, there will be destructive interference and no sound transmission will occur.

In conclusion, the minimum angle for no sound transmission through a doorway can be explained by the principles of wave optics, specifically Fraunhofer diffraction.