Question

The highest frequency that a healthy ear can typically hear is 2.0 × 104 Hz. Assume that a sound wave with this frequency travels at 346 m/s and passes through a doorway that has a width of 0.94 m. (a) Determine the angle that locates the first minimum to either side of the central maximum in the diffraction pattern for the sound. (b) Suppose that yellow light (wavelength = 569 nm, in vacuum) passes through a doorway and that the first dark fringe in its diffraction pattern is located at the angle determined in part (a). How wide would this hypothetical doorway have to be?

Answer 1

(a)
`1.05^(\circ)`

The equation that gives the location of the minima in a diffraction pattern from a single slit is:

`a sin \theta = n\lambda` (1)

where

a is the width of the slit

`\theta` is the angle of the nth-minimum

`\lambda` is the wavelength of the wave

Here we want to know the location of the first minimum, so n=1.

In order to find the wavelength of the sound wave, we must divide the speed of the wave by the frequency:

`\lambda=(v)/(f)=(346 m/s)/(2.0\cdot 10^4 Hz)=0.0173 m`

And the slit width is

a = 0.94 m

So we can now re-arrange the eq.(1) to find the angle of the first minimum:

`\theta = sin^(-1) ((n\lambda)/(a))=sin^(-1)(((1)(0.0173 m))/(0.94 m))=1.05^(\circ)`

(b)
`3.1\cdot 10^(-5) m`

This time, we have

`\lambda=569 nm=5.69\cdot 10^(-7) m` is the wavelength of the yellow light

`\theta=1.05^(\circ)` is the angle of the first minimum

So, re-arranging the formula for a, we find how wide the doorway should be:

`a=(n\lambda)/(sin \theta)=((1)(5.69\cdot 10^(-7) m))/(sin(1.05^(\circ)))=3.1\cdot 10^(-5) m`