Question

Miss Piggy is exercising her vocal chords by matching the frequency f = 686 Hz of her speaker 4 m away. Where should Kermit sit if he wishes to experience destructive interference between Miss Piggy and the speaker? List all possible answers where Kermit is between Miss Piggy and the speaker. You may assume that Miss Piggy and the speaker are in phase with one another and that Kermit is sitting on the line connecting Miss Piggy and the speaker. The speed of sound in air is 343ms−1.

Answer 1

`z=(2n+1)/(8)` for
`n=0,1,2,3,...,15`

Where z=0 m is the position of Miss Piggy and z=4 m is the position of the speaker.

Step-by-step explanation:

Assuming that Miss Piggy emits a sound wave that is in phase with the speaker, and that z=0 is the position of Miss Piggy and z=4 is the position of the speaker, we would have a superposition of two traveling sound waves. Furthermore let's assume that both waves have the same amplitude. The total resulting wave will be given by:

`\psi(t,z)=A\cos(\omega t-kz)+A\cos(\omega t +kz)` where
`\omega` is the angular frequency of the traveling wave and
`k` is the wave number defined as
`k=(2\pi)/(\lambda)`.
`\lambda` is the wavelength of both traveling waves (they have the same wavelength because they have the same frequency).
`\lambda=(v)/(f)` where v is the speed of sound.

By using the trigonometric identity
`2\cos(A)\cos(B)=\cos(A+B)+\cos(A-B)` we can rewrite
`\psi (t,z)` as

`\psi (t,z)=2A\cos(\omega t)\cos(kz)`.

In order for the resulting wave to have maximum destructive interference, that is to be zero for any time t, we need to have

`\cos(kz)=0`

`\implies kz=(2n+1)\cdot (\pi)/(2)\implies z=(2n+1)(\pi)/(2k)=(2n+1)(\pi)/(2)(\lambda)/(2\pi)=(2n+1)\cdot (\lambda)/(4)`

`\implies z=(2n+1)\cdot(v)/(4f)=(2n+1)/(8)`