Answer:
[tex]z=\frac{2n+1}{8}[/tex] for [tex]n=0,1,2,3,...,15[/tex]
Where z=0 m is the position of Miss Piggy and z=4 m is the position of the speaker.
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:
[tex]\psi(t,z)=A\cos(\omega t-kz)+A\cos(\omega t +kz)[/tex] where [tex]\omega[/tex] is the angular frequency of the traveling wave and [tex]k[/tex] is the wave number defined as [tex]k=\frac{2\pi}{\lambda}[/tex]. [tex]\lambda[/tex] is the wavelength of both traveling waves (they have the same wavelength because they have the same frequency). [tex]\lambda=\frac{v}{f}[/tex] where v is the speed of sound.
By using the trigonometric identity [tex]2\cos(A)\cos(B)=\cos(A+B)+\cos(A-B)[/tex] we can rewrite [tex]\psi (t,z)[/tex] as
[tex]\psi (t,z)=2A\cos(\omega t)\cos(kz)[/tex].
In order for the resulting wave to have maximum destructive interference, that is to be zero for any time t, we need to have
[tex]\cos(kz)=0[/tex]
[tex]\implies kz=(2n+1)\cdot \frac{\pi}{2}\implies z=(2n+1)\frac{\pi}{2k}=(2n+1)\frac{\pi}{2}\frac{\lambda}{2\pi}=(2n+1)\cdot \frac{\lambda}{4}[/tex]
[tex]\implies z=(2n+1)\cdot\frac{v}{4f}=\frac{2n+1}{8}[/tex]
