- #1

prodo123

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## Homework Statement

Two speakers A and B, 2.00 m apart, produce a sine wave at the same frequency and phase. A microphone is placed on the line BC perpendicular to AB, at a distance x from B. The speed of sound is v=344 m/s.

For a frequency ƒ low enough, there will be no destructive interference along the line BC. Find this frequency.

## Homework Equations

Let ##r=\sqrt{x^2+2^2}## be the distance from speaker A to the point along the line BC.

##y_{A}(x,t)=A\cos(kr-\omega t)\\

y_{B}(x,t)=A\cos(kx-\omega t)\\

k=\frac{2\pi f}{v}\\

\omega = 2πf##

## The Attempt at a Solution

Destructive interference occurs when ##y_{A} = -y_{B}##:

##A\cos(kr-\omega t) = -A\cos(kx-\omega t)\\

A\cos(kr-\omega t) = A\cos(kx-\omega t+(2n-1)\pi)\\

kr = kx+(2n-1)\pi\\

r=\sqrt{x^{2}+2^{2}}=x+\frac{(2n-1)\pi}{k}\\

r=x+\frac{(2n-1)v}{2f}##

For simplicity, let ##C = \frac{(2n-1)v}{2f}##

##r^2 = x^2+4 = x^2+2cx+c^2\\

2cx = 4-c^2\\

x(n) = 2/c - c/2\\

x(n) = \frac{4f}{(2n-1)v} - \frac{(2n-1)v}{4f}##

A previous part of the problem set ƒ = 786 Hz and asked to find points of destructive interference; values for which x(n) > 0 found points of destructive interference at n=1,2,3,4,5, which correlated with the answer in the back.

Since x(n) increased as n→1, I think n=1 is the first point of destructive interference from ∞→x. Therefore if x(1)=0, there are no other point of destructive interference other than the source B itself.

Finding the frequency ##f## for which ##x(1)=0##:

##x(1) = \frac{4f}{v}-\frac{v}{4f}= 0\\

4f=v\\

f=\frac{v}{4} = 344/4 = 86\text{ Hz}##

which is the correct answer in the textbook.I'm having trouble interpreting the equation x(n) that I derived.

1. There are also negative values of x(n) for ##n>5## and ##-4<n\le 0##, and positive values for ##n\le -5## when ƒ=786 Hz. Since x is a point along the continuous line BC, and r is defined for all x, why doesn't the textbook count all nonzero values of x(n) as points of destructive interference? Or if it's looking specifically for ##x>0##, why doesn't it count x(n) for ##n<1##?

2. If x(1) = 0 then isn't B itself a point of destructive interference?

3. x(0) as well as x(1) equals 0 at ƒ=86 Hz. What does this mean?

4. Is my interpretation correct that if the only point of destructive interference is x(1) = 0, speaker B is not able to output any sound since the source itself is destructively interfered?

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