Frequency at which no destructive interference occurs

[prodo123]

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?

 

[BvU]

Hello Prodo,

1.) There is indeed symmetry wrt the line AB. I don't have the exact problem statement for

A previous part of the problem

however, if it says 'distance from B' then all is well.

2. It sure is.

3. It means that the distance from B is zero when you are at B. That is not what you mean, I suppose . You mean to say something about $y_A+y_B$, but your equation for $y$ does not take into account that $A$ also varies with distance.

4. Therefore your interpretation is not correct.

 

[prodo123]

Hello Prodo,

1.) There is indeed symmetry wrt the line AB. I don't have the exact problem statement for
however, if it says 'distance from B' then all is well.

2. It sure is.

3. It means that the distance from B is zero when you are at B. That is not what you mean, I suppose . You mean to say something about $y_A+y_B$, but your equation for $y$ does not take into account that $A$ also varies with distance.

4. Therefore your interpretation is not correct.

Thanks for the reply. The chapter doesn't handle displacement amplitude decay over distance, which is why I held A constant.

The issue is that there's supposed to be reciprocity but the textbook ignores (or denies?) it. The previous part of the question simply says:

At what distances from B will there be destructive interference?

Values of $x(n)$ for n=1,2,3,4,5... give 9.01m, 2.71m, 1.27m, 0.534m, 0.026m which are the five values given as the answer to the problem. I'll give the textbook the benefit of doubt of looking for positive $x$ only, since the diagram shows BC going in the positive-x direction only.
But even so $x(-5)$ results in 0.372m, and $x(n)$ for all $n\le -5$ is positive. Looking at the graph of $y_A+y_B$ these are indeed points where the two waves cancel out.
I'm honestly starting to think the textbook got lazy and didn't bother to write down all the solutions...

 

[BvU]

The chapter doesn't handle displacement amplitude decay over distance

But you know from experience.

there's supposed to be reciprocity

reciprocity ?

At what distances from B will there be destructive interference?

It does not ask for $x$, it asks for distances. Distances are alsways positive. They can be to the left or to the right. If this is with 786 Hz, then I don't understand the given answers. Can you explain ?

 

[prodo123]

But you know from experience.

Yes, I know from experience but that's not what the question is asking for and including it will give the wrong answer to the problem.

reciprocity ?

It does not ask for $x$, it asks for distances. Distances are alsways positive. They can be to the left or to the right. If this is with 786 Hz, then I don't understand the given answers. Can you explain ?

Sorry, meant to say symmetry across $x=0$.

After some discussion and playing around with the graph of $y_A+y_B$ I concluded on the following:

• Diagram specifically points to the right only. Mathematically negative $x$ exists but the problem ignores that.
• The textbook is looking for positive distances from $B$ at which total cancellation occurs for all t.
  
• The given answers for 786 Hz (9.01m, 2.71m, 1.27m, 0.534m, 0.026m) are the only points on BC where this occurs under the given conditions and $f=786\text{ Hz}$.
• Graph of $x(n)$ (treating $n$ as continuous instead of an integer) shows it's a hyperbola, and the textbook's answers correspond to the positive values of the right half of the hyperbola only.
• If $f=86\text{ Hz}$, $x=0$ is the only place this occurs.
  
• Apparently $x\ne 0$ (the speaker B itself is not a part of BC?), therefore the question considers $f=86\text{ Hz}$ to have no such points on BC even though it occurs at $x=0$.

The solutions for $x(n)$ outside $[1,...,5]$ are zeroes (total cancellation) of $y_A+y_B$ at time $t=0$, but do not hold at other values of $t$. For example, $x(-5)=0.372$ is not a zero at time $t=10$.

I've decided it's just a really poorly worded question and moved on.

 

[BvU]

$x(n)$ for n=1,2,3,4,5... give 9.01m, 2.71m, 1.27m, 0.534m, 0.026m

I find these values if I take $f = 784$ Hz, not 786 ?!

Diagram specifically points to the right only. Mathematically negative $x$ exists but the problem ignores that

No. $x$ is a distance, a length. And a length is non-negative.

You wanted to solve $r=\sqrt{x^{2}+2^{2}}=x+\frac{(2n-1)\pi}{k}$ and in order to do so, you squared left and right. Did it occur to you that you introduced wrong answers that way ? Did you check that your $x(-5) = -x(6)$ satisfies the original equation (it does not) ?

 

[prodo123]

I find these values if I take $f = 784$ Hz, not 786 ?!
No. $x$ is a distance, a length. And a length is non-negative.

You wanted to solve $r=\sqrt{x^{2}+2^{2}}=x+\frac{(2n-1)\pi}{k}$ and in order to do so, you squared left and right. Did it occur to you that you introduced wrong answers that way ? Did you check that your $x(-5) = -x(6)$ satisfies the original equation (it does not) ?

Sorry, I got confused, it's for 784 Hz.

Showing that $x(-5)$ is indeed equal to $-x(6)$:

$x(-5)=\frac{4*784}{344(2(-5)-1)}-\frac{344(2(-5)-1)}{4*784}\\ x(-5)=\frac{3136}{-3784}-\frac{-3784}{3136}\\ x(6)=\frac{4*784}{344(2(6)-1)}-\frac{344(2(6)-1)}{4*784}\\ x(6)=\frac{3136}{3784}-\frac{3784}{3136}\\ -x(6)=\frac{-3136}{3784}-\frac{-3784}{3136}=x(-5)\\ \\ x(-5)=\frac{70065}{185416}\approx 0.37788$

as for $y_A=-y_B$ at time $t=0$:

$y_A+y_B=0\\ A\cos(\frac{784*2\pi}{344}x(-5))+A\cos(\frac{784*2\pi}{344}\sqrt{x(-5)^2+4})=0\\ \cos(\frac{70065\pi}{40678})+\cos(\frac{377393\pi}{40678})=0\\ 0.64329-0.64329=0\\ 0=0$

Graph of $y_A+y_B$ showing that 0.37788 is indeed a root:

 

[prodo123]

Here's a comparison of the roots for $y(x)=y_A+y_B, f=784\text{ Hz}$, assuming an arbitrary amplitude of 1 for both waves, found at different values of $t$.

The given set of roots include $x(4)=0.026$ and $x(5)=0.534$ (blue); the root in question is $x(-5)=0.37788$ (red).

at $t=0$:

at $t=0.8$:

at $t=1.3$:

which shows $x=0.37788$ to be a root only when $t=0$, while the given answers hold true for all $t$.

You wanted to solve $r=\sqrt{x^{2}+2^{2}}=x+\frac{(2n-1)\pi}{k}$ and in order to do so, you squared left and right. Did it occur to you that you introduced wrong answers that way ?

I think you're right about the squaring introducing false answers, but then how would one prove that a root at some time $t$ is also a root for all $t$?