How many places of maximum loudness?

[Sho Kano]

Homework Statement

the last problem on this page:

Homework Equations

$v= \lambda f$

The Attempt at a Solution

I'm guessing I'm looking for the maximum amplitude in a overlapping wavefront diagram like this?

$v=343m/s \\f=512Hz \\343/512=\lambda=0.67m \\3/0.67=4.5m$
So 4.5 wavelengths will fit between the two sources. I'm not sure where to go from here.

 

[TSny]

Can you relate the question to double slit interference?

 

[Sho Kano]

Can you relate the question to double slit interference?

I don't know what that is, we haven't covered that yet. How do I solve just using the diagram?

 

[TSny]

Have you studied interference from two point sources? Does the formula $d \sin \theta = n \lambda$ look familiar?

 

[haruspex]

"On a large circle centered between the sources"
That is not at all clear.
I think it means to draw a circle centered half way between the sources, in a plane containing them, and passing outside both sources. Count the local maxima on that. With that interpretation I get an answer matching one offered.

 

[TSny]

Hmm. I get one of the answers also if I assume that the center of the circle is halfway between the sources, but the radius of the circle is much greater than the distance between the sources. Making the radius much greater than the distance between the sources allows for the approximation that yields $d \sin \theta = n \lambda$.

 

[Sho Kano]

Have you studied interference from two point sources? Does the formula $d \sin \theta = n \lambda$ look familiar?

I have not seen that equation, what is n, d, and theta?

 

[Sho Kano]

"On a large circle centered between the sources"
That is not at all clear.
I think it means to draw a circle centered half way between the sources, in a plane containing them, and passing outside both sources. Count the local maxima on that. With that interpretation I get an answer matching one offered.

I'm having trouble with finding the local maxima, what exactly do I count?

 

[TSny]

$d \sin \theta = n \lambda$

$d$ is the distance between the two sources

$\theta$ is an angle specifying the direction from the center of the circle to a point on the circle at which you would get constructive interference if the radius of the circle is large. $\,\,\,\theta = 0$ corresponds to the direction along the perpendicular bisector of the line connecting the two sources.

$n$ is an integer 0, 1, 2, 3, ... corresponding to the different directions for constructive interference.

But, if you have not covered this formula, then I don't think you would be expected to derive it on your own just to answer the question. Here is a link that provides more information: http://webassign.net/question_assets/buelemphys1/chapter25/section25dash1.pdf

So, I'm sure how your are meant to answer the question.

 

[Sho Kano]

I'm having trouble with finding the local maxima, what exactly do I count?

ah ok the maxima in an interference pattern are the areas of maximum wavelength; loudness though corresponds with amplitude so why would this area be the loudest?

 

[haruspex]

ah ok the maxima in an interference pattern are the areas of maximum wavelength

No, max amplitude.
A local maximum occurs where the two signals are in phase. You are told they start in phase. If they arrive in phase at some point, what can you say about the distances from that point to the sources?

 

[haruspex]

the radius of the circle is much greater than the distance between the sources

That is unnecessary, and no equations are needed (just a little geometry).

 

[Sho Kano]

No, max amplitude.
A local maximum occurs where the two signals are in phase. You are told they start in phase. If they arrive in phase at some point, what can you say about the distances from that point to the sources?

The distances between each concentric circle is the wavelength isn't it? which for sound, I guess it's the dist btw each compression. So each ring is a compression, and each space is a rarefraction

 

[TSny]

That is unnecessary, and no equations are needed (just a little geometry).

True. The answer is independent of the size of the circle as long as the circle has a diameter greater than or equal to the distance between the sources. (The circle need not be centered on the sources. Any circle that encloses the two sources would have the same number of maxima.)

 

[Sho Kano]

The distances between each concentric circle is the wavelength isn't it? which for sound, I guess it's the dist btw each compression. So each ring is a compression, and each space is a rarefraction. Can you confirm if this is right?

Edit: actually it might be the amplitude; wavelength is one cycle of oscillation compression and decompression.

 

[TSny]

The distances between each concentric circle is the wavelength isn't it? which for sound, I guess it's the dist btw each compression. So each ring is a compression, and each space is a rarefraction. Can you confirm if this is right?

Yes.

Suppose P is some point on the "large circle". Let $d_1$ and $d_2$ be the distances from the two sources to point P. Do you know the relation between $d_1$ and $d_2$ in order for constructive interference to occur at P?

 

[Sho Kano]

D1 equal to D2.

But since sound waves are longitudinal what's the relationship between amplitude and the wavelength? Is the distance btw the circles also the amplitude?

 

[TSny]

D1 equal to D2.

This condition would indeed give you constructive interference. But there are other possibilities.
http://sciencealevelhelp.co.uk/?p=624

This is essential to answering the question.

 

[haruspex]

The distances between each concentric circle is the wavelength isn't it? which for sound, I guess it's the dist btw each compression. So each ring is a compression, and each space is a rarefraction

You can draw concentric circles that are centred on the sources and have radii which are whole numbers of wavelengths, but I consider that unhelpful in approaching this question.
Please try to answer the question I asked. If constructive interference occurs at a point which is d1 from one source and d2 from the other source what is the relationship between d1, d2 and the wavelength λ? (Hint: d1 and d2 themselves do not need to be whole numbers of wavelengths.)

But since sound waves are longitudinal what's the relationship between amplitude and the wavelength? Is the distance btw the circles also the amplitude?

For the purpose of the question, it does not matter whether the waves are longitudinal or transverse. They could just as easily be ripples on a pond. You can regard amplitude and wavelength as unrelated.