Resulting intensity of interference

[Maxo]

Homework Statement

Two speakers with the same power output generate sound waves in 360 degrees direction. Both speakers have the same freuqncy and the distance between the speakers is given.

How can we from this calculate the sound intensity at any given point around the speakers?

Homework Equations

$$I = \frac{P_{av}}{A}$$

The Attempt at a Solution

It is easy to calculate the intensity at a given point for only one speaker, using the equation above. The question is how we can calculate the intensity for any point of interference between two speakers? We need to know how the intensity will be for two interfering waves. How can we find that?

 

[berkeman]

Homework Statement

Two speakers with the same power output generate sound waves in 360 degrees direction. Both speakers have the same freuqncy and the distance between the speakers is given.

How can we from this calculate the sound intensity at any given point around the speakers?

Homework Equations

$$I = \frac{P_{av}}{A}$$

The Attempt at a Solution

It is easy to calculate the intensity at a given point for only one speaker, using the equation above. The question is how we can calculate the intensity for any point of interference between two speakers? We need to know how the intensity will be for two interfering waves. How can we find that?

Hint -- For a given point, calculate how many wavelengths away each speaker is. How is that calculation helpful?

 

[Maxo]

Ok, the wave speed and the frequency is given so thereby the wavelength. So it is easy to calculate how many wavelengths away each speaker is. But how does these number of wavelengths give the intensity?

 

[berkeman]

Ok, the wave speed and the frequency is given so thereby the wavelength. So it is easy to calculate how many wavelengths away each speaker is. But how does these number of wavelengths give the intensity?

Maximum intensity will be at those places where the phase of the two wavefronts is _______. Minimum intensity will be where the phase of the two wavefronts is _________.

And adding to the level of difficulty of the problem, remember that for isotropic radiators, the intensity decreases with the spherical surface area as you have shown in your equation. So that will attenuate each of the sound wavefronts according to how far the point is away from the sources.

So you need to combine the effects of interference with the effect of the reduction in intensity versus distance, in order to map out the full intensity space.

 

[berkeman]

Actually, what is meant in the problem statement by "360 degrees direction"? Is this supposed to be a 2-dimensional or 3-dimensional problem?

 

[berkeman]

I did a Google Images search on interference 2 point sources, and got lots of helpful diagrams. Here's one:

http://honda-cb750-s.456789.n3.nabble.com/file/n4029931/InterferencePattern.jpg

 

[Maxo]

That only gives the points of maximum and minimum interference, but not the exact intensity at any given point, which is what we are looking for.

 

[berkeman]

That only gives the points of maximum and minimum interference, but not the exact intensity at any given point, which is what we are looking for.

Right, which is why I said that you have to take both effects into account.

Perhaps the easiest way to approach it is to write the equations for each source seperately at first. That is, write the equation for the sound pressure as a function of time given the distance from the source. You can write it in cylindrical or spherical coordinates, depending on whether the problem is meant to be solved in 2-d or 3-d. Then add the two equations to give you the sum at any point p. There will be attenuation of each signal due to the distance from each source to p, and there will be a phase shift between the two signals due to unequal distances.

I don't know if the math will be easier with the origin at one of the two sources, or half-way between them. You should be able to tell pretty quickly by starting to write the equations.

 

[Maxo]

We suppose the problem is meant to be solved in 2D. We selected the origin at the left speaker.

What we need help with is rather which equations can be used? Equation for sound pressure? Which one is that? And how does it give intensity? Not to be impolite, but do you actually know how to solve the problem or are you just guessing?

 

[berkeman]

We suppose the problem is meant to be solved in 2D. We selected the origin at the left speaker.

What we need help with is rather which equations can be used? Equation for sound pressure? Which one is that? And how does it give intensity? Not to be impolite, but do you actually know how to solve the problem or are you just guessing?

No, I'm not guessing. What is the equation for the sound pressure as a function of time at any point in the 2-D plane that cuts through the spherical radiation pattern for one source? You should be able to write that equation in cylindrical coordinates. For the source that is at the origin, there will not be a variation with respect to theta, only with respect to r. For the other source, you will get a more complicated equation...

 

[berkeman]

Try this Google search:

spherical waveform interference +sound

There is a PDF in one of the first hits about a school physics lab with some background that should be a big help for you.

 

[Maxo]

What is the equation for the sound pressure as a function of time at any point in the 2-D plane that cuts through the spherical radiation pattern for one source?

Don't know. As a function of time? Doesn't sound familiar at all. Which equation is it?

 

[berkeman]

Don't know. As a function of time? Doesn't sound familiar at all. Which equation is it?

Did you do the Google search? :-)

 

[Maxo]

Of course anyone can make a google search and find equations that might be useful, but how should we know which one to use and how? Don't you understand the question is which equations to use and how for this particular task? So why don't you write that instead?

 

[berkeman]

Of course anyone can make a google search and find equations that might be useful, but how should we know which one to use and how? Don't you understand the question is which equations to use and how for this particular task? So why don't you write that instead?

Why are you refusing to show your work in this thread? You have shown work in other threads -- why are you not doing so now? It is part of the PF rules that you show your work on schoolwork problems. Please start showing some effort here, or this thread will be deleted.

 

[Maxo]

Well there isn't much to show at the moment, since we don't really know how to solve the task. We can write expressions for the distance to each speaker but that's only trigonometry. The question of the physics remain.

Is it perhaps this equation that should be used? $$I = \frac{1}{2}\rho v \omega^2 A^2$$
A being the amplitude... but then what is the amplitude of the resulting wave?
Still don't know how to go about it.

 

[berkeman]

Well there isn't much to show at the moment, since we don't really know how to solve the task. We can write expressions for the distance to each speaker but that's only trigonometry. The question of the physics remain.

Is it perhaps this equation that should be used? $$I = \frac{1}{2}\rho v \omega^2 A^2$$
Still don't know how to go about it though.

Honestly, by my posting the Google search terms that I did, I practically gave you the answer. No, that equation is not enough to be useful. The equation described in the PDF that you get to via the search has terms for time, frequency, distance, attenuation, phase shift, etc.

 

[Maxo]

In the school lab you are referring to there is no expression for the power output of the speakers, which is known in this case, so the equations must be made including some expression for power. How?

 

[Maxo]

Here is the exact problem formulation:
Two small speakers (omnidirectional) with power of 10 W are working in phase with each other and sends out sound with a frequency of 200 Hz. The speakers are placed 6 m from each other. If you walk in a circle with radius 4 m around one the speakers the sound intensity will vary. Assume that the speed of sound is 340 m/s. Draw a diagram that shows the total intensity in all the points on the circle, 0 -> 360 degrees, where the angle is 0 at the line between the speakers.

 

[berkeman]

In the school lab you are referring to there is no expression for the power output of the speakers, which is known in this case, so the equations must be made including some expression for power. How?

That power gets spread over the surface of the sphere of radius r...

 

[Maxo]

That power gets spread over the surface of the sphere of radius r...

That's obvious. The question is (still) how is the relation formulated in a equation?

 

[berkeman]

That's obvious. The question is (still) how is the relation formulated in a equation?

Are you asking how sound power, sound intensity, sound pressure, etc. are related? The wikipedia article is useful: http://en.wikipedia.org/wiki/Sound_pressure

What does your textbook say about the relations?

 

[Maxo]

Are you asking how sound power, sound intensity, sound pressure, etc. are related? The wikipedia article is useful: http://en.wikipedia.org/wiki/Sound_pressure

What does your textbook say about the relations?

No. Did you not read what I wrote above? The question is how to write the equation for the sound intensity at a given distance from two omnidirectional speakers, in terms of the given data.

 

[Maxo]

What is the resultant amplitude of two interfering waves, at any given point?

 

[haruspex]

What is the resultant amplitude of two interfering waves, at any given point?

Pick a point on the circle at angle theta to the line joining the speakers. How far is it from each speaker to the point? What does that tell you about the phase difference?

 

[Maxo]

Pick a point on the circle at angle theta to the line joining the speakers. How far is it from each speaker to the point? What does that tell you about the phase difference?

Well, there has to be some relation between the phase difference and the distance in wavelengths.

 

[haruspex]

Well, there has to be some relation between the phase difference and the distance in wavelengths.

Can you be more specific? Maybe start with the phase difference between each speaker and that point.