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Question
how can one calculate the number of minima if given the speed, frequency and distance between two wave sources especially if the frequency of both waves is increased when they superpose

Answer
Hello Vumile,

Please open this web site. I will refer to it in my discussion.
http://www.acoustics.salford.ac.uk/feschools/waves/super2.php
Scroll down to the 2nd simulation. This shows waves being developed in a water tank. Please also take a look at this website.
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Interference/text/Inte
Scroll down to Figure 3. The comments I make regarding the simulation in the 1st web site would also apply to a situation like this 2nd web site shows. The 2nd site's waves emerging from the double slits are equivalent to the waves caused by the vibrating sticks in the 1st site.

Your question refers to speed, frequency and distance between two wave sources. Using the formula
speed = lambda*frequency
you can calculate the wave length, lambda. I will talk about wave length and distance between two wave sources.

Referring to the 1st site, notice that the green line, indicating a maxima, that is perpendicular to a line between the vibrating sticks. The height of the peaks along that line are maximum and the depth of the troughs are maximum. There will always be a maximum along a line like that line. To make it clear which maxima or minima line I am referring to, I would like to number them from top to bottom. So the perpendicular line I have been talking about is line 4. There is constructive interference anyplace on that perpendicular line, line 4, because the path lengths, from their source, for the 2 interfering waves anyplace on that perpendicular line are equal. Therefore the waves are in phase anywhere along that line. That is the case required for constructive interference.

The 2 red lines numbered 3 and 5 on either side of that center green line, #4, represent a line radiating from a spot on the line between the vibrating sticks. I will be focusing on that line between the vibrating sticks. Any point p on those 2 red lines (the ones closest to that perpendicular green line) you would find that the path lengths from the 2 sources to point p differs by 1/2 of the wavelength. Look at the pair of red lines 1 and 7. The path lengths to the 2 sources of any point p you choose on lines 1 and 7 would be different by 1.5 wave lengths. In one case the upper source would be closer, in the other the lower source would be closer.

Notice that the simulation does not draw the red and green lines all the way to the line between the vibrating sticks. Please imagine that those line do go all the way to the left and make contact with the line between the vibrating sticks. The distance between the point where line 3 touches the line between the vibrating sticks and the point where line 4 touches the line between the vibrating sticks must be 1/2 of the wave length because line 3 is a line of minima. And the point where line 5 touches the line between the vibrating sticks must be 1/2 of the wave length from the point where line 4 touches the line between the vibrating sticks because line 3 is a line of minima.

So, how far apart are the points where lines 3 and 5 touch the line between the vibrating sticks? The point where line 3 touches the line between the vibrating sticks must be 1 full wave length from the point where line 5 touches the line between the vibrating sticks. And the point where line 1 touches the line between the vibrating sticks must be 3 full wave lengths from the point where line 7 touches the line between the vibrating sticks.

It appears that there is not another maxima. The 2 sticks are too close together for the condition that would provide another maxima. So what can we say about the distance between the 2 sticks? The condition that produces the maxima of line 2 is that the path lengths from the 2 sources (the sticks) to any point on line 2 must differ by 1 full wave length because that is necessary for constructive interference. So looking at the point where line 2 touches the line between the sticks must be 2 wave lengths from the corresponding point on line 6. The distance between the corresponding points on lines 1 and 7, must be 3 wave lengths.

Since there is not another maxima outside lines 1 and 7, the distance between the sticks must be less than 4 wave lengths. We have 4 minima and there appears to be about 3.8 wave lengths between the sticks in this simulation. We need a routine that will predict the number of minima for any width between sources. So it appears that if you do these steps:
1. determine the number of wave lengths between the sources,
2. add 1
3. divide by 2,
4. truncate any decimal, and
5. multiply by 2,
that would give you the number of minima.

This was harder than I expected. I thought I would find the rule online. I never did.

I am sorry, but I cannot understand your question about increasing frequency when they superpose. I invite you to send a followup.

I hope this helps,
Steve

Physics

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Steve Johnson

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I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

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I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

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BS Physics, North Dakota State University
MS Electrical Engineering, North Dakota State University

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