Chpt 11 With Videos(1)

8/12/2019 Chpt 11 With Videos(1)

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11SUPERPOSITION

CH

APTER

Fundamental Concepts in this Lecture

1. Waves in the same medium add together, we call this superpositionand describe it with the idea of interference

2. Superposition of waves with different frequencies (beats)3. Reected waves interfering with the original waves can form standing

waves for certain frequencies4. Interference from multiple sources5. Phase difference tells us how in sync two waves are

Waves that move

Waves are a movement of energy throughout a medium. It is not a surpriseto nd that waves move. Below we nd an animation of a traveling pulse on astring. Since the string is our medium, the red string particles actually go upand down.


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104 C H A P T E R 11 SUPERPOSITION

For a sinusoidal wave, we have to remember that the entire wave moves. In thisanimation, we have a green string, and we have marked a spot on the string.Notice that the red spot only goes up and down.

It does not move along with the wave. This is true of both light waves and soundwaves (except that in sound waves the air particles move back and forth).

Superposition - Wave Adding

Now, suppose we have more than one wave on a string, or more than one in-strument making sound waves in the same piece of air. How would these wavesmix?

The animation goes by fast, but lets see if we can gure out what is going on.We have two waves, each pulses. The pulses are traveling on the same string.


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C H A P T E R 11 SUPERPOSITION 105

The next set of graphs provides animations of the last gure showing eachpiece traveling.

And the next gure is an animation of all three, but with the original waves andsum one above the other so you can see them all at once.

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Lets see how we got the nal result. We start by picking a point on each wave,say, the point x = -2. From the rst graph, we see we have a value of y = 2 at x= -2. It is marked on the next gure with a little bar which shows the value ofthe height of the wave at that point.

We can also see that the second wave (green curve) has a value of y = -0.14 atx = -2. This is shown on the second graph with the small green bar showing the

height of the green curve at x = -2. If we add these two wave heights, we get2 0.14 = 1.86. We can see that this is just the height of the rst waves ampli-tude (red) added to the amplitude of the second wave (green) for the point x =-2. We could do the same for each x location. For x = -1, we have y = 1 for therst wave, and y = -1 for the second wave, yielding y = 1 1 = 0 for the result inthe third graph. In the gure, there is also a result shown for x = 0. We can seethat the waves addpoint for point along the x-axis.This point for point waveadding is called superposition.

This is really not too hard to see for a set of pulse waves. But what would thislook like for a sinusoidal wave? Here is an example. If you are able to see theanimation, you can focus on the red and green waves separately. Can you seethem moving by? Then notice that this makes a resultant wave whose ampli-

tude changes. Sometimes it is large and sometimes it is small.


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How about when the waves we add go in the same direction?

We can pause to recall what phase means. In this diagram, the red and greenwaves are perfectly in phase, so much so that we can only see one curve. Butthe result of adding these waves, the purple curve, is also in phase with theother two. The amplitude is now the sum of the amplitudes of the red and green

waves. So, the amplitudes are not the same. But the waves all rise and fall atexactly the same times. This is really what it means to be in phase. Suppose thered and green waves are not in phase. Then what?

From the above gure, you can see that the red and green waves are no longer

rising and falling together. The red wave is shifted down from the green curve.They still add to form a resultant purple wave, but that wave has a smalleramplitude, because when we add point by point, the peaks of the red and thegreen waves no longer line up. The maximum heights are not at the same placeto add. Can we add more than two waves? Of course! Think of all the wavegenerating devices (instruments) in a symphony orchestra! But lets try justthree for now.

So far, we get a new wave that looks a lot like the waves we are adding. But no-tice we have only changed the phase and amplitude of our waves so far. Whathappens if we add waves of different frequencies?


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Beats: Interference in Time

Until now we only superposed waves that had the same frequency. But whathappens if we add waves with different frequencies? Here are two such waves.

The picture is already pretty messy, so lets plot the new wave in another graph.

If we make the frequencies even more different we get this result:

If you have a version of this document that can show the animations, here is aportion of the last graph that is animated:

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What would this sound like? The amplitude rises and falls as the wave passesus. Since amplitude is related to how loud the sound seems to our ears, thiscombined wave just gets louder and softer. This change in loudness is calleda beat. When two strings, two clarinets, or violins, etc. are slightly out of tune(play different frequencies), the sound will seem to have a periodic change in

loudness, it will beat. The beats get faster with increasingly different frequen-cies. Thus, a quick beats mean you are more out of tune. Slow beats mean youare getting close to the same frequency. When the beats quit altogether, youare in tune! But be careful, at some point the beats go so fast that our ears can-not hear them. They are still there, but our ears cant pick them out. When thishappens, you are really not in tune! This is a good thing. If it were not true, we

would hear beats when a chord is played. As it is, the beats are quicker than ourability to hear when we play different notes. This is useful for tuning, but doesnot hurt when we play (isnt this world designed well?).

We can predict how fast the beats will go. The beat frequency (the frequencywith which we hear the loud pulses at a given location) is given by

For example, if we tune two instruments, one to 100Hz and one to 101Hz, wewould have a beat frequency of 1Hz.

Reection of waves

Can a wave bounce off of a boundary? Yes, we know from our look at light thatit can. If you live near the ocean, you might have seen waves bounce back froma breakwater. If you have thrown rocks in a pond, you may have seen the waves

bounce back from a reed or a rock that was in the pond. We have already calledthis bouncing of waves reection.

You might guess that calculating the shape of waves as they reect is math-ematically intense. You would be right. We would like to avoid this if we can.

First, you looked at two wave pulses coming together from opposite directions.One is upside down or inverted. If you are using the online version of this text,then click the animation.

The animation should give a series of pulses that interfere much like these onesecond snapshots that show the pulses coming together. The individual puls-es are red and green, and the superposition is purple. But mostly we see thepurple combined wave, except where the pulses are right on top of each other.

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Now lets consider just how a wave looks as it approaches the end of a medium(like a rope) that is tied down.

Or the 1 second snapshots:

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Lets look at both together. Notice that the top wave which reects from aboundary is just like the left half of the two waves interfering!

Here are the 1 second snapshots:

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We can see that in each case the top graph for the wave reecting looks just likethe left hand of the mixture of two waves.

This is an easy way for us to nd how waves should act at a boundary. Lets tryit for a jump rope tied to a ring on a post, but tied so that the ring can go up ordown. The end of the rope can be whipped upward.

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We can see this in the 1 second snapshots:

Whip upward is an apt name, because this is really how whip ends work!We have done simple pulses, but of course we could do other shapes as well.Lets go back to our sinusoidal waves.

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Notice that in our animation the purple combined wave had amplitude, andwavelengths, but notice that the wave does not seem to move! It stands stilleven though it is made from two traveling waves. Yes, spots go up and down,

but they seem to stay in the same horizontal place. In our easy way to nd what

would happen to a sinusoidal wave on a tied string, we again compare to thetwo waves interfering. We see the same shape!

This gives us an idea. Suppose we look at a pulse going toward a xed or tieddown end of a rope. It will reect back. We can view this as a new wave that willinterfere with the incoming wave. For pulses, this only lasts for a moment. Butfor sinusoidal waves, we can create a reected wave that will interfere with theincoming wave. And this strange thing happens. The resultant wave will seemto stand still. We will talk more of thesestanding wavesin the next reading.

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Waves that arent in the same line

What about sources of sound not in a straight line? In a musical group or or-chestra, not everyone sits in a straight line. So when we add waves we wont see

the easy charts that we have seen so far. Lets set up two speakers next to eachother and turn them on with a sound generator that creates just one frequency.They make big spheres of sound around them that have wave crests somethinglike what we see in the gure below.

Just like with our linear waves, we see that wave crests add to make a largercombined wave crest. The opposite of the wave crest is the low spot. We call it atrough (think of a water trough for horses in old western movies).

Wave troughs combine to form a larger wave trough; and where wave troughsand crests combine, we get zero. Zero motion is the same as no sound. This isa dead spot in your home theater or concert hall--something we want to avoid!

We call dead spots destructive interferenceand we call the loudest spots con-structive interference.

Lets see how to create a dead spot. If we take two waves that are half a wave-length out of phase and add them we get a zero.

Thus, anywhere in our area around the two speakers where the distance fromone speaker is more than the distance from the other speaker plus half a wave-length will be a dead spot.


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Really, there will be a dead spot when we can divide the distance to a spot bythe wavelength and we have a remainder of zero for one wave (say, the red

wave), and a remainder of half a wavelength for the other.

Dead Spot

Half a wavelength

farther from the green

wave center


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Chpt 11 With Videos(1)