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Lecture Comments (12)

1 answer

Last reply by: Professor Dan Fullerton
Mon Apr 13, 2015 6:38 PM

Post by SeungJoo Han on April 13, 2015

Example 4. I think the equation for wavelength is  Wavelength = 2l/n. I want you to explain how dod you get the 4l/n for figuring out the wavelength.

1 answer

Last reply by: Professor Dan Fullerton
Mon Mar 23, 2015 6:29 AM

Post by Wen Nguyen on March 22, 2015


0 answers

Post by Jamal Tischler on September 1, 2014

I found this nice example:

3 answers

Last reply by: Professor Dan Fullerton
Sun Apr 6, 2014 7:14 AM

Post by Jerry Liu on April 3, 2014

Example 7 is labeled as Superstition, it should be "Superposition".
Do you not have an editor?

2 answers

Last reply by: varsha sharma
Tue Mar 26, 2013 5:47 AM

Post by varsha sharma on March 24, 2013

Why at B a trough. Please explain.

Wave Interference

  • The total displacement of two or more interfering waves is the sum of all the individual displacements of the waves. This is known as superposition.
  • Identical waves traveling in opposite directions in the same medium can create standing wave patterns. Nodes appear to stand still, while antinodes vibrate with maximum amplitude above and below the axis.
  • Standing waves in string instruments and open-tube instruments create fundamental frequencies as well as 2nd, 3rd, 4th, … harmonics.
  • Standing waves in closed-tube instruments create fundamental frequencies as well as odd-numbered harmonics.
  • Two sound waves near-identical frequencies can interfere to create beat patterns.

Wave Interference

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Objectives 0:09
  • Superposition 0:30
    • When More Than One Wave Travels Through the Same Location in the Same Medium
    • The Total Displacement is the Sum of All the Individual Displacements of the Waves
  • Example 1: Superposition of Pulses 1:01
  • Types of Interference 2:02
    • Constructive Interference
    • Destructive Interference
  • Example 2: Interference 2:47
  • Example 3: Shallow Water Waves 3:27
  • Standing Waves 4:23
    • When Waves of the Same Frequency and Amplitude Traveling in Opposite Directions Meet in the Same Medium
    • A Wave in Which Nodes Appear to be Standing Still and Antinodes Vibrate with Maximum Amplitude Above and Below the Axis
  • Standing Waves in String Instruments 5:36
  • Standing Waves in Open Tubes 8:49
  • Standing Waves in Closed Tubes 9:57
  • Interference From Multiple Sources 11:43
    • Constructive
    • Destructive
  • Beats 12:49
    • Two Sound Waves with Almost the Same Frequency Interfere to Create a Beat Pattern
    • A Frequency Difference of 1 to 4 Hz is Best for Human Detection of Beat Phenomena
  • Example 4 14:13
  • Example 5 18:03
  • Example 6 19:14
  • Example 7: Superposition 20:08

Transcription: Wave Interference

Hi everyone and welcome back to 0000

I am Dan Fullerton and we are going to continue our study of waves today by talking about wave interference. 0003

Our objectives are going to be to explain how the law of superposition governs the interference of waves, to describe a standing wave pattern and how it is created, to identify nodes and anti-nodes in a standing wave, and determining fundamental and harmonic frequencies in wavelengths for strings, open tubes, and partially closed tubes. 0009

Super position -- When more than one wave travels through the same location in the same medium at the same time, the total displacement of the medium is governed by what is called the Law of Superposition, which is really a very simple concept. 0030

All it says is the total displacement is the sum of the individual displacements of the wave. 0044

The combined effect of the interaction of multiple waves, then is known as wave interference. 0049

It sounds very complicated, but very, very simple in practice. 0054

Let us take a look at an example. 0061

The diagram below shows two pulses approaching each other in a uniform medium. Diagram the superposition of the two pulses. 0063

That is a fancy way of saying just add them together when they cross each other. 0070

As this one is going to the right and this one is going to the left and when they hit each other, their displacements are going to add together, so you are going to get something that looks like that. 0073

If this one is 5 cm and that is 10 cm, then this must be 15 cm; they add together. 0087

You have the two wave pulses; they are coming together; they get closer and closer and when they start to interfere with each other, they are going to overlap, give you a bigger wave, and then as they move on, they are going to keep going as if they had never met. 0094

A split second later after they have passed, what this would actually look like then -- You will have the 10 cm one still going to the left and the 5 cm one still going to the right as if they had never met. 0106

How simple is that? 0120

Let us talk about types of interference. 0123

When two or more pulses with displacements in the same direction interact, the effect is known as constructive interference; you get an amplitude that is greater than the individual amplitudes. 0125

We call this in phase. 0136

When two or more pulses with displacements in the opposite directions interact -- well now they are going to start to cancel each other out; you are going to get an amplitude that is diminished, and that is called destructive interference or out of phase. 0138

Now once the pulses have passed each other, let me reiterate, they are going to keep going as if they had never met, even if for a second they completely cancel out and it looks flat, once they pass each other they keep going as if they had never met, whether it is constructive or destructive interference. 0152

Let us take a look at another example. 0168

The diagram below over here has two pulses approaching each other from opposite directions in the same medium. 0171

Which diagram at right best represents the medium after the pulses have passed through each other? 0177

After they have passed through each other, they keep going as if they had never met, so we need one where we have the smaller pulse traveling to the right and the larger pulse traveling to the left. 0183

That has to be this one right here, Number 2 -- they do not invert and they do not have anything else fancy going on. 0193

It is just that simple, they keep going as if they had never met. 0203

Let us take an example here with shallow water waves. 0208

The diagram below here shows shallow water waves of constant wavelength passing through two small openings (A) and (B) in the barrier, so the wave velocity must be in that direction. 0211

When it gets here we are going to see some diffraction around these and we will talk about that in more detail, but what we really notice here is that you have crests and troughs interfering with each other from (A) to (B). 0221

Which statement best describes the interference, here at point (P)? 0233

Point (P) is right there, so you will notice that that is occurring where we are on a crest from (A) that is meeting a trough from (B). 0236

If we have a crest and a trough, that is going to be destructive interference, therefore, our answer must be: It is destructive causing a smaller amplitude because destructive interference reduces the amplitude. 0248

Standing waves -- When waves of the same frequency and amplitude traveling in opposite directions meet in the same medium, a standing wave is produced. 0264

A standing wave is a wave in which certain points, which we call nodes, appear to be standing still and other points called anti-nodes over here tend to vibrate with maximum amplitude above and below the axis. 0274

This is the basis of a bunch of our instruments. 0285

In this case, as we look here, I see 4 nodes and 3 anti-nodes. 0289

Now do not get fooled. At any given point the wave is here or here, but you do not double count it. 0305

This is 1 anti-node, so 4 nodes and 3 anti-nodes and in any standing wave, you always have one more node than anti-node. 0310

So let us talk about how these occur in instruments. 0335

You can create a standing wave by holding a string in place at both ends and then introducing a disturbance, plucking it -- think of a guitar, a violin, a cello, a bass, a piano has a hammer that strikes those strings that are held in place. 0338

That is what is going to give you a sound that sets up a standing wave. 0353

Now, there is a lot to study about these standing waves though. 0357

Here we are going to look at a standing wave, the basic standing wave you get if you were to have a node on both ends, the strings held there and you have one anti-node here, so two nodes and one anti-node. 0360

This is what is known as the first harmonic and we will call that N = 1. 0372

It is the first harmonic or sometimes referred to as the fundamental frequency. 0376

You will notice that we only have half of a wave in there, so the length of our string is equal to half a wavelength. 0388

We could also have at the same time, a second harmonic (N = 2), where we have one full wave within that length (L) and that is called the second harmonic or sometimes referred to as the first overtone. 0396

In this case (L) is equal to the wavelength of our wave. 0415

We could have 1 1/2 wavelengths fit in there, and that is our third harmonic, sometimes referred to as the second overtone and now (L) = 1 1/2 wavelengths (3(λ)/2).0418

I could take this pattern here and I could try and generalize it by saying that (L) must be equal to N(λ)/2. 0439

We have 1(λ)/2, 2(λ)/2, 3(λ)/2, for N = 1, 2, 3, and so on. 0449

Well, if I wanted to solve then for the wavelength, I can rearrange this to say that λ must equal 2L/N. 0459

If I want the wavelength, it is 2 times the length of my string divided by the harmonic number. 0468

Or let us rearrange this a little bit more using what we know about the wave equation. 0474

If V = F(λ), I could rewrite λ as equal to velocity over frequency, so if I replace λ here with velocity over frequency, now I have velocity over frequency is equal to...0479

And I still have 2L/N and solving for the frequency, which is often times what we perceive as pitch, and we get that frequency is N (our harmonic number) times the velocity divided by 2 times the length of our string. 0494

So a couple of equations that can help us analyze what happens when we are talking about a stringed instrument. 0512

We have that one, we have the wavelength for N = 1, 2, 3 (whole numbers) and the frequency here. 0519

Now, not all instruments make music just by having a string. 0526

You can also create standing waves in open and closed types of tubes, like trumpets, pipe organs, flutes, clarinets, obo, and so on. 0532

Over here we are showing what we had for a stringed instrument and over on the right we are going to look at what happens when we have a tube that is open at both ends. 0541

You get the same pattern, but note here, we are showing the lines -- these are air pressure, not displacement. 0548

But over here, we now once again have half a wavelength in this open tube where we could have 1 full wavelength or 3-halfs a wavelength, 1 1/2 wavelengths. 0555

We have the same basic patterns, but the difference is that these are open at both ends. 0569

This is showing air pressure and not displacement. 0574

If we wanted to show the displacement, it would probably look something more like that, which is easy to see is still half a wavelength, but it is just off set a little bit, so open tubes and strings follow the same mathematical sequence. 0576

Let us take a look at standing waves in a partially closed tube. 0596

You can close the tube at one end only and you can still set up the standing wave pattern, but closed-tubes only produce odd harmonics. 0600

Once again, we will show air displacement here. 0608

We have a closed tube closed at one end and open at the other and now what we have is one-quarter of a wavelength inside this tube. 0611

Here we have three-quarters of a wavelength and here we have one and a quarter or 5/4 of a wavelength. 0622

This would be our first harmonic, this would be our third harmonic -- we do not have even harmonics when we have a tube closed at one end -- and likewise, this would be our fifth harmonic. 0630

I could generalize this with a pattern, that L = N(λ)/4 again, but now (N) must be odd, 1, 3, 5, and so on. 0647

Or rearranging again for wavelength, λ = 4L/N, where again N = 1, 3, 5 -- we only have odd harmonics. 0660

Pulling the same trick to find the frequency again, if V = F(λ), then λ = V/F and I can then solve for the frequency to find that frequency if NV/4L and again, N = 1, 3, 5, and so on. 0672

A lot to take in there, but I think it will get a little bit easier and we will do a sample problem with that here shortly. 0694

Interference from multiple sources -- this is a pretty cool effect. 0701

Waves from two or more sources reaching an observation point can lead to constructive or destructive interference depending on the position of the observer. 0707

Imagine we have a source (A) and a source (B) and if we are standing over here at point (C), because of the distances that these waves are traveling, they are reaching observer at (C), both at crest, so you are going to get constructive interference. 0715

It is the same sound wave, but an area where it is pretty loud. 0730

Move just a little bit though, so that now as far as the distances goes, (B) is here at a crest, (A) is here at a trough, which means you are going to get destructive interference, or for a sound wave, it will be quiet. 0734

Try this sometimes -- Take two speakers and try to put on a continuous tone at a set volume and then stand some distance away with the speakers pointed somewhat toward each other and walk in that sound field. 0746

As you move your head just a little bit, you may find areas where it is a little bit louder and a little bit softer. 0756

You can actually perceive that constructive and destructive interference pattern, which is pretty cool.0762

Beats -- When you have two sound waves with almost the same frequency, but not quite the same frequency, you can sometimes create what is called a beat pattern, almost a little bit of background ringing. 0770

You hear a slow, rhythmic change in amplitude. 0781

It is typically heard by humans when you have a frequency difference of 1-4 Hz and it is very useful for tuning stringed instruments. 0784

What you do is you try and play the same note on two different strings of a guitar for example. 0792

If they are exactly matched up, you hear one clear tone at a constant loudness. 0797

However if they are just off by 1 or 2 Hz, it is sometimes difficult to pick up that difference in pitch at first glance, but what you can hear as you play both notes at the same time is you will hear this rhythmic increase and decrease in the volume...0804

...the beat pattern that tells you, you do not quite have them in tune yet, so keep adjusting the frequency of those, the tightness of the strings until you get exactly what you were anticipating. 0820

By the way, what is happening when you are increasing the tightness of guitar strings -- well you are not changing the wavelength as you are set with that distance, but what you are really doing is you are changing the wave velocity because you are adjusting the medium. 0831

So you are increasing the tension; you are adjusting the velocity of the wave through that medium and therefore, you can get a different frequency. 0844

Let us take an example where we are going to look at how we use some of this new information on sound. 0853

The diagram below show 4 standing sound waves inside a set of organ pipes and we are going to assume the velocity of the sound in air is 343 m/s. 0861

We want to know the highest frequency for the wave shown, the lowest frequency, the longest wavelength, and the shortest wavelength. 0871

All right, so let us just do a full analysis of what we have going on here. 0878

Let us start over here where we have a tube closed at one end and it is pretty easy to see that if this is half a meter -- well, remember λ = 4L/N. 0883

Well, our (L) is 1/2 m, so that is going to be 4 × 1/2 m, so 4L = 2/N and that is 1 because this is a first harmonic, so our wavelength is 2 m. 0894

Or you could figure that out by realizing this is one-quarter of a wavelength and one-quarter of a wavelength is 1/2 m and if you want the full thing, you have to multiply by 4, so the wavelength is 2 m. 0907

Over here you have 1 1/2 waves in there and this would be a wavelength then -- if you have 3/2 in there of 2 m over that third harmonic -- 4 × 0.5 = 2 over the third harmonic -- is going to give you 2/3, so wavelength is going to be 0.667 m. 0918

Here you have 1 1/4 waves inside this half a meter, so as you go through the math here -- 4 × 0.5 = 2/5 (N) = 0.4 m (wavelength). 0940

Finally, we have a tube that is opened at both ends and we have one whole wave fit in here, therefore, the wavelength must be 1/2 m. 0960

Let us write that down -- wavelength equals 0.5 m. 0970

We found the wavelengths for these, so what is the longest wavelength for the pipe shown?0974

We can answer that -- That is 2 m. 0979

What is the shortest? That must be our 0.4 m. 0981

Now how about the frequency? To find the frequency, remember V = F(λ), therefore, frequency must equal V/λ, so the frequency over here of our left most tube -- let us call that tube 1, 2, 3, and 4...0986

For tube 1, frequency 1 is going to be equal to V/λ1 or 343 m/s/2 m (wavelength) for 172 Hz.1006

Frequency 2 is the same idea with velocity divided by wavelength, so we have 343/.667 or about 514 Hz. 1020

For this third tube, frequency 3, that will be our velocity 343 m/s/.4 m (wavelength) or about 857.5 Hz. 1034

Our fourth frequency here, F4 = 343/0.5 (wavelength) or 686 Hz. 1049

So what is the highest frequency for the wave shown? Well, that must be from tube 3, which is 857.5 Hz.1061

What is the lowest frequency for the wave shown? Well that is going to be F1 or 172 Hz. 1070

So we are putting into practice our open and closed tube sorts of problems. 1077

A musician is designing a custom instrument which utilizes a tube opened at both ends. 1083

Given the speed of sound in air as 343 m/s, how long should the musician make the tube -- so we are looking for (L) -- to create an (A) or a frequency of 440 Hz. 1089

As an instruments fundamental frequency, N = 1. 1106

Well, if you recall, frequency is going to be NV/2L and if we solve for the length, L = NV/2F, our harmonic number is going to be 1 because first harmonic is the fundamental frequency.1112

Our velocity of 343 m/s/2 × 440 Hz (frequency) will give us L = 0.39 m. 1131

If we make that tube 0.39 m long, we are going to get a 440 Hz A. 1146

A place of maximum displacement on a standing wave is known as...?1155

Well, if you recall on the standing waves, we have these areas with minimum displacement, called nodes and the areas with maximum displacement are called anti-nodes.1159

In this case we have 3 nodes and 2 anti-nodes and I get in a standing wave it is not at both places at once, it is going to be oscillating back and forth. 1181

At any given point in time, you might have it look like this and then a second later in time it is going to look like this. 1188

What are the ones called where we have maximum displacement? 1196

Maximum displacement is right there and that is an anti-node. 1200

One last sample problem. The diagram below represents two pulses approaching each other. 1207

Which diagram below best represents the resultant pulse at the instant the pulses are passing through each other?1213

When they interfere they are going to follow the Law of Superposition, which means their amplitudes are going to add. 1219

We have one positive and one negative and the negative is a little bit bigger than the positive, so what we are going to end up with is a net negative, but smaller than the initial, so 2 must be our correct answer. 1225

Thanks so much for visiting us with Have a great day everyone! 1240