Beats - Video Tutorials & Practice Problems

Beats

Hey, guys. So in the last couple of videos, we saw that waves can interfere with each other to produce special types of patterns. In this video, I'm going to show you a special type of interference with sound waves which is called beats. Just like the other videos, the most tricky part of this is actually visualizing what's going on with these interference patterns. So, I'm going to show you this step by step. I'm going to show you what exactly a beat is. So, let's check this out. We're going to come back to this in just a second here. What I want you to do is imagine that you have 2 sound waves that you're playing sort of like with 2 speakers at the same time. So I'm going to call this one sound wave a and sound wave b. Now, even though these are sound waves, we're still going to model them kind of like transverse waves because it's going to help us figure out or sort of visualize what's going on here. So, if you were to go ahead and calculate the frequency of this wave, remember frequency is cycles per second, you're going to find out that this frequency is 8 Hertz. You can pause the video and check for yourself. Basically, I have 8 sort of crest-to-crest patterns along this wave. If you do the same thing for wave b, go ahead and pause the video and count up the number of cycles that I have, you're going to get 10 Hertz for these 2 waves here. So what happens if you were to play both of these sound waves with these frequencies at the same exact time? They're going to interfere with each other. So what I'm going to do in this 3rd diagram is I'm actually going to overlap these two things on top of each other, these two waves. We end up sort of with some interesting sort of patterns here. You'll notice at the beginning of the wave, these waves are totally out of sync, but then there are other parts where the waves actually are going to basically line up to each other perfectly. There are other parts where they get out of sync, other parts where they get in sync again, and then so on and so forth. So these sort of alternating patterns are basically interference patterns. So remember that constructive interference happens when the waves are sort of overlapping on top of each other. So these are the points that I have highlighted in yellow here. So here are the points where you have constructive interference. Destructive interference is when the waves are canceling each other out. So destructive interference spots are basically the highlights in teal. So we have one here and then you have one here and then you have one all the way over here. So what we're going to do now is we're actually going to draw out what this net wave would look like. What would the superposition of these 2 sound waves look like? Well, to do this, the best way to do this is we have these two spots right here where the waves are going to interfere with each other. They're going to add up together almost perfectly to produce a bigger wave like this. So I'm going to have these 2 spots right here. And then in the teal spots, we basically have 0 displacement because these things are going to totally cancel each other out. So what happens in the middle? Well basically what happens is that from here to here, the wave is slowly getting in sync and the amplitudes are going to slowly start building on top of each other. So really, what this wave is going to look like is it's going to kind of look like this, where the wave size is getting bigger and bigger as you go to the right. Once you hit this point though, as you move to the right, the waves are getting out of sync. So basically the reverse is going to happen and this wave is going to get smaller and smaller until eventually it goes to 0. And then basically this whole pattern is going to repeat again. So you're going to have a wave like this. Whoops. I'm going to do this like that. And then here it's going to sort of get smaller and smaller and eventually go to 0. So we can see here that the difference between these sort of two patterns here and this net wave is that these two patterns always have the same amplitude. The amplitude is always the same along the same part of the wave, but in this part right here, this net wave has a varying amplitude. It goes from 0 up to a maximum value, and that's what a beat is. A beat is really just an oscillation in amplitude that happens for sound waves, and this happens when you have 2 sound waves with similar frequencies that interfere with each other. So the idea here is the only reason we weren't able to get this pattern is because we have 2 sound waves that had similar frequencies 8 and 10. If you had something like 15 and then 50, you wouldn't get something like this sort of wave pattern. Alright? So there are a couple of things that you need to know about these beats. The first is the amplitude. The amplitude is really just going to be twice the amplitudes of the waves that make it up. So the idea here is that if this amplitude is 1 and this amplitude is also 1, then that means that the net wave is going to have an amplitude of 2. These things are just going to add up to each other. Right? Now, the most important variable you need to know is the beat frequency. The beat frequency is the number of oscillations or the number of amplitudes that you hear in one second. So basically, it's the number of patterns that you get in one second. Remember, this whole thing happens in just one second. How many beats do we have? We have 1 beat here and then we have 1 beat here. So the equation to calculate this beat frequency is going to be the absolute value of the difference of the two frequencies of the waves that sort of make this, make this beat up. So the idea here is when we had 8 and 10, our beat frequency is just going to be the absolute value of 8 minus 10, and we get 2 hertz. Now basically, this 2 hertz should make some sense because we have 2 oscillations from 0 to amplitude and then down to 0 again. So it goes from 0 to max and then down to 0 again, and then up to max again, and then down to 0 again. So this happens twice in one second, therefore, we have 2 hertz. Alright. The last thing we want to talk about here is what the resulting sound would actually sound like. So if you actually have these 2 waves that were creating this beat, what would it sound like? Well, the resulting sound you hear is going to have a pitch. It's going to have a frequency of \( f_{sound} \), and this \( f_{sound} \) here is really just going to be an average of the 2 frequencies that make it up. But this \( f_{sound} \) is not the same thing as the beat frequency. The sound frequency is going to be the average. So basically, what we have here is \( 8 + 10 \div 2 \). So we're going to have a sound frequency of 9 hertz, but the loudness is going to vary at this beat frequency here. So the loudness is related to the beat frequency, but the pitch is related to the sound. And basically, this is just an average whereas this is a difference. So what this would actually sound like here is it would sound like 'wow wow' twice in one second. That's basically what a beat would sound like. Alright. So, let's go ahead and take a look at our example here. So, we have 2 musicians that try to play the same notes. One has a wavelength of 65, which has a wavelength of 65 centimeters at the same time. So \( \lambda \) equals 0.65. However, one of the instruments is actually out of tune, and so it plays a note with a wavelength of 65.4, slightly longer. So what I'm going to do is I'm going to call this 1 \( \lambda_{a} \) and then \( \lambda_{b} \) is going to be 0.654. I'm just converting this to meters here. Right? So I'm just moving the decimal place over. Now, we want to calculate the frequency of the beats that the musicians are going to hear. We're going to assume that the speed of sound is 343. So this is going to be our \( v \) here. This is going to be our \( v \). Alright. So how do we figure this out? Well, remember, the equation we have for the beat frequency is we're going to have to subtract. We're going to have to subtract these two frequencies and then take the absolute value. So let's go ahead and do that. So we're going to have \( f_{\text{beats}} \) is going to be the absolute value of \( f_{a} - f_{b} \). The problem is we actually don't have what these frequencies are, but we do know what their corresponding wavelengths are going to be. So how do we get from wavelength to frequency? Remember, we can just use the equation, the equation \( v = \lambda \text{frequency} \) here. So we have is sound waves. So we have the speed of sound which is 343. So what we have here is that we solve for the frequency. This is just going to be \( v \div \lambda_{a} \). So it's going to be 343 divided by 0.65. And what you're going to get is you're going to get 527.7, and this is going to be in hertz. Alright. So, the same thing for \( f_{b} \). If you go ahead and find out what \( f_{b} \) is, we're just going to get \( v \div \lambda_{b} \) and this is going to be 343 divided by 0.654. What you're going to get is 524.5 hertz. So we have these two notes that are being played, these 2 sound waves, and they're very very slightly off. Right? The difference between them is very very small, so they're going to create this beat frequency here. And so what happens is this beat frequency is just going to be the absolute value of 527.7 minus 524.5 and you end up with a beat frequency of 3.2 hertz. So basically, you would hear an oscillation from 0 to a maximum amplitude and you would hear that sort of pattern 3.2 times in one second. Alright, guys. That's it for this one. Let me know if you have any questions.