Standing Waves

by Patrick Ford
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Hey, guys. In this video, we're gonna talk about standing waves, which are types of waves that occur due to interference between overlapping waves. All right, let's get to it. Remember that waves along a string fixed their bounty at a boundary are gonna be reflected backwards as inverted waves. Okay, so these waves right here were a bunch of waves that had been produced by this hand sent forward and have already reflected their inverted. And remember that these inverted waves will have the same frequency as the incident waves as the initial waves that hit the boundary. Okay, now imagine holding the end of a fixed string. Oh, sorry. The end of the string fixed will wall and vibrating it right. We're going to go up and down. We're just gonna shake it, okay? We're going to produce a bunch of pulses that are going to travel from the free end towards the fixed end, and they're going to reflect backwards. Okay? These inverted reflections, they're gonna have the same frequency as they reflect off, and then they're gonna encounter new pulses that are being produced. Okay, forward pulses and the reflections since they're gonna be in the same space at the same time are going to interfere. Okay, Now, normally, what's going to be produced is some sort of uncoordinated vibration. Okay, Essentially nothing. Just a string that's doing a whole bunch of this. But it's specific frequencies. You are going to produce what are known as standing waves. Okay. I drew one instance of a standing wave here. In another instance of a standing wave here, the first standing wave that I drew is a wave that essentially looks like this. Okay. And then if you go faster, one part is gonna go down as the other is going to go up like this. That's the second one that I drew. Okay. Thes air called standing waves. Because they appear to be standing still. Instead of individual wave pulses that are moving forward, we now have just a pulse that's going up and down or going up and down, alternating like this. I don't have three hands, so I can't really do another standing wave which would be formed of three pulses. Okay. Two types of standing waves exist. Node node, standing waves and node anti node standing waves and different scenarios produce either node node or nodes. Anti node. What a node is is It's a point of no displacement. Okay, I highlight. I bolted and underscored this right there. So that will help you remember what a node is, Node. No displacement. Okay, An anti node is the opposite of a node is just a point of maximum displacement. So if we look at the standing waves drawn, this is a node. And this is a note. This is what we would call a node node standing wave because both ends of the string are notes, and then this point is an anti node. Okay, here we have three notes. A node, a node, another node. This is still a node node standing wave because both ends of the stringer at nodes and we have to anti notes. Okay, Now, let's look at the differences between node node and node. Anti notes, standing ways. Okay for node node, standing waves. As I said, Both ends or notes for node anti notes, standing waves, one end is a node. The other is an anti note. Okay, so I drew pictures off the three largest wavelength or lowest frequency of each. If you see here on either end, the wave returns back to the horizontal axis. It returns back to a node. Here on one end is a node, but on the other end, it's always gonna be at the amplitude. You can see here. This is the amplitude. It's the point of maximum displacement. Okay, Because of that, wavelengths and frequencies have to abide by equations. Okay, for node node standing waves. Theologian wavelengths are given by this equation. Where n is any integer, Okay. And the allowed frequencies are given by this equation. Where in is any integer? Okay, as long as your wavelength or your frequency agrees with those equations, it's a standing wave. If you have a frequency, for instance, if you whip your hand at a frequency that does not follow this equation, you will not produce a standing way. If you're just going to produce a bunch of uncoordinated vibrations, that don't mean anything. All right, I'm gonna minimize myself for node anti notes, standing waves. The allowed wavelengths obey this equation, but in has to be odd. This is crazy important to remember that in which is called the harmonic number we call in the harmonic number is any integer for node node standing waves. But it has to be odd. You can only have odd harmonic numbers for node anti node standing waves. Okay. And the allowed frequencies for note anti notes. Standing waves are given by this equation Where once again, the harmonic number has to be odd. Okay, that's crazy important that you remember that the harmonic number must be odd. All right, let's do a quick example to finish this off. You want to produce standing waves on the string shown in the figure below By vibrating the end, you're grasping. What is the third largest frequency? Produce a ble on the string. If it has a massive 0.5 kg and attention of Newtons, assume there's no friction between the ring and the end of the string. Sorry. The ring at the end of the string and the pole. Okay. What kind of standing waves do you think these are gonna be, guys? Your hand is gonna be a node. But this end, which is free to move, is gonna be an anti node. Okay, so this is a node. Anti node. If we want to alert the third largest frequency. We have to use this equation. What's the third largest harmonic number for a note? Anti node, Remember, In has to be odd. So it's 13 etcetera. The third largest is five. Okay, It's not three because two doesn't count. The third largest is five. Okay, So the fifth harmonic, which is the third largest harmonic in this case, is five V over four l and all we have to do is figure out what V is the speed of this wave on this string. We know that the string has a mass of 0.5 kg and attention of 150 Newtons and we know it's length is 15 centimeters. So the speed is gonna be the spirit of tea over meal which is the square root of 150 Newtons. The tension divided by 05 the mass over the length 15 right. And that's going to be 21 to meters per second. Now that we know the speed we can solve for the frequency, this is just gonna be five 21. over four. The length is 0.15 and so that is 177 hurts. Alright, guys, that wraps up this introduction into standing waves. Thanks for watching