Mechanical Waves

Ajouté : 2026-05-15

[00:00:00]is going to be based on wave especially mechanical wave. Five properties and five characteristics. Five properties would be wavelength, period, frequency, velocity and amplitude. And five characteristics of sound wave which is a mechanical wave would be reflection refraction refraction after effect and finally interference which has a phenomena called superposition.

[00:00:32]Now finally we're going to do a case study standing wave which which creates the harmonic. Let's dive in. We're going to start our conversation with pulse just a single disturbance. Pulse can be modeled by amplitude ai is equal to or a final is equal to a initial e rt it's the decay or growth.

[00:01:00]One of the standing wave phenomena is resonance. And resonance when something vibrate. When you accomplish a natural frequency of something then is amplify is amplitude.

[00:01:18]So that can be modeled by these equations. Pulse. Now a continuous pulse is what we call wave.

[00:01:26]So wave wave can be modeled by yx comma t uh is equal to a sin kx minus omega t. Here a is the amplitude.

[00:01:48]Uh k is the wavelength uh web number.

[00:01:55]Omega is um uh angular frequency.

[00:02:05]All right. And what is the relationship between angular frequency and frequency?

[00:02:10]So omega is 2 pi f. So frequency then is of course omega / 2 pi. And what is the web number? Web number is 2 pi over lambda.

[00:02:27]What is the angular frequency? 2 pi over big t. And the difference between big t and small t they both si unit is second and this one s a unit is second. But big t is right. If you have um u 360° starts from 0° to 360° then you can model it like this. Right? So is this a big t? No, this is a small t. Is this a big T? No. This is a small small T. Is this a big T? Well, it's close to big T, but not not yet. But is this a big T? Yes. So big T is the time it takes to complete 360°.

[00:03:07]All right. Or one revolution or one cycle. Okay. So now we going to talk about two types of work. So what is wave? Wave is a disturbance that carries energy and information from one location to other location. and it doesn't carry the mass. Do not forget. So we're going to talk about two type of waves. One is the first one. First one we're going to talk about mechanical wave and then the second type we're going to uh talk about electromagnetic wave mechanical waves. Uh there's medium. So we quickly write it this medium doesn't need medium.

[00:03:49]medium.

[00:03:52]No, it does not need medium. Mechanical waves, we're going to divide them in two part. Number one, you're going to say transversal and then longitudinal.

[00:04:08]The EM wave is just transversal.

[00:04:15]the transversal wave, a longitudinal wave and uh transversal wave the wave in general they do have uh five properties before we talk about the properties transversal wave look like this right so what happen a particle moves perpendicular with respect to the motion of the wave longitudinal wave look like is so a particle moves parallel with respect to the motion of the wave. If it is a longitudinal transversal wave, an example of transversal wave is water wave. An example of longitudinal wave is a sound wave.

[00:05:06]The wave has crashed and trapped.

[00:05:13]Now if you have a wave, so let's see. Let's count one, two, three. Three cycles, right? From here to here is what we call lambda web length.

[00:05:27]So lambda is a web length measuring as a unit is of course um meter but the definition of lambda is uh one point to other point in pace okay so I'm going to draw a few points and you want to see one point 2 point 3 point 4 5 6 uh 7 8 9 10 11 12 13. Right?

[00:06:00]So I'm going to quickly put a point over here. 1 2 3 4 5 6 7 8 9 10 11 12 13. Now number four makes a lambda with number eight. Why? They are equal in um well they are so four and five will not make a lambda. Why is that? Because four and five are not a 360°ree apart. Number one. Number two, they are not uh they are not they are not 360° apart and they are not in phase. So 4 and 8 they are in in phase because the velocity the magnitude of the velocity and direction of the velocity are the same. 4 and 8.

[00:06:43]What about 4 and 12? The magnitude of the velocity are the same. But hey they are more than 360° apart. So what is the definition of wavelength? they have to have 360° degree apart and they have to be uh what we call uh in pace. So again another example is three and seven and 11 are uh uh uh is 3 and seven wavelength are three and 11 wavelength.

[00:07:12]Well 3 and seven are webband because they are in pace and they are 360° apart and seven and 11 is a wlength. Why?

[00:07:19]Because they are in pace and they are 360° apart. Okay. So this is an example of the wavelength. Okay. So now I'm going to give you uh an equation for in the in this case the wavelength is this is compressed and this is compressed.

[00:07:37]Compression to compression is a wavelength. And if I give you a real fraction or less compressed less compressed to less compressed is also a wavelength. Now talking about uh the water wave and this is sound wave and this is light uh you know emw wave is the light wave. So if I put light over here I don't know let's say red light and blue light.

[00:08:01]Red light and blue light. And in this case I'm going to just put water or sound. Yeah I'm going to put both. I'm going to put water and sound. This is sound. This is water. An example of transversal wave. This is sound. an example of longitudinal wave and I have a red and blue an example of em waves.

[00:08:20]Now as you know water moves um with u um uh 1500 uh m/s right water moves 1500 m/s right sound moves through the water uh that's what I mean so if I put u three different uh uh three different medium one is uh let's say solid this is sound wave this is one is liquid liquid and one is gas the sound wave moves uh 5,000 uh k uh m/s and uh 1500 m/s and gas you have uh 340 331 331 m/s at 0° temperature. So the equation would be the speed of sound equation would be velocity is 331 + 6t and that is that means if it is 15° then velocity would be 331 + 615 and that would be velocity 340 m/s. What we're going to do now is next I'm going to give you the velocity of this one.

[00:09:31]The velocity of this one is going to be same. So in this case we're going to say that the red of course lambda is big and the frequency is small and for the blue lambda is uh uh uh frequency is big so lambda has to be small. Now lambda for uh for the red is big is 600 nanometer and lambda is small 500 nanometer and this one red uh the frequency is very small we going to say 5 5 * 10 to 14 htz and that is 6 * 10 to 14 htz. So we're going to come over here. We're going to write very small lambda which is 600 600 nanometer.

[00:10:18]So times 10 to9 and frequency 5 * 10 to 14.

[00:10:27]All right. So what is it? Um uh is 3 * 10 to 8 m/s. In this case big F big F is 6 * 10 to 14. and then the small lambda which is 500 * 109 which is 3 * 10 to 8. So in this case you see that the speed of light is also same. So what what what does that mean?

[00:10:53]speed of light. If you have the spectrum, in this spectrum you have the lambda. Lambda measures the wavelength and frequency measures the um the the f measures the frequency in hertz. Lambda measure the wavelength in uh in meter.

[00:11:10]So you have this is gamma ray and you have x-ray.

[00:11:16]This is ultraviolet. This is visible light.

[00:11:20]Uh this is uh you have the infrared, this is a microwave and this is radio wave. So what do you what do you see over here is big f. So this is over here you have a big f and then you have a small lambda very small lambda. Over here you have a big lambda and very small f. And over here you have a red and you have a violet.

[00:11:50]So red you have a big lambda. All right, very big lambda. So I'm going to make a very big lambda and you have a very small lambda. Small frequency, big frequency just like over there. So this is uh what does that mean? That means that speed of light is always going to be constant. But but the speed of light not always going to be constant when it moves through the uh the medium. The speed of light is constant when it's moved through the vacuum. So let's take uh understand that phenomena. So how can we understand that phenomena? So we're going to write over here the speed of light just like a speed of sound we wrote the speed of sound over here three different medium medium solid uh liquid and gas. And we're going to say that vacuum and we're going to say um uh air and we're going to say water.

[00:12:50]All right. For vacuum n is one. For air n is one. For water n is 1.33.

[00:12:57]And n is the index of refraction. Index of refraction. When light change the medium it bends. That is called index of refraction. uh the solid we say that this is 5,000 m/s. This is 1500 this is 1500 uh m/s and this is 331 m/s. This at 0° C uh the if the temperature go up of course the speed go up. This is the speed of sound. This is sound.

[00:13:31]Sound is a mechanical wave which needs medium.

[00:13:38]The light is uh not uh mechanical wave is electromagnetic wave. It does not need medium. It does not need medium. No medium needed. But it can go through the medium. Yes. Yeah. Now if it does go through the medium, it slow down. Let's see over here. N is C over V. So V is C / N. So V is C / N. So V is C / N. Now C, we already discovered what C is from our electricity and magnetism. The C is velocity is 1 / epsilon * mu. And then velocity is epsilon * mu which is 3 * 10 to 8 m/s. And when something move with this, this one come from electricity, this one come from magnetism, then we call it C. So then we're going to come over here 3 * 10 8 / 1 and then 3 * 10 to 8 / 1 and then we're going to say 3 * 10 to 8 / 1 not one this is 1.33 so in this case yeah speed of sound is 3 * 10 to 8 the speed of sound is so we're going to say b c 3 * 10 to 8 but this time we're just going to say b v is 2 * 10 to 8 so you see I'm I'm not sure whether this is 2 or 2.3 maybe 2.3 uh 2.3 * 10 8 but you want to check by calculator uh so this time we're going to call it C is because the speed of light is same this time we're going to say C because this is the speed of light when it is three * standard speed we don't call it B we call it C now the big picture is this now we're going to call we're going to talk about the properties and then we're going to dive into the five uh prop uh five characteristics of sound wave Now the the uh this the mechanical wave properties all the wave properties wave properties wave properties now let's say wave what is the wave wave is sinx comma t let's say you I said that a sin kx minus omega t so this is the wave moving to the right how do I know this is moving to the right because it's a negative sign so First put some number in it. Uh is equal to.1 sine and then put kx let's say 5 pi x minus 20 pi t. Now what does that mean? You have a very small amplitude. You have a um okay let's find out and then we going to be able to quantify how small how big. So we have amplitude. Amplitude is.1 mton.

[00:16:23]We have uh k is 5 pi.

[00:16:28]We have omega. Omega is 25 pi. Now we're going to find the lambda. What is lambda? So lambda is k is this is web number. K is web number. Web number is this is k.

[00:16:41]So this is this is k. So what is the k? K is 2 pi lambda. What does that mean? How uh how much a wave moves uh moves in 360° in this space and what is uh omega?

[00:17:01]Omega is so let's let's remember this then you will never forget 2 pi big t.

[00:17:08]So what is omega? This is omega and this is k. This is k. This is omega. This is k. What is omega? How fast? How fast wave is spin? How fast wave is spin? How fast a disturbance how fast a disturbance move 360° in the time and how fast uh a disturbance move in 360° in this space. How fast a wave or disturbance move 360° in the in time?

[00:17:37]How fast a wave or disturbance move 360° in space? That's all the k is the wave number. How fast a disturbance move 360° in this space? How fast a wave moves 360° in time? That's web and web number and and angular angular angular frequency. We going to find the web number right now. So k is 2 pi over lambda. So then lambda is 2 pi / k. So lambda is 2 pi and k is given 5 pi. I pi cancel. So two goes to this one. I don't know.

[00:18:18]Okay. So that's a hassle.

[00:18:21]Um this is five.

[00:18:26]This is two.

[00:18:28]So this is uh 04. Okay. 04. So lambda is 04 m.

[00:18:37]Okay. So lambda is 04 m. Now we're going to find the frequency. What is frequency? Omega is 2 pi f. Okay. So then f is uh omega over 2 pi. Omega is uh 20 pi 25 pi over 2 pi. I pi cancel this is 10. So frequency is 10. Of course you didn't have to find the frequency like this. You always can find the frequency one over big t. So frequency is one over uh big t is uh only god knows what is big t. We're going to find the big t. Now what is big t? Uh omega is um 2 pi over big t. So big t is 2 pi over omega. So big t is 2 pi. Omega is how much? 20 pi.

[00:19:27]So big t is.1. This this cancel this is 10. So 0.1. So big t is 0.1. Of course you can find this way. Frequency this way. No matter how which way you find it's going to be the same. Now velocity we're going to find different way.

[00:19:43]Velocity we're going to find uh we're going to find velocity is distance over time. We're going to find velocity is lambda time frequency. We're going to find velocity is uh omega over k. So let's find the most easiest way possible. D measuring distance lambda measuring distance. The t when t goes to be t. So lambda is how much? 04 big t is how much? Uh 0.1. So velocity is four.

[00:20:13]So velocity is four. Uh lambda is 04 and frequency is 10 which is also four. We already wrote that omega is 20 pi and k is 5 pi. So velocity is four. We already wrote four. So this is all the five properties of wave. amplitude uh wave number uh omega uh lambda frequency uh velocity. What else missing? I have one 2 3 4 5 6 uh did I put uh yeah okay uh uh frequency. Oh, we we forgot to put the period. Period is we found the period. period is I believe uh what what did I find the period? We found the period anyway. Uh period is um so omega is 2 pi / t.

[00:21:14]So period is 2 pi over omega. So 2 pi omega is 20 pi pi pi cancel 10 oh.1 so period is 0.1 second. So these are the properties properties of the wave.

[00:21:32]Uh now we're going to do the the the the characteristics of the wave and we're going to do only sound.

[00:21:41]So characteristics of the WAVE characteristics of waves. All right. Now number one reflection.

[00:21:53]Number two, uh, refraction.

[00:21:59]Number three, defraction.

[00:22:03]Number four, Dopper effect.

[00:22:09]Number five, uh, interference.

[00:22:16]number the case study uh this standing wave or we're going to say guitar and the phenomena we're going to study called resonance.

[00:22:31]Now reflection the equation for refraction is theta initial is equal to theta final. The refraction is theta initial is not equal to theta final.

[00:22:44]That's what makes it reflection.

[00:22:46]Defraction we're going to say uh the equation is if lambda is greater than whole or gap maximum defraction uh Doppler effect. Yeah, Doppler effect um high pitch means so the equation is uh uh this is uh then velocity of sound uh plus minus velocity of the observer uh velocity of sound plus minus velocity of the source.

[00:23:25]interference. Uh the equation for interference is uh you have uh uh y x comma t is equal to sine.

[00:23:36]Of course you you should write amplitude amplitude sign uh uh kx - omega t kx minus omega t. Um and the guitar equation uh there are few guitar equation uh we definitely going to derive them but yeah they are not I have to derive them they are not in the top of my head. So let's go let's go over uh and since I don't have calculator uh I'm going to hopefully not make mistake but hey I'm not going to be able to do 10 decimal place in my mind without calculator. So reflection. So what is reflection? I already wrote this one.

[00:24:18]Now we probably going to see two example. One example is let's say you have a man standing over here and the distance between these two is 500 m and now the man uh send a signal by saying something and the signal come back in 2 seconds. Right? And the outside temperature is 15° C. man want to find the distance between uh the wall one to the men which is location A let's say okay now very simple if it is 15° C plus find velocity so 331 + 6t and that gives you velocity very quickly the speed of sound now the second thing velocity is distance over time is no good 2D over t why is that because this is echo d1 D2 D1 is equal to D1 is equal to D2. So is equal just D. So D1's D twice. So D + D. So it's 2D. So now 2D is equal to VT.

[00:25:23]Uh is that right? 2D is equal to V_T.

[00:25:25]Yeah. D is equal to V_T / 2. Now V is 340. T is 2 and this is 2. So 2 to cancel 340. Now this is not answer 500 minus 340. So the distance of this one is 500 minus 340 and that is your answer. Okay, this is like the reflection. Another uh another one you can say that say this is the captain of the ship. This is a ship. Uh this is a Pacific Ocean, right? Uh the captain use the sonar technology to send the signal to the down bottom.

[00:26:05]Now this signal goes with 0°. Why is 0°?

[00:26:09]Because hey this is a normal line and this this angle is zero with respect to the normal line here which we're going to call incident ray and this we're going to call incident angle and this blue of course the reflected ray and we're going to call this one reflected angle and the incident angle is equal to reflected angle. The echo takes echo takes 6 seconds. Now we want to find the depth of this ocean which we call Pacific Ocean. Again velocity is 2D / T. Now 2D is V_T and then we're going to say D is equal to VT / 2. Now V since this is water and the temperature is 15° C or something like that close to you we're going to say 1500 * 2 and then divide by two distance is um well not time 2 * 6 I put somewhere but did I put it takes 6 second if I did not put I'm going to put now so time takes oh I wrote it I wrote over here 6 seconds so 6 seconds so 2 goes to six three times so 4.5 kilometer So what is the distance? Distance from ship to bottom is 4.5 km. So this is reflection. Now we're going to do refraction. Refraction is also very simple.

[00:27:37]Everything I'm going to use try to use two uh two u what do you call two examples so you understand hopefully you understand better. Uh so you have a water uh and you have I don't know uh air and this is a normal line and this is a fish. The fish uh the fish uh this is incident angle and this is this is incident angle. This is incident rate and the fish sends the information to other fish 30° angle. Now this angle I'm going to move toward the normal. Why is that is because that's how it refracted. So we're going to find the refracted angle. This is called refracted angle. So now let's find the refracted angle by SN sin theta 1 / V1 is equal to sin theta_2 over V_sub_2. So we're going to cross multiply or just plug it in. Sin theta 1 is sin 30. V1 is 1500 and this one is sin 32. I mean no.

[00:28:42]sin theta 2 is b2 is uh velocity is 340 assume that 340 m/s assume that this is 15° so we're going to say this is 340 now we're going to say fine theta 2 is equal to this is 340 and this is 2 so this is 170ide by 1500 0 cancel so theta_2 is equal to sine inverse 1750 and then s theta 2 is equal to 6°. So this is 6°. What does that mean? You going to be better off if you are on the tree so that you can uh enjoy the fish conversation. Other example you can give that u uh during the morning for example during the morning and during the night so this is 900 p p.m. and this is like 5:00 a.m. 5:00 a.m. you shout you shout with uh I don't know um 30° and 5:00 a.m. the temperature is not that much is cold over here and is hot over here on the top. So when you send the information right when you shout is going to bend toward what what to what a away it's actually going to bend away from the normal. So it is bending away to the normal and then if you go another 100 meter from the from the ground then it's going to bend away from the normal.

[00:30:12]If you go another 200 m from the ground and you're going to say it's going to bend again. So if you remove those thing if you remove those thing what are you going to see if you remove all the destruction then you're going to see that it is um it is what it is bending and it is bending and it is make a soar type of trajectory uh coming back that's why in the morning uh what do you see you see uh you see uh noise so much noise and then you prove it by say if you are 400 m per second.

[00:30:48]You prove that this is moving away from the normal. This is closer to the normal 30°. This must be more than 30°.

[00:30:57]Whatever it is, this is more than 30°.

[00:30:59]And you can do it by sin theta 1 v1 is equal to sin theta_2 v2 by plugging in sin theta 1 is sin 30. Now this is 300.

[00:31:10]Sin theta_2 you can definitely find by V2 is 400 because the here cold cold temperature has to be smaller than the hot in the other hand 900 p.m. If you shout that sound not going to make a parabola. That sound not going to make a parabola. It's going to go closer. It's going to go closer to this normal. So it's not going to come back. So that is another example of reflection. Now we're going to go to defraction. Defraction is sometime I often say hey I'm going to take you to the moon. Do you see me? You say no. Do you hear me? You say yes. Now sometime I say hey I'm going to take you outside the class and close the door.

[00:31:49]Ask you the same question. Do you see me? You say no. Do you hear me? You say yes. So why the two questions you don't see uh me here is because the light doesn't bend but you see me on the moon because light doesn't need medium. You hear me on the moon because uh there you uh you don't hear me on the moon because there is no medium. You hear me over here because sound can man find a hole.

[00:32:16]That's all about it. So sound can find a hole. Sound can escape if there is a hole. Now in a in a room like this there is always a hole. So sound always can find uh find find a hole a gap and escape. So now what we are talking about we are talking about uh a defraction defraction. So let's say frequency and then let's say I don't know lambda.

[00:32:39]Lambda is v over f and then let's say I don't know gap. And then let's say what what are you going to talk about um lambda versus frequency and finally you want to write defraction.

[00:32:53]Defraction is uh bending of sound wave.

[00:32:57]Bending of sound wave right? Bending of sound wave. Now let's say you give uh six 680 hertz and you give 340 hertz and you give 170 htz.

[00:33:11]Well, not you like a loud speaker. This comes from loud speaker. So if this comes from loudest speaker, you have 15° C. That means 340 m/s. That means 340 / 680 is going to be.5. 340 / 340 is 1.

[00:33:30]340 / 170 is 2. The gap is 1 m. 1 m. 1 m. Then over here you see is lambda is smaller than the gap.

[00:33:42]Obviously lambda is smaller than gap.

[00:33:44]Should I write gap like this? Otherwise you don't confuse with acceleration due to gravity. Lambda is equal to gap.

[00:33:51]Lambda is bigger than gap. So now gap.

[00:33:56]So now we're going to say almost almost no refraction minimum defraction maximum defraction. How do we draw it?

[00:34:08]If I say how many wave over here you're going to say let's count one two three uh four five. You're going to say five waves. Okay. So five waves. Then how many uh lambda over here? Of course five lambda. This is one lambda. Two lambda.

[00:34:23]Three lambda. Four lambda. Five lambda.

[00:34:25]So this is how you can draw five lambda.

[00:34:28]But this is also you can draw five lambda. This is one. This is two. This is three. This is four. This is five. So this one and this one are same. This one you call web. This one you call wave front.

[00:34:45]Okay. Great. Now to make it more visible, what you do? You put a line so that it is is it looks more visible. So this is wave frrons. This is wave frrons means this is the distance between two crust to cast. Then that is wavelength.

[00:35:01]And this is for scenario one. So this is one 2 3 4 5. This is scenario two. This is one 2 3. No I don't I don't have that much space. One two three four five. So spacing. You see that the spacing is one two three four five. So this is scenario two. This is scenario three. More or less because I am struggling with the spacing. So these are the web pron scenario one. These are the web fronts for scenario two. These are the web for scenario three. Now you're probably going to say where's the hole? Here's the hole. Hole. We're going to keep it con keep it constant. And that is one meter. Uh and then I'm going to change the color so it it doesn't get confused with the other thing. And I'm going to keep uh Okay. So here the this is almost no bending. This is almost no bending.

[00:36:00]This is little bit bending. Why? Why is that? Is because there is uh the gap and lambda are the same. So there is some bending some defraction here. The lambda is bigger than the gap. So this is maximum defraction. Maximum defraction means if someone is standing up over here, someone will be able to listen the music coming from the loudest speaker over here. If someone is standing over here, someone will not be able to someone will miss this the source. If the music come from the source, if someone is standing over here, forget about it. This one not going to work anyway. So this is the defraction. Now we're going to go to Doppler effect. Now if you check our list, you'll see that we are almost done with our list. So now what is Doppler effect? Doppler effect is uh we often say uh Doppler effect. I often say to my student, hey, this is Doppler effect.

[00:37:01]Okay, that's it. That's the Doppler effect. The blocking the ear is the Doppler effect. Why is that? I block my ear to save it from high frequency. I block the ear to save it from honking.

[00:37:10]block the year to save it from the uh the the the the the small lambda. Small lambda means big F.

[00:37:19]Big F means high frequency. High frequency means high pitch. High pitch means you plot the the the F. So that's what it is. Now how can you convert it to math? Well, you can first you're going to draw the diagram and convert it to math. You draw a truck on the road. You draw two observer. Observer one at rest.

[00:37:41]Observer two. This is observer one.

[00:37:44]Observer two. Observer two. Observer two at rest.

[00:37:51]Now the car is moving with velocity 40 m/s. The car is honking with with uh with uh with uh 80 hertz. Right. Car is moving 40 m per second. Honking with 80 hertz. Now this guy will block his ear block here. Why uh why this block uh this guy will block his ears is because of this. Hopefully you're going to try to understand it. So this is why he is blocking his ear. Now over here you see this is called lambda. If small lambda over here this is big lambda. Big lambda. I'm going to draw it over here.

[00:38:32]This small lambda means big frequency.

[00:38:34]Big lambda means small frequency. So that the speed of sound remains same at 15° C. The speed of sound remains same at 15° C. And um at at 15° C uh remember the honking needs the medium. Honking is a mechanical wave. Why is that? Because honking is sound. The honking needs the medium who is the ear to move. So this is the year uh speed of the speed of the speed of the air uh speed of the speed of the sound in the air. Air is the medium. Here is the medium. This is the speed of the sound. Okay. Great.

[00:39:17]Okay. Now this going to uh experience high pitch and this is going to experience the low pitch.

[00:39:26]Uh now high pitch is so this is observer two. is going to say uh this is uh uh after uh this is after this is the initial initial frequency. This is velocity of the sound plus minus velocity of the observer uh one velocity of the sound uh plus minus velocity of the source. Now remember top has to be what? Big in order for this one to block the ears or the high frequency or high pitch.

[00:40:04]Okay. So, so our math is very simple. 340 + 0 y 0.

[00:40:12]This guy is at rest. 340 y + 40. I have to make the bottom big so that my top is small. If my top um I have to make the bottom is small. So my top is big. If I have to in order to make the top big I have to make the bottom is small. So then this is 900 Hz.

[00:40:33]Okay. In order to make the top big I have to make the bottom small. So that this is 900 Hz. This is this guy perceive more than what the source really is. Okay. That's why this guy call it high pitch. That's why this guy is blocking the ear. This guy doesn't care. He doesn't block the ear. So this is frequency observer 2 is equal to frequency initial and of course v sound plus minus uh observer 2 uh and then uh this one is v sound and plus minus and this is the velocity of the source. Now this the top has to be small right because this guy doesn't care and it's going to be low pitch low pitch. So this is 800 this is 340 and this is plus zero because this guy is at rest and this is 340 and of course this has to be bottom has to be weak. Uh so the top is small.

[00:41:31]So then this guy is careless. So now this is 700 htz. The 700 htz he doesn't need to block his ear. He did not. So this is what we call doper effect. Now let's go to the interference and then we're going to do the guitar. Uh we're going to play some guitar and then we leave. Um and then I I cannot speak anymore anyway.

[00:41:54]Interference. So what is interference?

[00:41:56]Interference is like you know superposition. Interference when two object like uh at the same location at the same time right? when I and you talk at the same location. When I and you talk at the same time, then there would be a location where both sound wave uh are interference. Uh okay. So scenario one. Scenario one you're going to see uh like this. Oh, let's do it very nicely.

[00:42:23]Uh so this is let's say 2 m uh 2 m. This is a 4 m. This is say -2 m and so on.

[00:42:30]This is a 2 m. This is say 4 m. This is -2 m. So on you have a wave and you have a wave.

[00:42:40]Wave A moves this direction. Wave B moves this direction. And wave A moves this direction. Wave B moves this direction. Let's move four A and B.

[00:42:57]So what is A + B?

[00:43:01]A plus B would be A + B would be uh at some location where the two meet uh is going to be uh this is this this one is two this one is four is six six so a + b what is a plus b over here a + b if a goes this direction b goes this direction a plus b is zero it doesn't exist so this is interference to understand interference we're going to understand the standing wave finally we studing standing wave So, I'm going to erase the I'm going to Oh, wow. No.

[00:43:35]Yeah, I'm going to erase this. Um, so the standing web to do the standing web, we're going to start like this. So, this is a wire. You need a string to do the standing wave. So, so what are you going to do? Have to do have what are you going to have to do with the string?

[00:43:52]You're going to have to block it. You're going to have to use the barrier. If you don't use the barrier, this is a pre-string. Pre-string you cannot apply tension. If you don't apply tension, it's not going to make any notes anti harmonic. So you block it. You create a barrier. So they are the same string. So the length is same. We're going to say L1, you're going to say L2, you're going to say L3. Just because you move it up and down or you create different harmonics, you know, first harmonic is not L not going to change. So L1 is equal to L2 is equal to L3. So we're going to just say L. Now I'm going to say this one you know what is this one?

[00:44:30]This one is very simple half lambda. So what is plus harmonic? Plus harmonic is one half lambda. Second harmonic two half of lambda third harmonic three of lambda four half lambda and so on. So this is the first half lambda. Since this is not a free string it cannot keep going. This incident ray cannot go keep going going forever because there is a barrier. So he's going to bounce back with reflected angle right and keep going making a loop appears like making a loop. So what are we going to say? L is equal to half lambda. It is what it is. Now this is of course two half lambda. So we're going to say L is equal to 2 half lambda.

[00:45:13]2 half lambda. So that's how we're going to say two half lambda. So of course it cannot go on forever because this is not a free uh this is not a free string.

[00:45:23]There is a there is a barrier over there. This is what we going to call three half lambda. So we're going to say L is three half lambda. Three half lambda. So okay three half lambda. So of course this also going to make a loop.

[00:45:38]Now a few things we're going to do before we do uh this we're going to say node. Node means doesn't move. Node is n. Now if you have a free uh if you have a free string right the free string this one velocity is max.

[00:45:53]Now over here velocity is mean. Why?

[00:45:56]Because the velocity is equal in magnitude opposite direction over here.

[00:46:00]Right? This is the intersection between the incident ray and the reflected ray.

[00:46:07]So they cancel each other. So velocity over here is zero. So velocity here is zero. Velocity here is zero. Velocity here is zero. Velocity here is zero.

[00:46:14]Instead of saying velocity is zero, you say node. Node is where velocity is zero. How many node? Two. At two location velocity is zero. At three location velocity is zero. So you say node is three. At four location velocity is zero. So you say node is four. N for node. N is for node. Now here comes the loop. How many loop? One loop. How many loop? Two loop. How many loop? Three loop. Loop is another way calling anti-node. Anti node is easier than loop. Why that is? Because antiode is opposite to node. Node is where velocity is zero. Anti node is where the displacement is maximum meaning height is maximum meaning where it the amplitude is it is between negative a to a. So amplitude is maximum. So this is anti node. Anti node is being n. So being n is of course one. Bing n is of course two. Big n is of course three.

[00:47:10]Now k k is index. Index is zero over here. Index is one over here. Index is 0 1 2. Index is 0 1 2 3. Uh okay. So few things we're going to do before we uh do it. So we're going to make the ln. So what is ln? We're going to put one over here because we can. We're going to remove this two over here because we can. We're going to remove this three over here because we can. So that's how we're not going to say the standing wave is uh the first harmonic is this is first harmonic this is second harmonic this is third harmonic so a standing wave we're going to say what are we going to say we're going to say the first harmonic is half lambda the second harmonic is two up lambda third harmonic is three lambda fourth harmonics instead of saying this we're going to say n lambda n lambda instead of saying n lambda you're going to say n lambda / 2. That's it. That's the equation. Now, since we already know uh the relationship between the guitar and the L, we're going to try to find the relationship between guitar and lambda. So, lambda is of course you see that 2 L. In this case, lambda is 2 L / 2. In this case, lambda is 2 L / 3. What is N lambda? So lambda is 2 L. Lambda lambda 1 is 2 L. Lambda 2 is 12 / 2.

[00:48:42]Lambda 3 is 12 3. Lambda 4 is 12 4.

[00:48:45]Lambda 5 is 12. Now what is lambda n? If I put one over here, I made it obvious.

[00:48:51]You can say 2 L / N. N is for anti-node.

[00:48:59]and also is harmonic.

[00:49:03]Okay. Now, few things we need to uh take care. Velocity is this is equation one.

[00:49:10]We're going to say this is equation two.

[00:49:12]We're going to say this is we're going to say equation three. Lambda time frequency. So, lambda uh we're going to say that just equation three. So, we're going to say that lambda is we already called lambda 2 / n. So, we're going to say 2 L /n. And then uh don't forget f.

[00:49:29]So 2 l f is vn. So we're going to say f is 2 uh vn / 2 l vn / 2 l. So we're going to say this is equation number what? Uh four. That's it. Now two more equations. Spacing. Spacing. What is the spacing means? Spacing means there is a space between uh between k0 to k1 between n1 to n_sub_2 between uh being n1 to big2 there is a space and that is space is more than zero. So spacing is s is equal to l / big n. Now what is x?

[00:50:08]x is the position. Position is zero over here is more than zero over here. Okay.

[00:50:14]If position is zero over here, one over here, this is two, this is three. So what is position? Position is then k L / N for node one node. And we're going to keep it simple. We're going to just do node.

[00:50:29]We're not going to just do anti-node because we can, but I cannot even talk now. Uh uh. Okay. So here's we have what we going to do now? We're going to say that L is L is 20 m and velocity is 40 m/s. So that's what we're going to do.

[00:50:46]Now we're going to create a big travel.

[00:50:48]The standing wave we fit it in one um fit it in one uh what do you call one one page. That's very nice. Uh so we're going to say this is K. We're going to say this is small N. We're going to say this is big N. We're going to say this is F. We're going to say this is lambda.

[00:51:06]We're going to say this is spacing. We're going to say this is uh what are we going to say? This is x.

[00:51:16]We're going to say this is velocity.

[00:51:17]We're going to say this is acceleration.

[00:51:19]We're going to say this is zero. Uh we're going to say this is one. We're going to say this is two. This is three.

[00:51:26]And this is four. When k is zero, node is one. When k is one, node is two. And so on and so forth.

[00:51:35]Now what is our anti-node? Our anti-node to find our anti-node is very easy. We count uh uh let's do fourth harmonic.

[00:51:45]This one is for fourth harmonic. Fourth harmonic.

[00:51:51]So let's do fourth harmonic. So two fourth harmonic since there is a barrier is going to make a loop. Okay, this is terrible but I'm tired. So please uh so now this is node one two three four five so I have five node great now anti node 1 2 3 4 so I have four anti node I have frequency so let's do frequency where is the frequency equation frequency equation is right here so let's do the frequency frequency is going to go there so what is the frequency frequency is v is I gave you v is 40 n is I gave you the n is 4 2 * 20 so 40 * uh 40 40 cancel is just four so frequency is four so let's find the lambda lambda equation is right here where is the lambda equation right here very nice so lambda is 2 * 20 / 4 so 5 10 lambda is 10 m Now spacing spacing equation is right here. L is 20 20 / 4. Uh spacing is five. Uh so this is five five five.

[00:53:13]Distance between this distance between this one and this one is five. Dist.

[00:53:17]Okay. So position. So we're going to do the position. Position is x is equal to 5k because this is five. So 5k. So here's how we going to do it. Let's remove all the uh distractant. So we're going to say index is 0. 0 * 5 is zero.

[00:53:37]This is the position.

[00:53:39]1 * 1 * 5 is 5.

[00:53:48]1 * 5 is 5. This is position. This 2 * 5 is 10. This is position. This 3 * 5 is 15. This is position. This 4 * 5 is uh this and this is position 20. Velocity here is zero. Velocity here is zero.

[00:54:08]Velocity here 0 0. We said it before.

[00:54:11]The velocity of the node is zero.

[00:54:13]Acceleration is zero. All right. Great.

[00:54:16]So this is uh that's it. uh we going to do now the phenomena called uh resonance. So the resonance is one of the phenomena. So resonance.

[00:54:34]So resonance. So everything in the universe vibrate with a natural frequency. That include a wine glass or tacoma bridge or any other building in in the wall move or vibrate with natural frequency. However, if you can accomplish that vibration, then if you can play with that vibration, if you for example, you ask, you go to a wine store, ask the wine manager, wine store manager, hey, can you give me the national frequency of all the glasses that you have in the wine store and wine store manager will not going to give you. Why is that? Because you may come back to the wine store with a guitar and play that natural vibration with a natural frequency with a specific type of wine glass and this wine glass will start dancing and amplify the amplitude of the uh of the wine glass and then they will vibrate with higher amplitude and higher amplitude and higher amplitude until they shutter. This is what uh uh the resonance is. How can we write it in uh using mathematics? Okay, here is the mathematics of resonance. A final is equal to a initial e rt. So this is growth or decay in this case is growth. What makes it growth? The resonance. Why resonance make is growth?

[00:55:55]Because someone probably uh accomplish the natural frequency of something. Now if I uh if I if I say this is my eyeglass, right? If I hit it with this uh uh uh then this is probably not a natural frequency otherwise it would be vibrating. It is not vibrating. So it was not natural frequency. If I say that's probably not a natural frequency because this is not vibrating that's how I know whether I accomplished the national frequency of eye glasses. No. Okay. If that makes sense. Now 194 like many years ago 60 70 years ago there was a breeze in Washington called Tacoma Breeze. Now in the morning 11 in the morning the wind hit the breeze with the with this nest frequency. So the breeze started amplifying it amplitude.

[00:56:47]The initial amplitude was something and the final amplitude when it collapses 120 second later 10 times more than the initial amplitude. So it started densing it started vibrating with higher and higher and higher amplitude goes from one to two to three to four to five to five times six times. So 10 times higher and then spine will collapse. It cannot go in many times. So that's why we said growth. Now the a final is 10 * a initial. So that's how we're going to report. So that we're going to say 10 a i is equal to a i a initial amplitude initial e r. It collapses 120 second later. Now a i ai cancel. You see 10 is equal to e r 120.

[00:57:41]Now you're going to multiply both side by ln. ln is inverse of e. So hopefully uh get out rid of E. So 120. So they cancel. So you see that r 120 is equal to ln 10, right? So that r is equal to ln 10ide by 120 and that's probably going to give you 0.02 or something like that. Not uh not sure but you're going to try at home and see what it is. Now if you go over here we see that uh uh equation that uh the the amplitude the final amplitude is final amplitude or you just going to do this r * t so r * t so what is r 002 and what is t 120 uh and that gives you how much um let's see let's see the calculator uh 120 time. 02 and that is going to give you 2.4 is this actually. Yeah, that ln 10. So this is ln 10 / by 1 2 0 and that is 02. Yeah, that's right. So this is this going to give you 02, right? So what are you going to do this with this 02? You're going to come over here. E uh 2.4 4 E is 2.7 and then you have 2.4 uh so 2.7 2.7 raised to 2.4 four and that is 10 about 11. This is 11. So what is that mean? Um what does this 11 means? 11 means in the initial uh the final amplitude. So uh that's that's what it means. The final amplitude is 11 uh 10 ai right? So ai is 11 / 10. So what was the initial amplitude? initial amplitude is about one and final amplitude is about 11 or 10. Okay. So that's why Tacoma bridge collapses because this amplitude went higher and higher and higher until it collapses. This tutorial I thought is going to be 10 minutes but it ended up with one hour but as you saw that I took my time to cover everything we cover in the class last few days. You should take that bandage. I'm going to stop here because I can barely talk.