 Hi, I'm Zor. Welcome to Inizor education We continue the course called physics for teens and today we will start a new part It's called waves So it's basically about different kinds of waves Mechanical waves electromagnetic waves, whatever Electromagnetism which was the previous Part of this course is basically finished. I mean theoretical material is finished Maybe I will return with some maybe more problems or something like this exams. I'm not sure but anyway, we start the waves Now this course is presented on unizor.com website Together with a prerequisite course, which is called math for teens Now all the materials on the website are totally free. There are no ads. No strings attached So you're welcome to take the whole course actually if you by some chance Search engine or whatever You you came to this lecture on YouTube or any other Website. Well, obviously you're welcome to watch it as much as you want But the unizor.com Has the whole functionality of the course, which means lectures are logically related to each other Each lecture has a Textual description basically notes like a textbook So I do suggest you to use unizor.com as your main Gateway to this course and all lectures of it. There are some menus obviously, etc. So let's talk about waves now We will start with mechanical waves first of all mechanics is about movements, right so Certain movements have a property of repeating itself repetitive movements Now what kind of repetitive movements I have a few examples like carousel for instance when it's rotating The pendulum whenever the pendulum is going back and forth or I have an interesting example a tuning fork, you know, it's called comatone sometimes Whenever somebody wants to tune up the piano for instance, they're using this tuning fork They hit it against some hard Object and then it sounds specific note. So for every note, there is a special tuning fork. So as it vibrates It makes mechanical movements the legs of this tuning fork and Obviously, they are repetitive to a degree obviously now In some cases we can have a high degree of repetition So for example, if the clock has a pendulum While it's wound up the pendulum is moving really repetitively to a high degree because that's the measurement of the time like one second per One particular movement back and forth or something like this Sometimes movements kind of seem repetitive, but they are still kind of changing tuning fork Well, whenever you hit it it starts vibrating, but then the vibrating is Getting a little bit less and less the amplitude is diminishing although the frequency remains So the time of repetitive movement is maintained, but the amplitude is getting smaller So it's not exactly repetitive movement, but to a degree during certain Relatively short period of time you can say that this is a true repetitive movement now Sometimes the repetition is made in in different Periods of time for instance when the carousel is getting Faster and faster is rotating faster and faster. It's doing the repetitive movement. However The time during the it makes a circle is Actually diminishing when it speeds up or when it slows down the time gets Greater the time of one repetition So basically we are talking about certain period of time during which the repetition is occurring now The first very important is we can talk about periodic movement periodic movement is when this period of time when The object makes this repetitive movement is constant so the carousel is Making repetitive movement when it reaches its maximum speed and rotates during it works While it's speeding up or slowing down. It's not truly a repetitive movement I mean it's repetitive, but The time is changing and we have a special Another special word periodic movement Periodic movement is more When this time is maintained as a constant okay, so When time of One full circle is constant. We're talking about periodic movement. Okay? What's next? Okay, let's talk about mass just a little bit the object In our three-dimensional world World is characterized by three coordinates, right? And if we're talking about movement, we're talking about all three coordinates being functions of time, right and all three coordinates together They are making a point in a three-dimensional world, right we can actually say that this is a three-dimensional function of time Assuming that we have three different functions inside it so What is a periodic movement? periodic movement is that exist t constant Such that p of t is equal to p of t plus t for any moment of time t so this is a true periodic movement and Obviously, we were talking about this in the course called mass for teens when I introduce the concept of periodic function Well function which we were talking about was usually function which was a one-dimensional function in this case It's a three-dimensional function which means it's a three functions. So basically what this means it means that x of t plus Plus t equals x of t y of t plus Capital t is equal to y of t and z of t plus t equals z of t That's what it means. This Means this when p is a three-dimensional function. So this is a mathematical Definition of periodic movement Okay We are still kind of approaching the concept of waves and to Basically approach it the way how it's usually done. We have to introduce the concept of periodic movement Because waves are periodic movements of something of some some substance, right? So that's why I started from these Well, I would say more theoretical Than that than practical aspects you you might think about waves as waves on the surface of the water So let's just not talk about concrete waves Because they can be in sound for instance, it's air Oscillations and and some others Electromagnetic field the light. It's also waves. It's all waves so in any case we start with mechanical movement Which we can characterize as periodic movement and that was the definition of the periodic movement Now when we are talking about waves, you know back and forth back and forth We are talking about specific periodic movement when the object we are observing whether it's a Drop of water or something like this It's doing periodic movement, but not just any periodic movement Carousel is a periodic movement But it just goes round and round Pendulum is a different kind of periodic movement because it goes along the same trajectory But in different directions back and forth back and forth. So that's the difference Carousel is making Movements only in one direction Pendulum both directions. So we're talking about a concept called oscillation Okay Oscillation is a periodic movement, of course, but not just any periodic movement It's such a periodic movement when the object we are talking about is moving along the same trajectory in one direction goes to an extreme point then turns back and goes along the same trajectory To another extreme point and then back and forth back and forth. So this periodic movement is called Oscillation now What's important about oscillation is that we always need external force Now if you have somewhere in the space You have a carousel which somehow was turned And it's rotating right now. You don't need any more external forces for the carousel to continue Spinning around its axis if there are no gravity Now no no other forces. It still will be rotating by inertia If we're talking about oscillation, we cannot Avoid existence of the forces. Why? Well, since we have one direction Then it should slow down and then it should speed up again back into a Opposite direction. So we are changing the speed, which means there is an acceleration Which means there is a force Without the force This particular object would continue moving along the straight line With the same speed without any kind of restriction. So we need the force to slow it down stop and then Speed it up again into a different direction and then back and back and forth all the time So we always have the force Oscillation cannot occur without the force. It may be gravity It may be something else, but there is always the force Okay, that's important What's next? the next is There is a special position Where the If there is no force basically then there is a special position this particular object which makes Oscillation would not move it. It's a middle point. It's a it's a point of equilibrium basically so If the force is not actually acting on such a such an object, it would just stay in the equilibrium point So we need the force to start the movement and all the time to maintain the movement back and forth We need these forces without the force. It would just be in the in one particular Position which is an equilibrium point in neutral position and the obvious example which we always use is if you have a spring and Some kind of an object Now I Draw it horizontally because I don't want the gravity to interfere with elasticity of the of the spring So consider this is in in space when there is no gravity, okay? so if this is in space and This is a movement along some kind of a I know you can have it like a string for instance, but there is no Friction here. So the whole string it goes through the spring and then through some kind of a midpoint of this spherical object Along its diameter, but there is no friction. Okay, so it goes back and forth Without any friction. The spring always has some neutral position. So if we are in a neutral position and no forces are Acting on this particular object. It will basically stay still. That's an equilibrium point Neutral point for the spring. It's not stretched and it's not squeezed. It's a neutral point Okay, now if we apply certain force and Let's say stretch it to this position Now the spring is stretched and as we know whenever the spring is stretched There is a force which brings it back into a neutral position, right? So it will bring it back and then by inertia it will go and squeeze the spring and Then it will reach certain maximum and then back and forth back and forth. So that's how our object will move And I would like to basically investigate this particular movement We already did this in mechanical part, but it's important I will repeat this thing because this is Basically the beginning of how we deal with waves. All right, so We are talking right now about special type of oscillation. It's called harmonic oscillation So the characteristic point of harmonic Oscillation is sinusoidal movement and let me explain what sinusoidal in this particular case means Sometimes harmonic oscillation is defined as oscillation which is Characterized by the force proportional to Displacement from the equilibrium point. So this is a displacement and We know the Hooke's law, right? Remember Hooke's law that the force the force is Proportional, this is the coefficient of proportionality to a displacement Which is distance from the equilibrium point So the further we we stretch it the stronger the forces and it's proportional to this distance Where K is just a characteristic of the spring. It's called coefficient of elasticity Of this spring there are stronger springs obviously and obviously this particular coefficient is greater There are weak strings, etc. But this is what's important Now X is not just X X is X of t a function of time, right? Because we're talking about movement. So the X is the difference basically This is 0 this is x-axis and This is X of t So first we stretch this particular spring to certain distance initial distance So X of 0 is equal to a Initially at moment time t is equal to 0 we stretch it to a this is a And we let it go Which means we don't really push it or pull it into any other direction Which means that the speed which is the first derivative At point 0 at point time equal to 0 is equal to 0 So we're just we stretch it to the distance a from the equilibrium and let it go So there is no initial speed the initial speed is 0 basically But the force is acting so there is certain acceleration now. What is acceleration? Well remember this mass times acceleration. This is the second Newton's law So we have two two different things. We have this Now what is acceleration? acceleration as we know is the second derivative of time second derivative if X of t is distance From 0 as a function of time the first derivative is speed the second derivative is acceleration Again, if you don't remember these little things you really have to go to the mechanics of this course physics for teens and Whatever I'm going to do next would be related to derivatives and differential equation and Again, I can always refer you back to the prerequisite course called mass for teens on the same website. There is a very A big calculus chapter over there and in particular we were dealing with differential equations. So These two things This and this Allow us to make a differential equation This is F and this is F. It's exactly the same force, right? But in this case we are relating the force to Hooke's law and in this case It's the Newton's law that both are true. It's exactly the same force Which means that m? times x second derivative of time is equal to minus k X of t or X Second derivative equals to k by m X of t Okay, this is differential equation which has certain solution Actually, there are many solutions as I will show you to this particular equation But out of many solutions, we will choose only one which satisfies these initial conditions Okay, so what about solutions to this again? I can refer you back to Calculus and differential equations of the course called mass for teens, but I will just very kind of briefly Repeat something which you Should should really know you see this is the first the second derivative and the function itself Now if you remember Calculus, there are two trigonometric functions sign And cosine Which have second derivative actually looking very much Well considering some coefficients like original function because the Sign derivative is cosine And cosine derivative is minus sign So the second derivative Is derivative of the first derivative which is derivative of the cosine which is minus sign And second derivative of a cosine First derivative is minus sign so we have minus derivative of sine cosine So as you see function is the same just a coefficient minus here We have a very similar kind of situation here. So we might look for a solution as a combination of two Functions Oh, by the way, if instead of t I will put something like P times t or omega times t it will be exactly the same thing because the coefficient will be on both sides So we definitely have a little bit more general Solution here. So this is a General form Where I will look For a solution to this particular problem so I Don't know what C1 C2 and Omega are But I will look for certain C1 C2 and Omega to satisfy This differential equation so How can it be done? Well, very simply. I'll just Substitute this function into this equation and find Whatever the coefficients I need Okay So the first derivative Is equal to C1 Times derivative of a cosine is minus sign Times derivative of inner function, which is a proportionality Omega t so derivative is omega plus C2 Derivative of sine is a cosine Omega now the second derivative is Minus C1 derivative of a cosine of Sine is a cosine Times omega and the mega and derivative of inner function, so it will be omega square derivative of Cosine is minus sine and Again another omega so it will be omega square now. Let's let's compare this With equation which we have Now this is X of t X of t is C1 C1 cosine cosine C2 minus C2 sine sine and omega so as We can very simply see and We have here minus k over m. Omega should be omega square Omega square should be equal to k over m, right? Omega square is equal to k over m. You have k over m here, right and the minus sign so now we need to Think about C1 and C2, so let me just put X of t is equal to C1 cosine Omega is square root k over m t plus C2 sine square root k over m t Now what are C1 and C2? Well, let's use this one and this one If time is equal to zero my x should be equals to a Now if time is equal to zero, this is zero sine is zero This is zero so cosine of zero is one So I have basically a equals C1 X of zero is a right and This is one and this is zero so we have only C1. So this is my first now The second one Well, this is the second derivative, right? But I mean the first derivative if I will substitute zero here Now sine of zero would be zero. So this will be zero now this will be one and So the whole thing would be C2 times omega So X of zero is equal to C2 times omega But I have to be equal to zero which means C2 is supposed to be equal to zero So this is a and this is zero And my final function is the function which satisfies differential equation and Initial condition a cosine square root k over m T That's my function. This is a solution of an object on a spring So we consider this to be zero and this is initial Position at point a from zero stretched and that's how it will Oscillate Okay, what's next? Two things What is the period of this function? so What is well the smallest time period after which it will completely repeat its movement Well, we know that cosine Has a period 2 pi right? how about cosine of omega t Well, the period will be obviously 2 pi divided by omega Why well, let's talk about this cosine of omega So instead of t I will put t plus This is t So what is it? It's cosine of omega times t plus 2 pi over omega equals cosine of omega t Plus 2 pi and 2 pi is obviously period. So it's cosine of omega t so this is the proof that period t is equal to 2 pi over omega and Now the frequency Well, the frequency is how many oscillations This particular thing is making in one second Well, let's just talk about this One cycle One oscillation it's making in t. So one cycle In t seconds We need frequency is Certain number of cycles per second. So f cycles It's doing in one second. So we have to find out what is f. Well, this is a proportion obviously So f is equal to 1 over t Which is equal to 2 sorry omega divided by 2 pi Where omega is this? Square root of k over m. Well, that's it so this is 2 pi square root of Inverse so it's m over k And this is frequencies equal to 1 over 2 pi Square root of k over m So these are Well, obviously m is mass of our object this one and k is the elasticity of the spring So these are completely physical characteristics of what we have and based on these physical characteristics and the initial distance And the fact that we don't push it from that initial distance the initial speed is equal to 0 to This is a solution and these are two very important characteristics This is the time during which we are making one full cycle Which means from this point it goes all the way here Squeezing the spring and then all the way back back to this position and Frequency is number of these oscillations per second This is the formula and the whole thing is about Simple harmonic oscillation it's called harmonic because we're talking about sine and cosine Well, basically we started with harmonic oscillation as defined Based on the proportionality of the force at the same time I can say actually any a Movement which basically can be represented as a trigonometric function is harmonic because The word harmony is always related to trigonometry that's why So these are Basically a very very beginning Data about harmonic oscillations and obviously we will deal with harmonic oscillations all the time talking about waves Well, that's it for today. I recommend you to read the notes for this lecture And basically that's it. Good luck. Thank you very much