WAV02: The Scalar Wave Equation

Transcript

So you get in this synthesis, not only do you get all of electricity and magnetism, you get optics, you get radio waves, you get X-rays, you get everything. You basically get the way that we transport energy and information properly understood. the two not being that different from one another, energy and information in our universe. And and so that's where we'll start with the next topic, which is the wave equation. So, title this in your notes, the wave equation.

And I'm going to rewrite the differential form or the point form of Maxwell's equations, but I'm going to do it in the phaser domain. It might be helpful for you to go back and look at that lecture I did on the phaser because um you know that we're basically going to redust that dust off that exact same concept except we're going to use it on vectors instead of just voltages and currents. Remember you can do phasers with a vector just as easily as you can uh a scaler. In this case, we have things like an E field vector for example, which will be a function of space, threedimensional space, and time. And as is often the case in electrical engineering, we're just interested in time harmonic analysis.

A sinosoid that has been going from minus infinity to plus infinity. And the reason why we're interested in that is because a lot of radiation theory deals with things that look like sinosoids. If if they're not, they either are as sinyosoids or they can be approximated by syosoids. The change of their envelopes and phases is so slow that you might as well just analyze the system as it behaves for a sinosoid. And you've pretty much characterized that information signal that has wiggles on it.

So we're going to have an electric field. It's going to be a function of space and time. And like all time harmonic systems, we're going to use the phaser domain to remove the time dependence. So we only have to worry about space and that greatly simplifies the equations. We can then then use phasers differential operators become multiplication and much easier operations when you do that when you remove out that time dependence.

So if this is the case you can imagine this is a vector. I'm going to have e x cosine of 2 pi ft plus the phase on the x component. Stick it on the x plus ey. This is writing it out in longhand tedious form. And one thing I'm it could even be more tedious than this because what I'm not showing is that the amplitudes and phases that I've drawn out here are themselves functions of x, y, and z.

So there's a lot of positional dependencies floating around here, and it could be it's so tedious, and there's one of the reasons why we want to put this thing in the phaser domain. Ephaser we will write as e expp stick it on the x hat ey exppy stick it on the yhat E Z E E X P J F V Z stick it on the Z hat. Add all these components up together. Ey and and FY, EX, FX, EZ, FZ. Those are still functions of space, but I have removed the time dependence assuming that these are just the amplitudes and phases of sine waves.

And of course the inverse if you want to take this quantity in the phaser domain back to the original time domain form we use the standard expression that we learned three months ago E as a function of X Y and Z or our vector if we want to use a shortcut and it's also a function of time is equal to the real value of my E phaser vector as a function of space time ex J 2 pi ft. And again I'm reintroducing my carrier frequency of frequency f back into uh the phaser reconstituting it into a purely real vector that varies as a function of space and time. And I can do this for all of my phasers. I'm going to take this definition here. Make sure you all copy this down because I'm going to erase it in a second.

Um, I'm going to take this definition and convert the differential form of Maxwell's equation into a phaser form. And then we're going to add some more simplifications. And then we're going to study that chain reaction that I spoke of earlier that uh uh allows us to study electromagnetic waves as they propagate in nothing vacuous free space. So let's go ahead and do that. Let me erase this section here.

And let me start with Ampere's law, the modified AI's law. The divergence of H vector. And this is going to be a phaser. So I'm going to use the standard notation that I did earlier. It's only going to be a function of X, Y, and Z.

I converted it into the phaser domain. I'm going to put this till day or squiggle on top of it. And that must be equal to J with a squiggle on top of it. J vector with a squiggle on top of it. And I've got the time derivative of electric flux density.

Partial derivative with respect to time. Well, notice I I didn't have to modify this operator. Divergence is with respect to space and phaser is a linear operator that's only with respect to time. As a result, I can just leave this operator as such. I can't do that with the der derivative with respect to time though.

However, the beautiful thing about the phaser domain is what does differentiation with respect to time turn into as an operation. Think I heard it. Yeah, you multiply by the frequency. J 2 pif. That's one of the reasons we love the phaser domain because the act of taking the derivative is like multiplying by J2 pif or J omega if you're one of those controls types integration becomes dividing by J2 pif that's what's so beautiful about it and so this becomes plus J 2 pif dphaser dphaser let's see I go over Here again I've got the divergence of B field is equal to zero and that's going to be a phaser.

I put a phaser on that but zero is zero. The divergence of the magnetic flux density phaser vector is zero going down to here. Oh and this should be curl. This should be curl. Shame on you professor D.

Look at that curl. of E field phaser curl is a linear operator with respect to space. So I don't have to change anything. I can operate on the phaser as I convert this side of the equation to the phaser domain. I cannot do that for the time derivative of magnetic flux density vector.

Instead I turn it into the multiplication of J 2 pi F B field phaser and finally the divergence of my electric flux density vector phaser. Divergence is only operating with respect to space linear operator. I can just pass the phaser transform right on through and take this as a um turn this into a phaser vector. And this should be equal to charge density phaser. So does anybody have any questions? Anybody suspect me of shenanigans in my conversion of the differential point form of Maxwell's equations into into phaser form? Everybody buy this so far? Okay.

I'm going to make some more simplifications. We are going to operate in a simple sourcefree medium. Now there are four four criterion that that this medium is going to this place where we operate our equations is going to satisfy. If it is simple, there are kind of three subcategories that we can we can spin off at. We said this is linear.

This medium is going to be linear. And all that means is that if you double the flux density, you double the field. Double the H, you'll also double the B. You double the E, you also double the D. Just about every medium that you guys have dealt with thus far has been linear.

whether they stated it or not uh in your in your classes linear. The other thing that we're going to do going to say that the the source is uh the medium is homogeneous. What does that word mean? The same. That's right. Yeah.

The same with respect to space. Homogeneous just like milk, right? I don't know why, but but in the sciences and in electromagnetics, it's always homogeneous. But when you go to the store to buy milk, it's homogeneous. It means the same thing. It's the same word, but the pronunciation is universally different.

I don't know why. One of the mysteries of life. But what is homogenized milk? And you probably don't even see, do you even know? Have you even heard of that term? Like back back when I was a kid growing up, they used to still advertise that on milk because it was a big deal, you know, back in my parents' youth to have homogenized milk. If you normally got milk from the dairy, uh they would, first of all, refrigeration was kind of like at a premium 60 or 70 years ago. So they the milk guy would come by regularly with your milk um to to feed your family uh because it would spoil if you didn't.

and every day or so you'd come by and leave leave these milk jars on your uh front porch and they were not homogenized which means that uh there was kind of a milky milk at the bottom and then there was kind of a thick layer of cream at the top and they were kind of like different strata in your milk. Yeah. So yeah. Yeah. is like just dip.

You mean you used to be able to kind of dip off the the top and make some butter out of it and then you drink the top half and then maybe you save the bottom half for your dieting cousin or something like that. I don't know. The good stuff was at the top though. So, um but you know it was a big deal that that you could actually homogenize milk like shake it up and and get the the milk distributed and it wouldn't settle out over time. That was homogenized milk.

Now, I think people were to take take it for granted that of course milk is homogenized. If it wasn't, you'd think it was bad, right? Curdled stuff at the top. But that was actually natural back then. And this means the same thing. This means that the material, the electric properties of the material are the same everywhere in space, at least over the region that we're solving our problem.

The permeability is the same, the permitivity is the same. if we want the conductivity is the same but we're going to get rid of the sources anyway so that part doesn't matter. The other condition that you should be able to spout is that this is isotropic. Now iso also means the same and trope means side uh in the the Latin root means side there samesided and what that means is that the permitivity or the permeability or any material parameter for that matter is the same independent of the orientation of the field or the polarization of the field. The E field is in the Z direction.

the the constant that that uh of proportionality between the electric field and the electric flux density in the Z uh direction is the same as if I look at that constant of proportionality in the X or the Y or any combination thereof samesided same in space samesided and linear these three together mean that we are allowed to write Our material parameters with these simple constants of proportionality. We can relate fluxes to their equivalent field uh vector quantities using just constants of proportionality. There are media out there that these one or more of these three conditions can fail. And in that case, what you actually have to do if if it's nonlinear, that means that this constant of proportionality is actually going to turn into a function of the magnitude of E field or H field. Mu is going to turn into a magnitude of uh uh is going to depend itself on the magnitude of H field.

If it's not homo homogeneous then e epsilon and mu become functions of position themselves instead of simple constants and if they're not isotropic then believe it or not instead of simple constants of proportionality epsilon and mu become matrices. Those are nasty problems to solve. But there are actually media that are anisotropic that are actually very interesting and important. The ionosphere for example where there's charges and a magnetic field present is an anisotropic medium and it has that causes certain radio waves for example to twist around in polarization as they travel through the atmosphere. It's kind of cool.

It's a neat phenomena but it's just one example of an anisotropic medium. And then we're also going to have a source free medium which means there are no charges present in the medium. So now looking at these four conditions how may we simplify Maxwell's equations? Well if it's linear homogeneous and isotropic we can turn all of our B fields into muh fields where mu is just a constant. So let's do that. Muh field vector.

This is also a muh field vector. And in fact, I can take mu and just put it out there because it's a constant. I pull it right out of my differential operator. It's not a function of position. I can do the same thing with my D field.

I can turn them into epsilon E fields. In fact, let me divide through by each and let me see let's do it up here too. This is going to be epsilon E field. So that that takes care of the first three. source free means that there are no charges and if that is the case well this is already zero I don't have any charges present in the medium free charge that would give rise to electric field lines electrostatic field lines and then over here if I don't have any electric charges I can't carry any current so I'm going to set that equal to zero as well and here we go Maxwell's equation differential form using phasers in a simple sourcefree medium.

What would be a good simple source free medium? Outer space. There's nothing there. And and epsilon and mu are truly perfect constants. You have epsilon kn and mu not. Uh and you don't have to worry about nonlinearities and all that other stuff.

Air is actually a pretty good homogeneous linear source-free uh isotropic medium as well. It's almost like free space for most type of optical or radio wave type problems you encounter in life. U you can essentially treat it as uh true free space epsilon not muon. Usually if you're in a dialectric medium like inside maybe the center of a co uh um well coaxial cable or a fiber optic cable you can treat that as a as a homogeneous um homogeneous isotropic nonlinear source free medium as long as you don't expand your your solution space beyond the boundaries where things start to transition. If you go deep deep enough underwater, can you consider the same conditions? Yeah.

In fact, once you go below the surface of seawater or lake water, any type of water, as long as you're solving beneath that, but above the land, that area there is for all intents and purposes a a simple source free medium. It has a permitivity, permeability, has a conductivity also um that is relatively constant. Some of those things can depend on the material properties which in turn depend on things like temperature and pressure. So you might get slight grad uh gradations gradations but uh you you don't there are a lot of problems where that doesn't really matter. It's such a slight change you don't care.

There are some that it is important. So what does a problem like this type of stuff look like? Well, what we will do is take our phaser form of Maxwell's equations when we come back next time and we will synthesize the wave equation which tells you how energy travels around in space and then we'll be able to say what is the basic form of a wave traveling that what does it look like which direction is it traveling in how big is it I'll give you an example problem and that's the only thing I could possibly ask her about this on the desk something simple like what does a wave look like in this medium? Okay, everything good? Excellent. Let's go ahead and continue our lecture from Tuesday. We've got to finish off our discussion of the wave equation. And this is how we this is the point at which we left off.

We had put Maxwell's equations on the board. And then I showed you a really nice simplification of Maxwell's equations that you can use when you are dealing with time harmonic sinosoids, waves traveling around and in simple source free media. And let me go ahead and put those back on the board so you see what we were talking about with the end goal today. We're going to show how all that stuff that we've been learning about during the course of the last several months, uh, electrostatics, magnetoatics, Faraday's law, how those can all be combined to get energy and information transmission in nothing, which is the cool part. So we had Maxwell's equations in a simple source free media.

And this of course is phaser form. And let me go ahead and recount what they all were. The curl of h phaser vector is equal to j 2 pi f epsilon electric field phaser. or this is this is Maxwell's fudge factor right when you say J 2 pi F in the phaser domain that's differentiation in the time domain so it's the derivative of the the electric field and we know that a sinosoidal electric field will have a swirling magnetic field around it the same frequency um whose amplitude will be proportional to frequency and the amplitude of the original E field was oscillating there. Then we have the equivalent of Faraday's law which said that E fields circulate around changing magnetic fields and this was minus J 2 pi F mu H vector and we said that if there are no sources no currents no charges no nothing then the sourciness or the divergence that is of H and E vector phaser in the simple le the simple source free medium is equal to zero.

So these are simplifications to Maxwell's equations given the fact that you got a really simple medium with no sources and you're operating with time harmonic sinosoids. A very elegant little set of equations and we can see very readily where uh the wave might come from. L H field is proportional to the change in E field which is itself proportional to the change in H field which itself is caused by a a changing E field and so forth and so forth. That's the chain reaction that gives rise to waves. And so what I'm going to do now is kind of put all these equations together into something called the Helmholtz wave equation.

Very elegant little formula that shows how things propagate in space. and we'll we'll be able to close the loop with something that we saw at the very first week of class. Um, and it'll all make sense and you will be set up for prime success if you go in to take the follow on class to this which is 3065 electromagnetic applications. There you start off with kind of advanced sinosoidal transmission lines and you do some RF circuit design and then you get into wave propagation and antennas and sparameters and all this really cool stuff. It's very useful and increases your starting salary by about $5,000 if you go through that.

But hint hint. So let's see. We got Maxwell's equations here. What are we going to do? Well, what I would like to do is to make one equation that doesn't have these two co, you know, we got basically two coupled equations here. I want to make one equation using a little bit of vector calculus magic.

Uh, that allows us to to study just E or just H. So, we can actually see what waves look like mathematically, what form they have. So I'm going to go over here and I'm going to start with let's see I'm going to start with my Faraday's law curl of electric field phaser minus j 2 pi f muh phaser faraday's law and I'm going to take the curl of both sides of this equation whatever I do to a one I got to do to the other right so if I do that I've got double curl of electric field is equal to J2 pi f mu that's all a constant that's all a constant so I can bring my double curl or in this case a single curl on this side in through those constants and operate it just on the vector h that is going to be itself a function of three-dimensional space that's not a constant so what I have is minus j 2 pi I f mu all these constants curl of h everybody with me so far makes sense that's a logical step right okay and what's more I've got an equation for this right this is j 2 pi f epsilon eph phaser vector vector, right? So, let's simplify this. Let's group all of our constants. I got a J * a minus J.

Minus J^2 is pos1. That just goes away. I don't even have to write it. I got 2 pi F and I got 2 pi F here and 2 pi F there. So, that's going to be 2 pi fantity squared.

And I got a mu and I got an epsilon. And then I got my Ephaser vector and that is equal to double swirliness. Double swirling. What is double swirliness? Does that hurts my head. I don't even have a really nice cute thing I can call that to help you visualize it.

I I have trouble visualizing double swirliness in vector calculus. Uh but we're going to change that. We're going to do a we're going to apply some mathematical identities to unpack that and simplify it. I'm going to show you something really cool though. Uh first first of all this quantity 2 pi f * the square of mu epsilon and of course this is just that quantity squared that occurs so often in electromagnetics that we give it uh give it its own variable k.

So basically k is equal to 2 pi f square of mu epsilon. It has units of radians per meter and we call it the wave number. We've seen a wave number before in this class. What what what variable did we give it? Beta. Beta.

That's right. When we talked about sinosoidal transmission lines, for some reason, when we're on a bounded wave, we often use the term beta, like in a plane uh a wave guide or a transmission line. When we're in free space and our waves are propagating without bound, we often like to use K, which is the wave number. It has the exact same geometrical interpretation in that it is 2 pi over the wavelength of radiation. Wavelength is the distance in space you got to travel for the wave to do one cycle if you freeze it in time to see two two pi phase change on your wave.

Uh and so that'll help kind of simplify our expressions as we develop them. Okay, you first Justin. Um by double swirliness, do you mean that the swirl is doubled or do you mean that you take the swirl of the swirl? The swirl of the swirl. Does that warp your mind? No. Okay.

So then I knew he was. So is that's like a like this type of swirl. Yeah. So if you think about it, if if swirling in one direction and it's swirling in the other plane or something. Yeah.

You take you figure out what the curl is, the swirliness. So if something is swirling around, there's going to be a ve right-handed vector in magnitude and direction that points to the swirl. And then you're looking at that vector and seeing how swirly that is. the copper wire wrapped around a copper wire wrapped around kind of kind of. We're going to simplify this.

We're not done with this yet. Don't try to visualize it too much. You might hurt yourself. This is actually something called uh Well, let let me show you something really cool, too. If if you want to really impress people uh in in the mathematical community.

So, we can move this all to one side. And in doing so, we would get double curl of my E field minus K^2 the constant wave number squared times my E phaser vector is equal to zero actually the zero vector. So double curl of this minus a constant times that same vector is equal to zero. And this is actually called the vector wave equation. And if you are super cool, if you are the sly mathematician, you'll see a lot of times people will write this as an operator double double curl minus k^ squ operating on the e vector gives you zero.

Isn't that slick? Do you do you see what they're doing there? They're saying, "Oh, just distribute E to each of these terms and out pops all the operations that you need." It's kind of a shorthand vector wave operator. You've condensed it into one really cool mathematical operator. So anyway, if if you are so inclined, you may use that on any any expression that you want in this class or in the future. Everybody will think you're very sophisticated. Okay.

Now, let's unpack this concept of double swirliness cuz it's hurting my head just trying to explain it. Well, it turns out we can apply a vector calculus operator. This is going to be a purely mathematical operation that I'm going to do and you can find this in any vector book. Look in the back with all the the relationships that you see. And uh you'll find that any vector's double curlininess can be actually written as the following.

Let's let me just use uh a to just to note that any vector could be uh written in this fashion. This can be written as the gradient of the diverence of a. Remember a is varying with respect to three positional dependencies x y and z. It's a function of 3D space. So you can take the gradient or excuse me the divergence that gives you a scalar and then you operate with the gradient and that gives you a vector again.

So we've got vector equals to vector. Remember curl gives you a vector curl over curl will also give you a vector. So vector equals vector minus lelassian of a vector. Now we've seen leloian before too. Remember when we did leelass's equation? Lelassian is a shorthand notation for the divergence of the gradient.

The opposite of what we're doing over here. This is the diver gradient of the divergence. This is the divergence of a gradient. And if you wanted to write it out in longhand, it's the partial differential operator. second partial derivative with respect to x plus second partial derivative with respect to y plus second partial derivative with respect to z.

And when we used it, we operated it on what what quantity in lelass's equation when we studied that with physical quantity? Voltage. Voltage. That's right. That's right. Voltage.

And voltage is a scalar. Here I'm using it on a vector. Is that legal? Well, it actually is as long as you're careful with things, right? What this really means is I'm going to take the leloian of my x component, which is a function of 3D space. I'm going to stick it on the x hat unit vector. the leloian of the y component of a stick it on the y plus the lelassian of the z component of a stick it on the z.

You see why vector calculus is so elegant, right? We have these cute little shortcuts for something that would probably take off the entire board if I wrote it out longhand. So if you if you so want to try this out, it's a fun exercise if you're really into math. You don't have to, but in the privacy of your own home, go through and and write the longhand expression of all this out and all this out and confirm to yourself that they're the same thing. It's purely mathematical. You're just getting the derivatives in the right order and grouping terms and you'll find this all pops out.

Basic mathematical equation. Okay. So, how does that help us? because it looks like now instead of double swirliness I've got gradient of sourciness minus leassian that doesn't help me that doesn't help me at all however if we apply this to our E field in the simple media something should jump out at you a simplification that can be made you can divide up I can substitute this slightly more complicated in thing for there but it's not really more complicated because what simplification can I make to this this is zero Yeah, Maxwell's equation said if there aren't any sources present, the divergence of the E field should be zero. So really, this should just be equal to minus the leloian of my electric field phaser vector. Okay, let's see where that gets us.

I'm going to plop this minus lelassian into here. And since there's a negative side on both sides, I'm going to just uh um divide that out. And what I will get is the following. It's called the Helmholtz wave equation. It is the leloian plus the constant wave number squared operating on electric field is equal to zero and this thing operates on each component of the E field on E subx that's going to be zero on E suby that's going to be zero on E subz that component that phaser component will also be equal to zero.

And this is often called also the scalar wave equation. Now you might scratch your head when you first see that term because clearly this is a vector. However, this is really three independent uncoupled differential equations, scalar wave equations. It's three, it's three uncoupled scalar differential equations. The vector different uh wave equation is not that.

If you don't make that simplification where the divergence of the E field goes to zero, then the components couple with one another and it's nasty. It's nasty. This is actually a much easier system to solve. Len things that are not coupling here. Oh, okay.

So, let me write it out longhand. This is actually saying well this equation here is actually if I wanted to write it out a little more longhand leloian plus k^ squ operating on the x component of e phaser the y component of e and the z component of e and these are themselves functions of threedimensional coordinates. So I'll write everything out longhand. And of course this is going to be equal to zero and zero and zero. I'm saying like how do they have nothing to do? How do how are they not? Oh, so so this operates only on the x component and you and it is equal to zero and there's no y component in that component of the equation.

There's no z component. So in other words, e subx, e sub y, e subz appear in their own separate equations, not in the math side like you in talking about the picture of like what you're Oh, okay. Hold that thought. Hold that thought. Well, we'll get there.

We'll get there. I got to unpack this and then we'll talk about what does it physically mean. Yeah. Okay. Good.

Good. Good. Okay. Now, if we wanted to, we could have started our analysis with the magnetic field instead, done the exact same steps, and arrived at this exact same formula. Yeah, I don't I'm not going to do this on the board cuz the lecture is already too mathematical as it is.

We got to finish this up. So what you would find you could also get that the Helmholtz wave equation to solve for all the components of the H field as well. So actually this is what this combination of Maxwell's equation says. I take all of my electronamics and electrostatics and and ampers law magnetoatics formulas put them together and I get a wave equation and what's more I get this in the presence of nothing in vacuous free space waves can propagate in vacuous free space and what this means is they're actually six separate scalar equations this operating on the xyz Z of E field phaser and XYZ component of Hfield phaser all those must be equal to zero. Now what does this mean physically? Let's take let me just take the X component of E field and write this operator out longhand for you and then we'll discuss So here we have the longhand form of our lelassian operator plus k^ squ operating on my x component of electric field and that must be equal to zero.

Even in this form, it's kind of difficult to see physically what's going on unless I do this. What's left? Second partial derivative with respect to Z plus a constant K^2 is equal operating on a phaser vector time hormonic phaser vector is equal to zero. Where have you seen something like that before? I'll give you a hint. Got what does that look like? This is the partial differential equation that governs voltages and currents on a transmission line. This is what we saw this when we studied sinosoidal transmission lines.

This is actually if you wanted to do this in the time domain without the phasers V and I, this would become a partial derivative with respect to time squared which is part of the telegraphers's equation. So if you ignore this part, this looks like the transmission line formula for syosoidal voltages and currents. What did that formula, that partial differential equation physically mean? That meant that the only voltages and currents that you could support on a transmission line were ones that were traveling with constant velocity and a constant impedance between the two down the Z direction of the transmission line. So this is constant velocity sinosoidal propagation a traveling wave in the Z direction. If I add now these terms, what does that mean? Constant velocity propagation of a sinosoidal wave in every direction X, Y, and Z.

And that's all it means. A transmission line in a way is a lot like the people mover at the airport, right? It's a constant velocity device. There are two two conveyor belts moving at the same velocity but in opposite directions. Little kids have a reflection coefficient of one, right? Cuz they just play around and go back and forth and back and forth. And if you can imagine it, what this says is that when I make an electromagnetic disturbance in space is like throwing a rock into the three-dimensional surface of a pond or a three-dimensional people mover at the airport.

The ripple is carried in all directions at a constant velocity. And that's physically what the Helmholtz wave equation means. Now it's actually a very simple equation. It's a simple equation to solve. What gives it difficulty is the boundary conditions.

The boundary conditions. And you can spend your the rest of your life studying this equation with various boundary conditions. All the complexity and and stuff in our world interacts with this and changes the boundaries and uh depending on what you call yourself. If those boundary conditions involve very high frequencies, you're an optics person. If it's lower frequencies, you're a microwave per person.

If you're even lower frequencies, it's an RF person. If there's always metal around, you're a circuits person. All we're all we're ever doing in our profession in electrical and computer engineering is solving some level of abstraction of Maxwell's equations. You may be doing voltages of current. You may be doing ones and zeros and transistors, but in the end it's all boundary conditions on Maxwell's equations.

And this is one of the most common forms of it that you'll you are allegedly solving whether you know it or not. Any questions so far before we talk about specifics? the specific solution. Professor, how did you jump from that question to the one on the bottom? This one? Oh, yeah. I didn't jump. This is actually this is a separate equation that's in your notes from the syosoidal transmission line content.

So, this is a sinosoidal the telegraphers equation for sinosoids basically. Now, there's some powerful um analogies that you can can work between these two to kind of get some um um um in intuitive insight. If you're familiar with syosoidal transmission lines, this really isn't that much harder than that. There's some extra dependencies in space that you got to deal with, but the concepts are all the same.