# A Further Introduction to the Wave Equation

Moving on to the wave equation
So you guys see how it's similar except the derivative is the second derivative
What does this describe well we've talked about of this a little bit before but let me remind you
It's kind of like the temperature excuse me the heat equation
but with overshoot
And I'll explain what I mean by that this kind of tells you that velocity the rate of change is
proportional to the Laplacian and this one says that
Acceleration is proportional to the laplacian
You guys there with me on that
So when acceleration is proportionate to the laplacian the function will starts going will start going to the place where laplacian is zero?
That's where it wants to go it wants to go to the equilibrium
but when it reaches the equilibrium by the time it reaches the equilibrium, it will have picked up some speed and
so it'll overshoot it and
Go as far perhaps to the other side, and then it'll say oh now
I've overshot the laplacian now. The whole relationship is inverted. I used to be higher than all of my neighbors now
I'm lower than all of my neighbors, and it'll start bringing you back
Acceleration will reverse it reverses as soon as you pass through the equilibrium, and then you'll come back
And so that's why you get life
That's why the second derivative gives motion and therefore life you overshoot the equilibrium, and you come back and you begin to isolate
Which you can never do here because this tells you the velocity
The velocity is degree
Or is proportional to the degree by which you deviate from the equilibrium
so the closer you get to the equilibrium the slower you go and
So you never really reach it. It's sort of like exponential decay as you will see and this is all about
overshooting and therefore coming back and
creating this wave
oscillation and
So this is called the wave equation once again. I'm not gonna write it down
So that's why the wave equation looks like this, and we've postulated it for guitar strings, and in this case you will think of drums
because drama is like a two-dimensional string and
Why does it sound so much worse?
How come when you pluck a guitar it just has this beautiful?
Tone that lasts and when you street, and when you hit a drum it just goes like
okay, I'm I
Will use a better sound effect in the video. You know why is it so much different. Do we have the answer to that?
Yes, we do PDE's these simple PDEs will hold the answer to that
Wave equation let's talk about its initial and boundary conditions think of a drum
well, you certainly need to know its shape and what happens at the boundary so you once again need to know all of the
values of the function at the Boundaries or
if the drum is moving
But that's prescribing the value at the boundary you can't just say I don't know you have to know where it is
but you can say I don't know what the function does on the boundary note that won't be enough to really specify the
evolution of the drum
So you need to know the boundary and you of course need to know where you start?
And this is where the od analogy will be perfect
Because where you start here is not enough
This is again an evolution, but it's a second-order evolution you're given not the velocity but the acceleration
the ODE analogy is U double prime equals u
You need two conditions here you need to know the initial position
And how fast you're moving if you think of it as an evolution that you March through
Then that's what you'll need to know you it won't be enough to know the initial position
You will also need to know how fast it's moving
Its rate of change in the initial moment
makes intuitive sense by analogy with ODE's
I think it does so yes from the point of view of at least I have an idea on how to start
The wave equation is simpler than laplace's equation. Which is sort of as this simultaneous snapshot. You've got to satisfy here
You start somewhere you have this additional piece of information for velocity and then you know how to treasure hunt your way along?
Maybe it's easy. Maybe it's not
But at least you know what's going on. So wave equation all of the
phenomena associated with vibrations
Ultimately come down to the wave equation
everything vibrates
the glass the Bridge the table the Ruler the straightedge the chalk your body to degree a
drums musical instruments
elastic bodies plastic bag bodies metal bodies
everything vibrates
currents
So whenever there is this phenomenon that?
that describes
That entails going to the group equilibrium
But of course overshooting it because guess what inertia is the most is one of the most fundamental facts of life
Even if it wasn't recognized for three thousand years as such you know, but whenever there's inertia
And then a restoring force and somehow you have a sense that energy is conserved
If you let it be then wave equation plays a role
if the system is all about relaxation slow relaxation to where it wants to go then it's a heat equation and
It finally want to describe the geometric final relaxed state
its laplace equation, and if there is something that doesn't let it be
relaxed
persistently, then it's Poisson's equation and
These three equations
Describe basically describe I don't want to say most of the natural phenomena
But there's always an element in any natural phenomenon that somehow Corresponds to one of these behaviors