hey what's up you got Parthia back at
you with another video this time my hair
and my beard are even longer than my
previous video because somebody said
that my hand beard were getting too long
no such thing in today's video I want to
talk to you about something known as the
wave equation look something like this
but don't worry you don't need to have a
mathematical background to understand
what I'm gonna be talking about as long
as you've studied physics to high school
level or even maths to high school level
then you should follow exactly what I'm
saying and by the end of this video
hopefully you will have some
understanding of what this equation is
trying to tell us and why is called the
wave equation before we go any further
though I'd quickly like to say if you
enjoy this video please do leave a
thumbs up and subscribe to my channel if
you haven't already leave me a comment
down below telling me which equation
you'd like me to explain next because I
really like making these equation
explain the kinds of videos any other
side let's get into it let's start by
labeling a few variables that we see in
the wave equation here specifically the
variables X and T let's get the easy
ones out of the way first X refers to a
spatial direction let's say for our
purposes that the X Direction is toward
the right what this means is that along
this line we have different values of X
X is equal to 0 X is equal to 1 X is
equal to 2 and so on and so forth the
units don't really matter we just need
to know that this is a spatial direction
and T as we expect stands for time this
is because we'll be studying the
behavior of waves as time progresses so
we can choose a particular instance when
we're studying away we can choose T is
equal to 0 to B now for example and then
we can work out what happens to a wave
that we're studying maybe I don't know 5
units of time later maybe 5 seconds
later and so basically just labeling
these two quantities we'll be able to
understand one simple thing about the
wave equation might seem like a
relatively obvious point but the wave
equation studies the behavior of waves
in space and in time
in this case we've got what's known as
the one-dimensional wave equation
because we've only got X which is the X
Direction one dimension now other
letters that we can see in this equation
are u D and C out of these three letters
C is the simplest one to explain here C
is simply the speed of the wave that we
happen to be studying so if we're
studying an electromagnetic wave maybe a
light wave then C is simply the speed of
light moving on to you
you is known as the displacement of the
wave or specifically the displacement of
different parts of the oscillating wave
because for example if we think of a
wave that moves like this and if we just
focus on one particular point of that
wave one particular point in space of
that wave we see that even though the
wave is moving from left to right the
point is moving up and down and so when
this point is let's say at this position
we say that U is equal to zero as it
moves upward we say that u increases and
as it moves downward we say that u
decreases in other words U is the
displacement of this point in let's say
the up-and-down direction the only
letter that we haven't yet discussed in
this equation is D D is not actually a
variable there is not something that
takes a value in this equation the DS
are simply there to tell us that there
is something known as differentiation
going on in this equation so what's
differentiation well one way to think
about differentiation is that it's a
really useful tool in helping us
understand the rate at which objects do
things for example we could plot a graph
showing us how a car moves from one
point to another over some time so what
we've got here is the position of the
car on one axis and the time on the
other axis what we're seeing here is
that at T is equal to zero at the
beginning of the cars journey the car is
at a position X is equal to zero and as
we progress through time as we move from
left to right on this graph its position
changes in other words like we said
already at T is equal to zero the
position is X is equal to zero and then
later on its position changes and what
we've got here is a curved graph but
instead of just being able to tell what
the car is doing in terms of how its
position changes over time we can use
differentiation to tell us something
about how fast the car is moving because
specifically on this graph the gradient
or the slope of the graph at any point
tells us the speed at which the car is
moving at that point in time in other
words if we were to find the gradient or
slope of this graph at this point here
corresponding to T is equal to five
seconds
then what we would find is the speed at
which the car is moving at T is equal to
five seconds and what we see is that
this car is constantly speeding up it
initially starts out very very
slow because the gradient is flat and
then as we go further along to the right
the gradient increases so the speed of
the car is increasing over time it's
speeding up so to help us understand
this better what we can do is to
actually plot the gradients of all of
these slopes as time progresses rather
than the position of the car as time
progresses at the beginning like we said
the gradient was zero so at t is equal
to 0 we get gradient is equal to zero
and then as time progresses the gradient
increases in other words as time
progresses the gradient is getting
larger and larger and larger so this is
the graph that we end up plotting what
we've actually plotted is time on the
horizontal axis and like I said earlier
is the speed of the car that we've
plotted on the vertical axis but
technically this speed is actually DX
divided by DT or as it pronounced
normally DX by DT what we've done is
we've differentiated the cars position
with respect to time and specifically if
we differentiate an object's position
with respect to time then we find its
speed and actually interestingly in this
case if we differentiate the
differentiated quantity if we
differentiate speed now with respect to
time then we find the cars acceleration
in other words if we were to find the
gradient of this graph at every point in
time and produce a new graph where we
plotted all those gradients as time
changes then we would see that in this
particular case the gradient of this
graph stays constant for all times it
doesn't change which means that when we
differentiated the speed we find a flat
line because the gradient is constant at
all times and here what we've actually
found is the acceleration of the car
many of you might be familiar with this
idea where we take the distance traveled
by an object and then we produce a graph
that shows the speed of the object and
we produce another graph that shows the
acceleration of this object however you
may have only done it in terms of an
object that is say moving at a constant
speed so you don't have to worry about
changing gradients well this is where
calculus and specifically
differentiation comes into play
differentiation can deal with the fact
that the gradients of our initial
function are changing but the important
thing to take away here is how we took
gradients at different points along the
graph and plotted the values of those
gradients to give us the differentiation
quantity and this has been a very very
brief overview on differentiation I'm
not gonna go into too much detail here
if it is confusing by the way let me
know in the comments down below because
understanding the rest of what I'm about
to say kind of hinges on the fact that
you've sort of understood what I've just
said so far so I really want to make
sure that what I've said so far is clear
and understandable and for that reason
I'll leave some useful resources in the
description box down below so that you
can understand differentiation a bit
better but please do feel free to ask me
in the comments below if I haven't
explained anything clearly so with all
of that in mind let's consider a wave
now let's say it's a really simple wave
like we said earlier a transverse wave
where it's moving like this from left to
right and a particular point along that
wave is actually oscillating up and down
we can even think of this as a wave on a
rope for example let's say one person is
holding on to an end of a rope and they
just wiggle it up and down then what we
see is this kind of transverse wave
anyway that's enough world building
let's just imagine that we've got this
mathematical function that looks like a
sine curve if you're familiar with sine
curves rather than thinking about waves
on a string or anything physical like
that this is a wave that's moving from
left to right over time well let's
assume that we can take a snapshot maybe
we take a camera and we take a snapshot
of this wave at a particular time that
snapshot of that wave looks like this
now what we can do is to be really nerdy
be physicists basically an assign
coordinate axes to this wave in other
words the X Direction is toward the
right and we'll say its displacement its
vertical displacement is up-and-down
because we're going to assign X and u
values to every point along that wave
and we're going to use these values to
mathematically calculate something about
the waves behavior now notice what we've
done we've plotted U on the vertical
axis and X on the horizontal axis let's
differentiate U with respect to X
remember what that means essentially
finding the gradient of our wave at
every single point and then plotting
that gradient so let's start with X is
equal to zero that's this particular
position in space so we see that at this
particular point the gradient is quite
steep therefore it has quite a large
numerical value and then let's say at X
is equal to I don't know one meter we
see that the gradient is zero because
what we've got here is a flat line so we
plot that onto a new graph where we've
got the differentiated quantity on the
galaxias and X on the horizontal axis
and then we repeat this process for
multiple different points along our wave
but realistically we do this for every
single point along a wave what we get is
something that looks like another sine
wave but slightly shifted this is
technically a cosine wave but now this
is not a wave in real space anymore this
is not something that's actually
oscillating this second graph that we've
created is just a mathematical function
whereas the original wave is actually a
wave moving through space we had just
assigned it an x-coordinate and a
vertical displacement new coordinate but
then at this point what we can do is we
can take our graph of D u by D X and X
and then differentiate it again with
respect to X this gives us what's known
as D 2 u by DX squared is basically just
doing the differentiation process again
or some people like to say d squared u
by DX squared which I'm not used to so
I'm not going to say that anymore and so
what we've done is we've taken our
original wave and we've generated a
mathematical function from this way
that's equal to D 2 u by DX squared on
the vertical axis and X on the
horizontal axis this seems to be getting
quite abstract and mathematical but bear
with me things boil hopefully start to
make sense soon what we've done so far
is we've taken a snapshot of the wave
remember we did that earlier we took a
photo of it as it was moving through
space so we could see how it was
behaving at one particular point in time
that's the time at which we took the
snapshot instead let's now do the
opposite thing let's now consider the
behavior of one particular spatial point
as time progresses if we take one
particular point in space let's say I
don't know X is equal to 8 then let's
see how this particular point behaves as
the wave moves from left to right we see
that initially it starts out at U is
equal to 0 but then as the wave moves
from left to right we see that point
moving up and down now what we're going
to do is to plot another graph this time
we've got time on the horizontal axis
and the vertical displacement U on the
vertical axis and we need to be careful
here because this graph we're plotting
is not an actual physical picture of the
wave this is just a plot of what this
particular point this red point along a
wave is doing as time progresses and we
see that at T is equal to zero it starts
out at U is equal to zero and then as
time progresses let's say at T is equal
to one second for you
sample we see that the you value has
decreased to the lowest possible value
and then as the weight continues to move
we see that we've produced another
sinusoidal pattern now remember like I
said this graph that we've produced off
U and T is not actually a physical
picture of the wave is just a
description of what our red point along
our wave is doing over time and it might
be confusing to see so many different
kinds of sine curve or at least
sinusoidal curves but remember the only
one that is actual physical picture of
what was going on is the original wave
here anyway so we've got this picture of
what this particular red point is doing
as time progresses and the wave moves
from left to right let's differentiate
this graph with respect to time not just
once but twice what we're finding is D 2
u by DT squared now you might be seeing
where we're going with this here we
found D 2 u by DX squared earlier and
we've just found D 2 u by DT squared
right now as it turns out both of these
mathematical functions remember both of
these mathematical functions are
sinusoids they look like sine curves and
so we could in principle by some basic
mathematical processes equate d 2 u by
DX squared and dt u by dt squared
because these two functions are
sinusoids because they essentially have
the same shape you should be able to see
that we could in theory equate these
mathematical functions either by
stretching or shrinking one of them or
doing something to both of them and
specifically what the wave equation is
telling us is that when we take d 2 u by
DX squared and multiply it by the square
of the speed of the wave then we get D 2
u by DT squared in other words what
we're finding here is that a sinusoidal
wave is a possible solution to our wave
equation another way to think about this
is a sinusoidal wave moving at a very
specific speed through space the speed
being C is a solution to the wave
equation it bays the wave equation and
there are some very important
constraints to be made here about
exactly what we put as an argument of a
sinusoidal wave function basically it's
not as simple as having just a sine X or
a sine of T function remember that the
oscillation RP down enos of the wave
depends on not only position but time as
well because earlier what we did is we
took a snapshot at a particular time and
we also looked at the movement of one
particular point in space at all times
but if we want to describe this whole
wave
for all particular points along the
entire x-axis and for all different
times then we need to worry about the
wave being a particular function of both
X and T and exactly what function of X
and T we have is going to be related to
the speed of the wave for example here
is a possible solution of the wave
equation I'm not going to talk through
it because it's quite advanced but if
you understand differentiation pause the
video and check it out when you do
you'll see that we've got these
quantities Omega and K in our argument
of the sine function we're specifically
unknown as the angular frequency and the
wave number of the wave these are very
important quantities for a wave and in
this particular case we can see that
they also relate to the speed of the
wave but like I said that's way too
advanced I'm not gonna go into huge
detail here now here's the thing this
wave equation that we've been talking
about so far is quintessentially known
as the wave equation but there are lots
of different wave equations specifically
this is a classical wave equation in one
spatial dimension classical because it's
not accounting for any effects of
relativity or quantum mechanics this
describes all sorts of things such as
mechanical waves light waves sound waves
and so on you just switch out the value
of C for the particular speed of the
wave that we're thinking about but like
I said this only describes classical
waves waves that have been known about
for years and years an example of a
different wave equation is actually the
Schrodinger equation the Schrodinger
equation describes how the wave function
of a quantum system behaves over time in
space
and like I said that's a wave equation
with very very different qualities so
this is my brief description of what the
classical wave equation is about certain
things that I haven't gone into are the
fact that is known as a second-order
linear partial differential equation
second order simply because we're
differentiating our quantity twice both
with respect to X and with T linear
because our equation only deals with you
it does not deal with any powers of U
such as U squared or u cubed or anything
like that
remember the d 2 u by DX squared only
tells us that we're differentiating u
twice we're not having to deal with
differentiating like u squared twice or
anything like that and it's a partial
differential equation because
technically these DS are partial
derivatives remember what I said earlier
about the vertical Wibbly Ness of a wave
being dependent on both what position of
the wave we're considering and
what time along the waves journey were
considering but partial differential
equations kind of split this dependents
up into separate chunks we see on one
side we're only dealing with the
dependents on time and on the other side
we're only dealing with the dependence
on position and these curly d's tell us
that we're dealing with just
differentiating with respect to X and
ignoring completely anything to do with
T and then vice versa so this is a
second order linear partial differential
equation described in classical waves as
if I haven't thrown enough words at you
yet listen like I said earlier if
something doesn't make sense to you if
I'm just throwing was that you that
don't make any sense whatsoever then let
me know in the comments down below and
I'll try and clarify it and of course if
I've got something wrong then please do
let me know down below as well I'd like
to correct this by leaving a comment
down below and pinning it so that you
guys can read it and not go away with
incorrect information for the time being
though that's all I have for you on the
wave equation it's a very basic very
hand wavy overview of the wave equation
or rather the classical wave equation
but I'm hoping that for those of you who
aren't undergraduate physics students
you'll get a bit of a sense of what the
wave equation is trying to do and
remember that all those sinusoidal waves
the ones that we've talked about in this
video are a very valid solution of the
wave equation they're not the only
solution to the wave equation so with
all of that being said let me know in
the comments what equation you'd like me
to try and explain next before now I'm
gonna end this video here thank you so
much for watching and thank you so much
for all your support I really appreciate
it I'll see you really soon
buh buh buh buh bye