Hi I'm Toby and welcome back to the Joy
of Mathematics, I'm so glad you could
join me today. I'll run some ideas across
the bottom of the screen that we are
going to encounter during a story today.
I'd also like to thank this episode's
sponsor Brilliant.org.
So today's story is another one about
our friend Bob,
you may remember Bob from a previous
episode, he lives in a little hut
in the hills and he cares very deeply
about his animal friends. So this story
starts early in the morning when Bob is
in fact asleep, he is awoken though by
the sound of an acorn hitting his little
tin roof. You see, Bob's hut is surrounded
by several tall oak trees. The frequency
of the acorns falling on his roof is
getting less and less as the days get
colder and autumn comes to an end. Bob walks outside and collects the
acorn that woke him as well as a few other acorns and he puts them in his basket of
acorns which he has been collecting.
These baskets are overflowing so Bob
knows that it is time for his yearly
acorn adventure. Bob doesn't have much
use for the acorns but knows of a large
family of squirrels who need them to
snack on during winters hibernation. It
is time to take the acorns to the
squirrels, the only problem is---
There's a
giant lake in the way.
Bob does have a
small rowboat which he's going to use to
take the acorns across the lake. It took
him several trips back and forth to load
the baskets of acorns into his boat and
as he was doing so he noticed a sign. It
was a warning sign
and Bob will need to understand the sign
to know whether it's safe to go out on
the lake today. Bob looks out onto the
water and he can see a duck out there, it
is bobbing up and down smoothly with the water. This motion is actually the same
motion that Bob moved with when he was
loading the acorns into the boat. He
was jogging backwards and forwards
between the boat and the acorns, jogging
fastest in the middle section but
slowing down to pick up a basket or
slowing down when he reached the boat as well. If he didn't he would trip over it.
Smoothly going back and forth is what is
meant by sine. It's a natural sway, the
epitome of smoothness. A gentle back and forth rocking that starts fast, slows
down, turns around and speeds up again. It is the way that a spring would bounce or
a pendulum would swing. A different way
to move backwards and forwards would be
abruptly like something that bounces off
a wall at full speed without slowing
down. That would not move in the motion
of sine. If we traced the path of the
duck over time it would trace out a sine
wave.
It's the same smooth motion of sine but
with a horizontal component.
The warning sign was there to say that a
sine shaped wave was moving through the
water and that was what was causing the
duck to bob up and down. This is a sine
wave that I've drawn in terms of its
width and height but it also moves
forward in time because the wind is
blowing it to the right. I can only draw
a snapshot of the wave, this is a picture
taken at t equals zero, the time at
midnight, but it is always moving. How
high and how low the wave gets depends
on the first number on the sign here,
this is the waves amplitude. For this
wave it is four so the wave will get up
to 4 meters high and 4 meters low that's
kind of lucky because the dangerous sea
creatures lurk at about 6 meters in
depth so Bob's not going to encounter
them on the boat today. If there was no
number here it would be our stock
standard sine wave and it would get up
to 1 meter high and minus 1 meters low
down the bottom. The second number on the
sign is inside the bracket, it's in front
of the x and it has to do with how
stretched out or squashed the wave is. If
there was no number here then the wave
would reset after its natural cycle, that
would be how far the wave travels before
it gets back to where it started which
was zero. I'll make it a little bit
clearer here that the wave starts at
zero goes up gets to zero goes down and
gets back to zero. So to about here is
one cycle, but what is the natural cycle
of sine? Well it might be helpful to
think about a circle. Now circles are
often used to explain sine but a circle
is really nothing more than combining
two sine waves, one in the horizontal
direction and one in the vertical
direction offset by a quarter of a cycle
and you could call that cosine. To
complete one cycle of a circle
you would have to go around 360 degrees
and that is the natural cycle of sine -
but we don't have to work in degrees
those are just a human construct. If we
think about a standard sine that goes
from 1 to minus 1 that would trace out a
circle of radius 1 now we can define a
Radian as the distance traveled around a
circle divided by the radius of that
circle so in one full cycle of sine we
will go all the way around the circle,
always when you want to find out the
circumference of a circle it will be
twice the radius times pi so for us we
will have twice the radius that's 2
times pi divide that by our radius which
is 1 and we could just as well have
written that as 2 pi. 2 pi is the number
of radians in our circle, a different way
of writing 360 degrees. 2 pi is the
natural cycle of sine, it's how far we
would need to travel in the x-direction
before a wave gets back to where it
started. Now all that is true unless we
have a number here in front of the x
then we're either going to be squished
or a little stretched out. The secret
lies in the number that we have here on
the bottom, for us it is a 5, we have two
pi over five and five is actually our
wavelength. It tells us that for our wave
it takes a distance in the x-direction of
five meters before it gets back where it
started. Now that's not 2 pi, if it had
have been our natural 2 pi for our
standard sine wave then we would have
had a 2 pi on the bottom there and the 2
pi on the top and bottom would have
cancelled to leave us with nothing
written there essentially a 1 in front
of our x, but we have a 2 Pi over 5
so we know that this entire distance
will be five meaning that each bump or
each trough is going to be two and a
half meters across that's kind of useful
information to try and figure out if
it's safe for our little boat which is
two meters long to travel along it, and I
think a two meter long boat on a two and
a half meter bump, that's okay, that
should be alright for Bob. So now we know
things about the size of the wave but
how fast is the whole thing moving
across? Well we can look to the last term
in our equation here to find that out.
If we added any number in here inside the
brackets it would shift the wave along
but a wave on the water doesn't just
shift a bit and then stop it continues
moving so that's why we have multiplied
a number in here by t, this offset will
continue to grow as time goes on. Now in
a very similar fashion to what we did
with our first term we can look at the
number on the bottom of this 2pi
fraction here to find out how long it
will take our wave to essentially reset
this time it's not resetting in space
but in time. For us it would take 10
seconds for our wave to move such that
it's moved along and it looks like it is
where it started
so this swell would move over to here
and a different swell would replace it.
That 10 seconds is the period of our
wave for us it takes 10 seconds to move
a distance of 5 meters since that was
our wavelength, giving us the speed of
our wave as 0.5 meters per second. Note
that there is a negative sign in here
for this term which does move the wave
to the right and a positive sign in
there would have moved it over to the
left. A wave speed of half a meter a
second is actually pretty good for Bob
it will help him get to the other side
of the lake pretty quickly.
The thing with Bob is that everything in
his life tends to work out to be just
right, almost suspiciously convenient and
since today is no different
Bob sets off across the lake smiling at
the sign knowing that even though it
will be a little bumpy he's going to
have a joyous journey. He enjoys every
high swell and laughs on the way down. It
was a very sunny day and as Bob stepped
off his boat on the other side of the
lake he noticed he had developed a bit
of a tan. He delivered the acorns to the
squirrels who were very thankful, Bob
knew he had given much more than they
needed but if they lost some or couldn't
eat all of the ones that they buried the
acorns would grow into oak trees and
help to provide food for the future
generations. It is now time to thank our
sponsor and to deliver a special message.
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This is Bob.
Bob collects acorns.
Bob cares for the
squirrels.
Bob is a good guy.
Be like Bob.
Bob's actions as well as
helping the squirrels, will help to plant
new trees. Bob is part of team trees and
encourages you to also plant some trees
or donate to the Arbor Day Foundation
at teamtrees.org to help plant 20
million happy little trees by 2020.
I'd like to
wish you happy
studying and I hope that you have an
absolutely mathematical day.
