mr.p: Good morning. Today we are going to review
the mechanical wave topics covered
on the AP Physics 1 exam.
Bo: Hey guys.
Billy: Hey Bo.
Bobby: Hi Bo.
♫ Flipping Physics ♫
mr.p Let's start with the concept of a single wave pulse.
Here I've drawn a rope, a spring, or a string
through which a wave pulse is travelling.
Actually it could even be the side view
of the surface of a lake.
The important piece to understand here is that a wave
is a disturbance travelling through a medium,
it is not the motion of the medium itself.
That disturbance or wave pulse
is energy travelling through the medium,
and the overall displacement of the medium
after the wave pulse has passed by is equal to zero.
As the wave goes through the medium,
the rope goes up and down and returns
back to where it started.
So a wave is not a physical object,
it is energy moving through a medium
as a disturbance of that medium.
Sending multiple wave pulses down at specific,
even time intervals will create periodic waves.
There are two types of mechanical waves.
One is called a transverse wave.
In a transverse wave, the disturbance of the medium
is perpendicular to the direction of wave propagation.
For example in this transverse wave
which I've drawn on the board,
the direction of wave propagation,
the wave pulse is moving to the right,
and the disturbance of the medium is up and down
which is perpendicular to the direction of wave propagation.
Examples of transverse waves are
ripples in a pond and waves on a rope.
The second type of mechanical wave
is a longitudinal wave.
Bo, how is a longitudinal wave
different from a transverse wave?
Bo: In a longitudinal wave the disturbance of the medium
is parallel to the direction of wave propagation.
mr.p: Correct. The only difference between
these two definitions is that single word.
For transverse it is perpendicular,
and for longitudinal it is parallel.
Examples of longitudinal waves are
sound waves in air, and the waves that travel
through the surface of the planet
after an earthquake or a large explosion
which are called seismic waves.
So far every wave we've talked about
requires a medium to travel through.
However there is one type of wave
that does not require a medium.
Bobby, what is that wave?
Bobby: Electromagnetic waves do not need
a medium to travel through.
Billy: And electromagnetic waves are transverse waves, right?
Bo: Yeah, they are alternating electric and magnetic fields.
mr.p: Electromagnetic waves do not require
a medium to travel through so they can
propagate through the vacuum of space
and they are transverse waves.
Now, let's take a closer look at this graph
of a periodic transverse wave.
The top part of a wave is called a crest,
the bottom part is called a trough.
And the distance between two successive crests,
Billy what is that called?
Billy: The distance between two successive crests
is called the wavelength.
Bobby: And the symbol for wavelength is a lower case lambda,
and the height of a crest or the depth of a trough
is called the amplitude.
Billy: Right, and the higher the amplitude,
the larger the amount of energy in the wave.
mr.p: And Bo, if I replaced the horizontal axis with time,
what would the time between two crests be called?
Bo: The time between two crests...
Oh yeah, that's the period.
mr.p: Right, because the period is the time it takes
for one wavelength to pass by a point.
And Bobby, this is a graph of a transverse wave.
What change could I make to the graph
to make it a graph of a longitudinal wave?
Bobby: Uh...
Oh, change the y-axis to density.
mr.p: For a longitudinal wave,
you have density on the y-axis.
Therefore a crest is a location
of high density or compression,
and a trough is a location
of low density or rarefaction.
That is why a longitudinal wave
is sometimes called a compression wave.
Bo, what is the equation for velocity?
Bo: Velocity equals change in displacement
over change in time.
Billy: Actually, x stands for a position.
Delta x is displacement.
So velocity equals change in position over change in time.
Bo: Right.
mr.p: Velocity equals change in position over change in time.
And if the wave has gone through a distance
of one wavelength, how long did that take?
Billy?
Billy: Um...
Bo: The wavelength is the distance a wave travels
during one full cycle, and the period
is the time it takes for one full cycle.
So the time for one wavelength is the period.
Billy: Yeah.
mr.p: And frequency equals one over the period, therefore...
the velocity of a wave equals
its frequency times its wavelength.
Now, a lot of the terms we're using
and the graphs we're drawing are very similar
to the ones we use for simple harmonic motion.
Because simple harmonic motion and waves are related,
absolutely true.
However, simple harmonic motion
does not have a wavelength.
So please do not find wavelength for simple harmonic motion
and do not use this equation for the velocity of a wave
for simple harmonic motion, please.
Now Bobby, is a wave a physical object?
Bobby: A wave is not a physical object, it is energy
in the form of a disturbance of a medium.
mr.p: Which means two physical objects
cannot occupy the same space at the same time.
However, two waves can occupy
the same space at the same time.
And we determine what happens using
something called superposition.
Superposition is the description of how waves
interfere with one another when they occupy the same space.
Basically, we add the amplitudes together.
Let's say we have two wave pulses in a rope
moving toward one another.
Now, I've identified them as blue and red
to better visualize what's happening,
but realize it's all one rope.
Let's take a look at what happens a few moments later
when both wave pulses occupy the same space.
The amplitudes or energies add together
to create one larger wave pulse.
This is called constructive interference
and it is important to understand what happens
after the waves interfere with one another.
The two waves continue on their way
unaffected by the constructive interference
that occurred when the two occupied the same space.
Now let's look at what happens when the amplitudes
are not on the same side of the rope.
Because the two interfering wave pulses
are of equal magnitudes and opposite directions,
they end up cancelling one another out completely.
This is called total destructive interference.
Now you can have destructive interference
or partial destructive interference
if the magnitudes of the amplitudes are different.
Again it is important to realize
that the two wave pulses continue on afterwards
unaffected by what happened when the two
occupied the same space.
Even though when they did occupy the same space,
the rope was entirely flat because
they completely cancelled one another out.
Now let's talk about standing waves.
What I've drawn here is a rope
that is fixed at one end, that end is stationary.
And I'm sending wave pulses down
that when they run into that fixed end,
they are going to be reflected,
meaning they'll come back,
and they will be inverted, which means they switch sides.
And if I continue to send wave pulses down the rope,
those wave pulses will interfere with
the reflected and inverted wave pulses
both constructively and destructively
to create what are called standing waves.
This is how we draw standing waves.
Notice L, the length of the rope,
and I've set up different standing wave patterns
in the rope.
These dotted lines represent locations
the rope could be at some point in time.
So the rope is moving up and down here,
up and down here, up and down here.
Notice where it's not moving up and down,
these locations right here which are locations
of total destructive interference,
which are called nodes.
And we have locations of constructive interference
which are called antinodes.
Now let's determine the wavelength
of each of the waves in these standing wave patterns
in terms of L.
Billy, in the first standing wave pattern
which has one antinode and two nodes,
how many wavelengths equals L, the length of the rope?
Billy: With one antinode there is only half a wavelength
in the distance of the length of the string, L.
mr.p: Correct, and therefore the wavelength of the wave
equals two times L, the length of the standing wave.
And we already know the velocity of a wave
equals its frequency times its wavelength.
Therefore the frequency of the standing wave pattern
is equal to the velocity of the wave
divided by its wavelength, or the velocity of the wave
divided by two times the length of the rope.
Bobby, could you please do the same thing
with the next standing wave pattern,
which has two antinodes and three nodes?
Bobby: Let's see.
One wavelength equals L, so the frequency
equals the velocity divided by the wavelength
or the velocity divided by the length of the string.
mr.p: Very nice. And Bo, could you please do
the last standing wave pattern which has
three antinodes and four nodes?
Bo: Okay, with three antinodes and four nodes,
there are one and a half wavelengths
in the length of the string,
which means the wavelength equals
two times the length divided by three.
And therefore the frequency equals
three times the velocity divided by
two times the length of the string.
mr.p: Thank you. Believe it or not,
we're creating a pattern here which is easier to see
when I rewrite the equations this way.
They are one, two, and three times
the velocity of the wave on the string
divided by two times the length of the string.
Which gives us the equation for the frequency
or the sound we hear coming from
a stringed instrument like a guitar or a cello,
and this N is called the harmonic number.
When N is equal to one, it is called
the fundamental frequency or the first harmonic.
When the harmonic number is equal to two
it's called the second harmonic.
When N is equal to three it's called the third harmonic,
and so on and so forth.
Just so you know, we hear the frequency of sound as pitch.
For example concert pitch, the A above middle C,
is typically 440 Hz.
Alright, now let's do the same thing we just did
only with wind instruments,
and let's start with what is called
an open pipe instrument which is
open at both ends, like a flute.
Actually we don't really need to do anything,
because the equation for an open pipe wind instrument
is exactly the same as the one for a stringed instrument.
Remember, an open pipe instrument open on both ends.
And the reason for that is because
an open end creates an antinode.
So the pictures look almost identical
except they are inverted.
In other words, for example, for this first one
we end up with antinodes on either end
and one node in the middle.
But again, it gives you the exact same equation.
So we're going to move on and talk about
a closed pipe instrument which is
closed on one end and open on the other.
In a closed pipe wind instrument,
the closed end creates a node
and the open end creates an antinode.
Therefore for the fundamental frequency
of a closed pipe instrument,
one fourth of the wavelength fits
for the length of the pipe,
and therefore the wavelength is equal to
four times the length of the pipe.
And we know the velocity equals
frequency times the wavelength,
and therefore the frequency is equal to
the velocity divided by the wavelength
or the velocity divided by four L.
So notice the difference here.
Rather than having two times the length of the string
or the length of the open pipe
for an open pipe instrument,
we have for a closed pipe instrument,
it is divided by four times the length
of the closed pipe instrument.
Which gives us the equation for a closed pipe instrument.
m times v over four L instead of two L
and m instead of N because m,
the harmonic number for a closed pipe instrument,
can only be odd, 1, 3, 5, et cetera.
And please realize that standing waves
can only be set up in string and wind instruments
at specific wavelengths, and that's determined
by the length of the string or pipe.
Now as long as we're talking about music,
let's talk about beat frequency.
If you play two notes that are
close to one another in frequency,
then the interference of those two frequencies
with one another will cause periodic changes
in the amplitude of the sound
which sounds like beats.
In other words, if we played one note by itself
which is 440 Hz, we hear just that note.
(tone plays)
And if we play another note by itself which is 441 Hz,
we will also hear just that one note.
However if we play both together,
we hear a beat frequency.
And that beat frequency is equal to
the absolute value of 441 minus 440,
or one cycle per second, or one beat per second.
If we play 442 and 440 Hz,
we hear two beats per second.
Again this is caused by two frequencies
that are near one another
and the interference of those two waves.
One last concept, the concept of the Doppler effect.
This is a visual representation of the sound
coming from a single frequency, stationary sound source.
And each one of these lines is called a wavefront
and represents a crest.
Therefore Bobby, what is the distance
between each wavefront or the distance
between each crest called?
Bobby: The distance between two successive crests
is the wavelength of the wave.
mr.p: Now let's draw the same sound source moving to the right.
As the sound source moves to the right,
the wavefronts to the right of the sound source
are closer together and therefore
the wavelength is decreased.
[mr.p] What we really want to know is what we hear.
Billy, what happens to frequency?
Billy: Well the velocity of a wave
equals frequency times wavelength,
so the frequency equals velocity divided by wavelength,
so then frequency and wavelength
are inversely proportional.
So as the sound source moves towards the observer,
the wavelength will be decreased
and therefore the frequency will be increased
and the pitch will sound higher.
mr.p: Notice that the reverse is true as the sound source
is moving away from the observer.
As the source is moving away from the observer,
the wavelength is increased and therefore
the frequency is decreased.
Billy: So it sounds like this.
(imitating car speeding by)
mr.p: (laughs) Yes.
And this change in frequency heard by the observer
is called the Doppler effect.
We have reached the end of my review lesson
about mechanical waves.
My next review lesson is about electrostatics,
you are more than welcome to enjoy that video.
You can also visit my website flippingphysics.com
where you can find all of my AP Physics 1
review lessons organized with lecture notes.
Thank you very much for learning with me today,
I enjoyed learning with you.