Mr. p: Good morning. Today we are going to look at graphs of waves
Billy: Yeah, I Iove graphs.
Bobby: Right.
Bo: Of course you do.
Mr. p: Specifically, these four graphs of mechanical waves, which at first glance,
might look identical, however, it is very important that you understand the differences between them.
[intro]
Bo: They all look the same to me.
Billly: Yeah, they all have the same exact sine curve.
Bobby:... They many have same sine curve, but they have different labels on their axes.
Bo and Billy: Oh..
Mr.p: Right. Notice on the y-axis we have either position or density.
On the x-axis we have either time or position.
Let's start with understanding the position as a function of time graph.
If we have a mass-spring system moving in simple harmonic motion,
this could describe the position of the mass in a mass-spring system as a function of time.
Bobby: uh, mr. p?
Mr.p: Yes Bobby?
Bobby: I feel like the yellow part of the graph indicating the position of the mass
in a mass spring system in simple harmonic motion,
it should be moving at a constant rate along the time axis, however, it is not quite doing that. Why not?
Mr.p: Good point Bobby.
So, in my demonstrations I tend to use real objects and not ideal ones or ideal animations.
So this is not an ideal spring and will not create a perfect sine wave graph.
If it were an ideal spring, it would be a perfect sine wave. That's why it's not perfect.
Uh... In fact none of these demonstrations are going to be perfect, but they are all going to be very close. Okay.
Bobby: Sure Mr. p, that makes sense.
Mr. p: Okay, this position as a function of time graph
could also describe the movement of a mechanical wave as a function of time...
More specifically, this describes a transverse wave. Remind me Bo, what is a transverse wave?
Bo: Uh, a transverse wave is where the direction of wave propagation is....
perpendicular to the direction of the disturbance of the medium.
Mr.p: Right. So this graph represents the position of a specific point
on the transverse wave as a function of time. And it is tracing out a sine curve
because the source of the wave is an object moving in simple harmonic motion.
Actually, this could be a graph of the location of the source of the wave,
my hand, or any other point along the spring.
Bo: Wow, ok I can actually see that.
The point on the spring plots out the sine curve as a function of time. That's pretty cool.
Billy: It is cool.
Bo: Yeah
Mr. p: Agreed.
But now let's change the graph to a y-position as a function of x-position graph.
Notice this graph can no longer describe the motion of a mass
in a mass-spring system moving in simple harmonic motion,
because a mass-spring system only moves in one dimension.
Billy: Well that makes sense, the mass-spring system has a constant position in the x-direction, so this graph,
which shows multiple positions in the x-direction, cannot be a graph of simple harmonic motion.
Mr.p: That is correct Billy. Realize this graph can describe the motion of a mechanical wave.
This graph simply describes the location of all of the particles
of a mechanical wave at one specific moment in time.
In fact, if we pause the wave on the spring video,
you can see how the physical location of the wave is represented by the graph.
Billy: Uh, I'm sorry, however I do not see it.
Mr.p: Okay, let me add a curve which adjusts the x-axis scale of the graph
and I think you will be able to see it better. Does that help?
Billy: Yes it does. Thank you Mr. p.
Mr. p: You are welcome Billy.
Again, this graph is different because it describes the location of the whole mechanical wave
at one specific time, unlike the previous position as a function of time graph
which describes the location of one piece of the mechanical waves as a function of time.
Bo: Sure. It's an instantaneous graph.
Bobby: Right.
Billy: Instantaneous graph. Yeah. Okay.
Mr. p: Okay, now let's go back to the position as a function of time graph
and discuss the measurement which extends between successive crests and successive troughs.
Bobby, what are we measuring when we go from one crest to the next
or from one trough to the next on a position as a function of time graph?
Bobby: Well the x-axis is time. So, we must be measuring a time of some sort.
Billy: That makes it the time for one full cycle.
Bobby: Right, so that measurement is the period, the time it takes for one full cycle of the wave.
Mr.p: This is true for both simple harmonic motion and mechanical waves.
However, Bo, if we change the graph to a y-position as a function of x-position graph,
what is the measurement between successive crests or troughs now?
Bo:..Well because the x-axis is position, that measurement must be some sort of distance
So, it's the DISTANCE between two crests, that means it's the distance for one full wave.
Bobby: That is the wavelength.
Bo: Right, the wavelength.
Mr.p: Yes, the wavelength is the length on one complete wave cycle
measured from crest to successive crest or from trough to successive trough.
In other words, meausured from one point on the wave to the next corresponding point
on the wave which is going in the same direction.
The symbol for wavelength looks like the letter y mirrored
over the x-axis and is the lowercase Greek letter lambda.
Also, another definition of wavelength is the distance traveled by a wave during one period.
Bo: Uh, Mr. p?
Mr.p: Yes, Bo?
Bo: I thought the sine curve on this graph was an instantaneous graph
representing the position of the full wave all in one specific time.
Mr.p: It is.
Bo: Then why do you have the spring moving on this graph?
Mr.p: Okay I can see how that can could be confusing.
What I mean to show here is that this y-position as a function of x-position graph
can show the position of this wave at any moment. In other words, we can pause it to show a sine wave
or a negative cosine wave
or a negative sine wave
or we can pause it show a cosine wave
or even something in between those. Does that help?
Bo: Yeah, thanks.
Billy: Mr.p?
Mr.p: Yes, Billy
BIlly: I can see that mechanical waves have a wavelength
and we know there are a lot of similarities between waves and simple harmonic motion.
Does simple harmonic motion have a wavelength?
Mr.p: Interesting question. Recall that simple harmonic motion cannot have a graph of y-position
as a function of x-position, therefore, simple harmonic motion does not have a wavelength.
Billy: Okay, thanks. Simple harmonic motion does not have a wavelength.
Mr.p: Let's now change the graph to density as a function of time.
Bobby, what type of wave does this density as a function of time graph describe?
Bobby: Well, last time we learned about transverse and longitudinal waves.
The other two graphs were transverse waves, so I bet... I bet this graph describes a longitudinal wave.
Mr.p; I like your logic Bobby. You are correct, this graph does describe a longitudinal wave.
Billy: In a longitudinal wave the direction of the wave proprogation
is parallel to the direction of the disturbance of the medium.
Bobby: Yeah Billy. That must mean that a crest on the graph represents a location of higher density
in a longitudinal wave or a location of compression.
A trough on the graph must represent a location of lower density in a longitudinal wave or a location of...
what is the opposite of compression?
Bo: Rarefaction. Rarefied air has low density. So a location of low density is a location of rarefaction.
Billy: But you could also call it expansion, correct Mr.p?
Mr.p: That is correct. A crest is a location of a high density or compression,
and a trough is a location of low density or rarefaction or expansion....
Now realize, this graph of density as a function of time represents the density of a specific location
of a longitudinal wave as a function of time.
The specific location for this graph is where the hand is pointing.
Again, because the graph represents the density of one specific location on the spring as a function of time,
the location on the spring which the hand is pointing at starts at equilibrium density,
then the density of the location increases...
and decreases back down to equilibrium density,,
and continues to increase to minimum density...
and then increases to equilibrium density...
and continues to increase to maximum density...
and then decreases back down to equilibrium density.
Billly: Wow
Bo: Woah.
Bobby: Yeah
Bo: Okay
Mr.p:.. Now let's talk about the last graph, density as a function of position.
This graph still describes a mechanical, longitudinal wave;
however, it represents the density and position of the whole wave at one specific time.
Bo: Right, this is an instantaneous graph.
Billy: Instantaneous graph.
Mr. p: Yes, this represents the density of the whole wave at one specific time,
unlike density as a function of time,
which represents the density of one point on the wave as a function of time.
Billy: Oh right. You can see the two crests which are the locations of high density.
Bobby: Those are labeled compression.
Billy: And the one trough, which is a location of low density.
Bobby: That is labeled rarefaction.
Bo: I like what you did with the subtle text spacing for compression and rarefaction there Mr. p.
Mr. p: Thanks everybody. Now, realize the terms period and wavelength
are still applicable for the longitudinal wave.
We measure period on the density as a function of time graph.
And we measure wavelength on the density as a function of position graph...
To review, we have four graphs here: Position as a function of time, y-position as a function of x-position,
density as a function of time, and density as a function of position.
The only one of these graphs which is applicable to simple harmonic motion
is the position as a function of time graph.
Both of the graphs with position on the y-axis describe a transverse mechanical wave.
Both of the graphs with density on the y-axis describe a longitudinal mechanical wave.
Both of the graphs with time on the x-axis describe a specific point on a wave as a function of time
and have period as the measurement of one full wave on the x-axis.
Both of the graphs with position on the x-axis describe the whole wave at one specific time
and have wavelength as the measurement of one full wave on the x-axis...
Thank you very much for learning with me today, I enjoyed learning with you.