Where up and down or back and forth do not add up to zero...
To this point in this course we
have discussed the motion of material objects, particles or
collections of particles moving through otherwise empty space so
that they are in different places at different times. Now we want
to study a related but different sort of motion. Imagine a long
string stretched between a fixed point and a point that can be
moved up and down. If we pulse the movable point quickly up and
down once, then hold it in place, we will see something move down
the length of the string. Run the Stretched
String Pulse display to see this. In this display we only
look at part of the string. It is attached way off the screen to
the right. The Action button produces the initial impulse.
What this display illustrates is that
although the particles that make up the string only move in the y
dimension, the disturbance moves along the string in the x
dimension. This idea of a "disturbance", as opposed to
a particle or other object, moving through space is something new
to us. What is actually being transported down the string is
energy, in the form of kinetic energy of the string's mass
and potential energy due to displacement of the string. Energy
transmitted in this way is said to be carried by a traveling
mechanical wave.
In order for a mechanical wave to exist there must be some
source of disturbance, a medium that can be disturbed, and some
physical connection through which adjacent particles in the
medium can influence one another. We distinguish here between
mechanical waves where there is a tangible medium in which the
wave travels, and other types of waves where the medium is not so
evident. Sound waves, water waves and waves in solid objects are
all mechanical waves. For the rest of this section I will use the
term waves to mean mechanical waves.



The kind of traveling wave we show
above is called a pulse. It is a disturbance that does not repeat
itself. This particular pulse is very well behaved in that it
does not change its shape as it travels along the string. Real
pulses on real strings tend to spread and flatten out over time.
Another characteristic of this pulse is that it may be combined
with other similar pulses by simply adding the amplitudes at any
instant in time. The amplitude is the height of the pulse. This
sort of combination of waves is called superposition. The
Pulse Superposition
display shows two
pulses, one starting at the left and one starting at the right,
traveling in opposite directions. Waves that add in this fashion
are called linear waves. Linear meaning in this case that the
waves follow a simple superposition rule, not that the waves
themselves involve straight lines.

Waves may be transverse, as the pulsed string example was, or
longitudinal, as is the case of a sound wave where the
disturbance is in the form of changing density of the medium. An
example of a longitudinal wave is the disturbance we saw
traveling the length of the little nine atom string in the
Atomic Solid
display when Action was
first clicked. Look at that once again if you please.
Notice that when you halt the display with the Cut button you
may find some of the atoms bunched together and others spread
apart. Since density is the number of atoms per unit length in this
system, what you are seeing is variations in density as the
energy of the initial poke travels down the string of atoms and
bounces back and forth between the ends. In longitudinal waves,
the regions of high density are called condensations, those of
low density called rarefactions. In our transverse wave example
we avoided the complication of the energy being reflected at the
ends of the medium by dealing with a very long string and only
looking at the left 20 meters of it. In this case our medium is
only nine atoms long so the energy is trapped in that short
length and must be reflected at the ends, greatly complicating
the motion of the atoms.



If we use a string that is
attached to a rigid support at the right end of the screen we
will see this reflection phenomena take place on the string as
well. Look at the
Inverted Reflected Pulse
display to see this effect. Notice that the reflected
wave has a negative amplitude whereas the original wave had a
positive amplitude.

If the right end of the string
was attached to a frictionless massless ring on a vertical post
so that it was free to move up and down, then the reflection
would be without the phase inversion. The reflected wave would
come back with the amplitude of the same sign as the initial
pulse. The
Uninverted Reflected Pulse
display illustrates this situation.



Both the examples we have looked at so far were of one
dimensional waves. That means that the energy moved in only one
dimension, even though in the case of the transverse wave the
string motion actually involved a second dimension. Let's go
back to the pulsed string example and look at it in a slightly
different way. At any instant in time, we have y depending on x,
y=f(x). Run the Stretched String Pulse
display and halt the run with the Cut button when the pulse is
near mid screen. Then you may use the cursor to show the value of
y for any x at the instant you stopped time.
In the
Pulse Sliced Along Time
display what you see is a whole series of y
vs. x pulses at different values of time, as though 20 snapshots
were taken and laid out along the time axis in accordance with
the time they were taken. Click on the Action button to see the
plot develop. You can change the viewing angles to better
visualize the (x,t) surface.
This display is sort of a "Where is the pulse?" view.
It shows for the selected values of t, how far along the xaxis
the pulse has moved. Notice that the displacement of the pulse
between any pair of time values is the velocity of the pulse
times the difference in time. By replacing x in f(x) with xv*t
like this,
y=f(xv*t),
we can include the time dependence in our pulse function. In fact
any function, f(xv*t), represents a traveling wave. In the case
of a single pulse the function must be one that yields a single
peak. The actual function used here is
y=3/((xv*t)^{2}+1).
Whenever (xv*t) is zero, the function has its maximum value of
3.0. When x is much less than or greater than v*t, the
denominator becomes large and the function approaches zero.

Now if instead of freezing time at several points and examining
y=f(x) over the range of x values, we will fix several values of
x and look at y as a function of time at that x position. At any
position on the xaxis we have y depending on time, y=f(t). A
series of plots of y vs. t pulses corresponding to different
values of x is shown in the Pulse Sliced
Along x display. This is sort of a "When is the
pulse?" view. If I take a fixed position on the x axis, when
does the pulse come by?



All together then y is a function of both position, x, and time,
t, y=f(x,t). As we have already seen, the particular function of
x and t in this case is
y=3/((xv*t)^{2}+1).
If you imagine an (x,t) plane,
then for every point in that spacetime plane there is a unique y
so that the values of y form a surface over the plane. The Pulse SpaceTime Surface display
illustrates that idea.
I
know this seems like a lot of bother, studying a onedimensional
wave in three dimensions... and not even proper dimensions at
that, one of them being measured in seconds rather than meters.
This idea of a function of more than one variable is a logical
extension of the function discussion we had early in the course.
It will be convenient for us to be able to visualize a wave as a
function of position and time when it comes to understanding some
of the underlying principles that make waves work.

Let's look at another example of our wave on a string, one
perhaps more in line with our intuition about waves. The pulsed
string served its purpose in making clear the traveling aspect
of the wave but we are accustomed to waves which are periodic,
which actually look wavelike. Instead of giving the string one
shake and holding it still, now we will move the end continually
up and down in simple harmonic motion. Run the String Wave display to see this. This time
the Action button turns on an oscillator to which the string is
attached. You will see that our string wave travels down the
string just as the single pulse did, with a certain velocity. The
difference here is that there is a whole train of pulses
alternating between negative and positive y values.
Again by replacing x in y=A*sin(x), by (xv*t) we get the
function
y=A*sin(xv*t),
which describes the traveling sine wave.



The
Continuous Wave SpaceTime Surface
display illustrates the continuous wave started at time zero.
Notice that part of the spacetime plane is not covered by the
disturbance since distant x at early time was not accessible to
the wave.
Clearly continuous waves of the sort illustrated here have a
definite frequency , f,
established by the driving mechanism. One cycle of the driver produces one
cycle of the wave. The velocity, v, of the wave is determined by
the nature of the medium. We will work on that relationship some
more later. The period, period
, T, of the wave is obviously 1/f based on the definition of f.
The wavelength , l , of the wave is determined by the period
and the velocity. The wavelength must be the distanced traveled
by the wave in one period. In the form of equations we have
T = 1/f and l = T
* v.

Now
if we look at a wave on the spacetime surface which has a higher
velocity we would expect to see more of the plane covered because
the wave could reach distant x earlier. Also we should see that
the number of fluxuations seen at a fixed x are more per unit
time since the wave peaks and valleys are passing that spot
faster. Look at the Fast Wave (x,t)
Surface display.
Now, getting back to the velocity of the wave, Newton's
second law tells us that acceleration is proportional to force.
If we think of a wave as a disturbance in some medium, like a
string, then the more force a certain displacement produces in
the medium, the more quickly things happen. In other words, the
undisturbed portion of the string will get the news about the
disturbance in less time if the string is stretched more tightly.
If a second string which is more massive than the first had the
same tension on it, the wave would travel more slowly down that
string. For waves in strings then the velocity depends inversely
on the mass per unit length (m) and
directly on the tension (F) in the string. So a candidate
expression for the velocity would be
v = F / m.
But look at the units of F / m. They
are mass times length over time squared divided by mass over
length, or M*L/T^{2}*L/M, or L^{2}/T^{2}.
In order to get units of L/T or velocity, it is necessary to take
the square root of our candidate expression. If F and m are the only quantities involved, as we
suspect then
v = (F /
m)^{0.5}.
You should be noticing a pattern developing
here. Dynamical system properties that are inversely dependent
on time, like velocity, frequency and angular frequency always
seem to be proportional to the square root of springiness over
inertia. For the pendulum w =
(l/m)^{0.5}. For the simple harmonic oscillator w = (k/m)^{0.5}. For a wave
traveling down a string v = (F /
m)^{0.5}. I used the term "springiness"
here because it captures the idea of stiffness or resistance to
displacement without being so specific that it applies in only
one situation. Likewise the term "inertia" in general
represents the persistence of motion. These two concepts cut
across the boundary between mechanics and electrodynamics so that
the ideas we are developing here can be applied later.



Now let's go a bit farther with the mathematics of
traveling waves on strings. We have talked about the first and
second derivative of functions of one variable. How would those
concepts apply to functions of two variables such as the
functions representing traveling waves? It turns out that we can
apply the derivative concept one variable at a time. We can hold
time constant and look at the rate change of y with respect to x,
and the rate of change of that rate of change with respect to x.
Likewise we can work with the first and second derivative of y
with respect to t, holding x constant. Derivatives like these are
called partial derivatives.
Go back to the
Fast Wave (x,t) Surface
display and follow one of the red lines, imagining its slope and
rate of change of slope. That would be the first and second
partial derivative of y with respect to x. Then follow one of the
green lines looking at the slope and rate of change of slope.
That would be the first and second partial derivative of y with
respect to t. The symbol for the first partial derivative of y
with respect to t is y/t, similar to the ordinary derivative
being dy/dt. For the second partial derivative you see
^{2}y/t^{2}.
In the case of our string, the second partial derivative of y
with respect to t is the acceleration in the y direction of the
bit of string located at a particular point on the xaxis. We
know from Newton's second law that this acceleration is
proportional to the tension in the string that we called F. The
second partial derivative of y with respect to x is a measure of
the sharpness of the peak of y=f(x) at a particular instant in
time. If the mass per unit length of the string were very small,
the shape of the y=f(x) curve would be rather broad and flat
since distant parts of the string would be responsive to small
forces. With a very massive string, the peak would be rather
narrow and sharp since distant parts of the string would be
unresponsive to small forces.
By sort of a hand waving argument we have arrived at a point
where it at least seems plausible that the velocity of a wave on
a string could be expresses in terms of ^{2}y/
t^{2} and ^{2}y/ x^{2},
since
^{2}y/ t^{2} sort of plays the role
of springiness and ^{2}y/ x^{2}
sort of plays the role of inertia. In fact it is true that for
any linear wave, the wave function will be a solution to the
partial differential equation
v^{2} = F /
m = ^{2}y/t^{2}
/
^{2}y/x^{2}.
If you feel up to it you can verify that the two wave functions
we have been using,
y=3/((xv*t)^{2}+1) and
y=A*sin(xv*t),
both satisfy this equation.
We will save more detailed discussion of traveling waves for
another course. This introduction to the topic will serve as an
anchor point, tying those later courses back to mechanics.
For the last time, are there any questions?

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