**ÒPortal-BouncingÓ and Oscillations **

* Scott H. Hawley, Ph.D.*

**E****lementary School:**

Open two
portals, next to one another, on the floor. Then drop a storage cube – or yourself –
inside one of them! See what
happens? Whatever gets dropped in
one portal, comes out moving upward in the other portal. Then the object – or you –
reaches a maximum height, and starts falling, and eventually comes back upward
from the original portal!
This process repeats itself over and over.

When
something moves repeatedly back and forth (or up and down), we say that it is *oscillating*. Many things in nature oscillate. Can you think of any others?

One
example is a pendulum, such as a girl on a swing, going back and forth. In this case, the girl is *oscillating*. Oscillation is one of the most basic processes in the
universe. From piano strings, to
tree branches in the wind, even to stars within galaxies, so many things in
nature oscillate!

Imagine
a little red ball that can roll along a hill. If you start the ball in the place shown, it will roll down
and to the right, into the valley and back up on the other side until it comes
to a stop.

Then it will start rolling back down and to the left,
into the valley and up the other side.
If thereÕs not much in the way to slow the ball down, this *oscillation* will continue for a very
long time.

The
time it takes for the ball to make it back to its original position is called
the *period* of the oscillation, and it
stays the same throughout the motion. Many things in nature have some period of their motion
that we use to time things. The
time it takes for the moon to go around the earth and come back to its starting
point, for example, is called a month. If you or your parents have a digital watch, thereÕs a
tiny crystal inside, and the crystal oscillated with a certain period that is
used to tell the time!

Exercise:
Now go back to the Portal game.
Using a clock or a stopwatch, can you find the period of the boxÕs (or
your) oscillation? Make sure it
stays the same as you keep watching the motion!

**High School Physics:**

The
most common type of oscillation youÕre likely to study is called *Simple Harmonic Motion (SHM) *or *Simple Harmonic Oscillations (SHO)*, or
more generally, *Small Oscillations*. For just about every system in nature
that oscillates, oscillations with small enough *amplitude* (the size of the oscillation) will follow SHM. SHM is a consequence of *HookeÕs Law*, which describes a *restoring force F*, which is *linearly proportional* to the
displacement from equilibrium*:*

* *

(The
minus sign is the ÒrestoringÓ part.)
HookeÕs Law can also be represented graphically, by a line with a slope
of *-k*:

The
basic system for understanding HookeÕs Law and SHM is a mass attached to a
spring. If the mass is *m* and the spring constant or
ÒspringinessÓ of the spring is *k *(in
Newtons/meter), then the period of oscillation *T* is given by

.

That
is to say, smaller masses will produce shorter periods of oscillation, as will
stiffer springs. Note that
in the above formula, *the period of
oscillation does not depend on the amplitude,* but it *does* depend on the mass.

Now
let us turn our attention to Portal. If you drop from rest at a height *h*, the time it takes to reach the ground is found (using ) to be

.

The
time it takes to fall from height *h* and
go through, say, a blue portal in the floor, come out the orange portal in the
floor and reach a height *h* again is
thus 2 *t*_{Fall}. But this is only *half* an oscillation cycle.
The full cycle is when you fall *back*
through the orange portal, come out the blue side, and reach height *h*.
Thus, the period of oscillation for jumping-through-portal oscillations
is

.

Now
*h* is the size of the oscillation,
i.e. it is the *amplitude*. We see here that the period of
these oscillations *does* depend on the
amplitude – jumping off a higher point will produce a longer period of
oscillation. Furthermore we see
that the *mass* of whatever is
oscillating (e.g., your body) is *not *a
factor in the oscillation, and does not affect the period. This is because, instead of HookeÕs
Law, the force of gravity on your body is constant with respect to displacement,
and scales with the mass:

.

Another
way of saying this is that all objects accelerate at the same rate in a
constant gravitational field, and this is quite different from HookeÕs
Law. (HookeÕs Law would be like the
acceleration your body experiences being greater the higher up you are.)

Exercise
1: Show that , where *h*_{ft}
is the initial height measured in feet.

Exercise 2: Go ahead and estimate the height from which you are about to drop into a
portal, and use that to find the time it will take to reach the floor. Then multiply that by 4 and see if the
time you get is the same as the time you measure by playing the game with a
stopwatch in hand!

**College Physics:**

Visually,
oscillations in and out of the floor in Portal *look* very much like SHM.
The oscillating object (or body) has a maximum speed at the floor, and
gradually slows as it reaches a maximum.

To
show how similar this motion is to SHM, letÕs plot a graph. If we graph the height of the object/body
as a function of time, we get a *waveform*. To help in the comparison with SHM, weÕll
assume you drop into a blue portal, and regard any heights reached on the
Òorange portal sideÓ as *negative*
heights. If we do this, the
comparison with SHM is striking:

The red line is sinusoidal and shows displacement vs. time for simple harmonic
motion. The blue line is a graph
of oscillatory motion in and out of two portals on the floor in a uniform
gravitational field. (The
amplitudes and periods have been normalized for the purposes of comparison.) Note how close these curves are! No wonder the oscillation in and
out of portals *looks like* SHM!

ThereÕs
more to these curves than meets the eye. The red curve is sinusoidal, whereas the blue curve is
*piecewise parabolic. * Both are continuous functions, and have continuous first
derivatives.

LetÕs
compare the first derivatives, i.e. the *velocities*
of the motion:

See
how the red line switches phase by 90 degrees, whereas the blue line becomes a
triangle wave? LetÕs take another
derivative and look at the accelerations:

Now
we can see the crux of the differences in the motion. The red curve is still sinusoidal and continuous but the
blue one has become a square wave, which is *discontinuous*. This is due to the discontinuity in the
gravitational field that occurs as you go through the floor: it instantly
switches from pulling in one direction to pulling in the other. And, as we said earlier, the force
(/acceleration) doesnÕt depend on the displacement from equilibrium, so larger
amplitudes donÕt result in larger accelerations, they simply result in longer
oscillation periods.

Exercise:
Instead of a creating a portal, imagine you can create a hole that goes all the
way through the earth. You jump
from the earthÕs surface into the hole and come out the other side.. Regarding the earth as a sphere of
uniform density, show that the resulting motion is simple harmonic, with a
period of about 80 minutes.

**Afterward: What it ÒFeelsÓ Like**

So
far we have neglected a fine point: We have been regarding the body in motion as
a single particle of zero extent, centered at the personÕs Òcenter of
mass.Ó This is likely how the game
developers employed the physics engine when writing the game. The trouble is that the concept
of Òcenter of massÓ becomes meaningless when the body is disjoint and subject
to *discontinuous* gravitational fields. When you are ÒinÓ the portal, such that
parts of your body (from your waist down, say) are sticking out of the orange
portal, while the rest of you is sticking out of the blue portal, you have no
single center of mass, you have *two: o*ne
for each piece of your body which is outside a portal.

Furthermore,
these two masses are both being attracted back *inside* the portal.
Your upper body is getting pulled into with a force equal to its weight* *--- in the direction of your feet ---
while the lower half of your body is getting pulled back into the portal with a
force equal to *its* weight,* *in the direction of your head.

This
is the kind of thing you experience all the time, just standing on a
floor. Your body gets pulled
down by its weight, and the floor pushes back back up with exactly the same
amount of force, so your net acceleration is zero. What you *feel* is
the floor pushing up on you, and/or your legs pushing down onto the floor.

The
same thing happens as you are going through a portalÉsort of. In this case, it is the weight of
however much of your lower body is sticking out of the portal – not your
entire weight -- which pushes on your upper body.

You
feel this precisely at the spot where your body is in the plane of the floor,
i.e. wherever the portal ÒcutsÓ your body. Thus, as you start to go into the portal, you feel a slight
pushing at the bottom of your feet.
This sensation rises along your legs, getting stronger (in a more or
less linear fashion) as the plane of intersection progresses up your body as it
moves through the portal. The
force reaches a maximum when you are halfway through, and the sensation
continues traveling up toward you head, getting weaker and weaker until your
head is finally through, and you travel upwards (feet first), completely out of
the orange portal.

Note
that when you are exactly half-way through the portal, such that half your
weight is pushing in one direction (relative to you) and the other half is
pushing in the other, the net force on you is zero.

With
these observations in mind, the acceleration graph becomes no longer a square
wave. Instead, the vertical sides
of each ÒsquareÓ are no longer vertical but instead ÒslantedÓ, resulting in
what one might call a ÒtrapezoidÓ wave.

Curious
to experience the feeling of one Ò*gÓ *of
acceleration occurring in a thin slice of your body, moving up and down as you
go through each portal? Submit a feature
request to Valve! ☐

Scott Hawley is a computational astrophysicist at Belmont University in Nashville, TN, where he advises the Society of Physics Students and teaches classes for audio engineers.