# Let ε < 0.

## 09.19.09

### Oh! The places waves go!

Filed under: Harmonic analysis, Science humor — Travis @

–Kate Carlisle, Haverford College

Modeled on “Oh! The Places You’ll Go,” by Dr. Seuss.

Congratulations!
You’re off to learn many

You have math in your brains.
You have waves all around.
And soon you will find
That oscillations abound.
You’re not on you own if you know what I know.
But YOU are the one who’ll learn how these waves go.

Waves go up and down sine curves, so graph them with care
Though sometimes equations make life less hard to bear.
F equals ma helps you find Diff. EQs
And select t of zero so phi you will lose.

Simple harmonics are the
Pulse of all things.
We study them using
Complex numbers, and springs!

But springiness changes
If life rearranges…

With damping and driving
The amplitude GROWS
If the frequency to
Omega-s goes.

And when resonance happens,
Don’t worry. Don’t stew.
Just know that the max
Is related to Q.

OH! THE PLACES WAVES GO!

Oscillations in springs
And on strings and of light!
With such simple motions,
The waves can take flight!

They don’t lag behind if you add a phase shift:
Delta, we call it, is pi-over-two
Whenever the drive frequency is the square root
Of k-over-m, (omega-s, to you).
But if it isn’t
Then delta is different.

I’m sorry to say so
That nasty
Equations
Can happen to you.

And amplitude, too,
Of the damped-driven kind,
Is less messy and stress-y
With these things in mind.

Tan delta is gamma
Times drive over both
Frequencies squared-minused.
And you’ll then not be loathe

To find amplitude which
Is not so much fun
And deriving this one
is not easily done.

Soon you’ll come to a place where the springs are combined
With pendulum bobs — it will boggle your mind
And the beats mesmerizing will make you cross-eyed.
How can you solve this? Can you even provide
A solution to this question so wide?

Can you split these behaviors into left in right?
Or breathing and pendulum? Or, maybe, not quite?
Can you make any waveform with these normal modes?
The math does work out, and so I suppose
And orthogonal is as orthogonal goes.

You can get so confused
That you’ll start in to race
Through reams of scratch paper at break-pencil pace
And grind on for miles across weirdish wild space,
Headed, I see, toward a most useful place.
The Hilbert Space…

…for waves superposing.
The simple components
Of a pendulum, or a mass-on-spring
Of a water wave, or a loaded string
A cat’s meow sound, or the phone’s shrill ring.
Combining to make waves that go
Wherever it is waves want to go.

Some oscillations in the breeze
The oscillations of the seas
Even the buzzing of the bees.
They all have their own Hilbert Space
Of normal modes which lend some grace –
At least when we’re trying to work out the math
So the gods of normality don’t send their wrath
For wasting so much paper.

Yes! That’s just the thing!

Then these normal modes
help us with waves on a string.
Beaded? Continuous? A solution I bring!

With wave numbers k
We can find all those modes.
Now we’re ready for anything under the sky.
And so we’ll see that waves travel and fly!

Oh, the places waves go! Traveling left! Traveling right!
We can find all the frequencies, even for light.
Because magical things E.M. waves sure are
The travel so easily here, there, afar.
Plane waves! Self-sustaining, move forward at c,
And this is the same ratio as E over B!

Everywhere waves will go
And you know they’ll go far
And you’ve learned all about them,
Whatever they are.

You’ll get mixed up, of course,
You’ll get mixed up
With many strange waves as you go.
So be sure when you guess
A sine-omega-t
To remember ol’ Euler,
Who makes things quite easy.
Just never forget to be dexterous and deft.

And will you succeed?
Yes! You will, indeed!
(98 and three-fourths percent guaranteed.)
Kid, you’ll move SINUSOIDS!

So…
Be your name Buxbaum or Bixby or Bray
Or Mordecai Ali Van Allen O’Shea,
You’re off to more Physics!