### Oh! The places waves go!

#### Kate Carlisle

Haverford College

*(Modeled on "Oh! The Places You’ll Go," by Dr. Seuss)*

Congratulations!

Today is your day.

You’re off to learn many

Great things about waves!

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

But, sadly, it's true

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 wavenumbers 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,

As you already know.

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 never mix up your right-hand rule with your left.

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!

Today is your day!

And Quantum is waiting.

So... get on that wave!