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!
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 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,
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!

09.17.09

Lines inspired by a lecture on extra-terrestrial life

Filed under: Harmonic analysis, Science humor — Travis @

– J. D. G. M.

Some time ago my late Papa
Acquired a spiral nebula.
He bought it with a guarantee
Of content and stability.
What was his undisguised chagrin
To find his purchase on the spin,
Receding from his call or beck
At several million miles per sec.,
And not, according to his friends,
A likely source of dividends.
Justly incensed at such a tort
He hauled his vendor into court,
Taking his stand on Section 3
Of Bailey “Sale of Nebulae.”
Contra was cited Volume 4
Of Eggleston’s “Galactic Law”
That most instructive little tome
That lies uncut in every home.
“Cease,” said the sage, “Your quarrel base,
Lift up your eyes to outer space.
See where the nebulae like buns,
Encurranted with infant suns,
Shimmer in incandescent spray
Millions of miles and years away.
Think that, provided you will wait,
Your nebula is Real Estate,
Sure to provide you wealth and bliss
Beyond the dreams of avarice.
Watch as the rolling aeons pass
New worlds emerging from the gas:
Watch as brightness slowly clots
To eligible building lots.
What matters a depleted purse
To owners of a Universe?”
My father lost the case and died:
I watch my nebula with pride
But yearly with decreasing hope
I buy a larger telescope.

From The Observatory 65, 88 (1943).

09.15.09

Mathematical limericks, vol. 7

Filed under: Harmonic analysis, Science humor — Travis @

Nonexamples!

pi goes on and on and on …
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed?

Chebychev said it and I’ll say it again:
There’s always a prime between n and 2n.

Man has pondered
Since time immemorial
Why 1 is the value
Of zero-factorial.

Three jolly sailors from Blaydon-on-Tyne
They went to sea in a bottle by Klein.
Since the sea was entirely inside the hull
The scenery seen was exceedingly dull. [FW]

Little Willie was a Chem-E,
Little Willie is no more.
What Willie thought was H2O
Was really H2SO4. [CR]

When calculating polynomial degree,
The minimum value it can be
Is “1″ plus the number of bends.
And remember this too my friends:
The polynomial’s degree is only even
If its graph enters the same side its leavin’.

References

[CR] by Crifton Robinson.
[FW] by Frederick Windsor, from The Space Child’s Mother Goose, 1958.

09.1.09

Epigrams

Filed under: Harmonic analysis, Science humor — Travis @

Nature and Nature’s laws lay hid in night.
God said, “Let Newton be!” and all was light.
Alexander Pope

It did not last: the Devil howling “Ho!
Let Einstein be!” restored the status quo.
Sir John Collins Squire

05.28.09

The electron in gold

Filed under: Harmonic analysis, Science humor — Travis @

–Arthur H. Snell

There was an electron in gold
Who said, “Shall I do as I’m told?
Shall I snuggle down tight
With a brief flash of light
Or be Auger outside in the cold?”

Said the K-shell electron in gold,
“I’m thinking of leaving the fold
To be hit like a hammer
By an outgoing gamma.
In freedom I’ll live till I’m old.”

Said the K-shell electron in gold,
“I wonder if I might be bold
And make a slight shift
From this circular drift
And change this damned atom to platinum.”

If your physics needs a little help, the three stanzas refer to fluorescent yield, internal conversion, and electron capture, respectively.

05.19.09

The thermodynamics of Hell

Filed under: Science humor — Travis @

A thermodynamics professor had written a take home exam for his graduate
students. It had one question:

Is Hell exothermic (gives off heat) or is it endothermic (absorbs heat)? Support your answer with proof.

Many of the students wrote proofs of their beliefs using Boyle’s Law (gas cools off when it expands and heats up when it is compressed). One student however gave the following answer:

First we need to know how the mass of Hell is changing in time. So we need to know the rate that souls are moving into Hell, and the rate they are leaving. I think we can safely assume that once a soul gets to Hell, it will not leave (that is, after all, the point of Hell). Therefore, no souls are leaving.

As for how many souls are entering Hell, lets look at the different religions that exist in the world today. Some of these religions state if you if you are not a member of there religion, you will go to Hell. Since there are more than one religion out there that has this belief, we can now assume all souls go to Hell. With birth and death rate as they are, we can now expect the number of souls in Hell to increase exponentially.

Now, we look at the rate of change in the volume in Hell because Boyle’s Law states that in order for the temperature and pressure in Hell to stay the same, the volume of Hell has to expand as souls are added. This gives two possibilities:

(1) If Hell is expanding at a slower rate than the rate at which souls enter Hell, then the temperature and pressure in Hell will increase until all hell breaks loose.

(2) If Hell is expanding at a rate faster than that of the souls entering Hell, then the temperature and pressure will drop until Hell freezes over.

So which is it? If we accept the postulate given to me by Ms. Theresa Banyan during my Freshman year, “That will be a cold night in Hell before I go out with you,” and take into account the fact that I still have not succeeded in getting her to go out with me, then #2 can not be true. So, Hell is exothermic.

There was only one ‘A’ given on the exam.

Clearly, some people are better at taking tests than I. Also along these lines, you might consider the thermodynamics of Heaven.

05.18.09

The thermodynamics of Heaven

Filed under: Science humor — Travis @

The temperature of Heaven can be rather accurately computed. Our authority is in the bBible: Isaiah 30:26 reads “Moreover, the light of the Moon shall be as the light of the Sun and the light of the Sun shall be sevenfold, as the light of seven days.” Thus Heaven receives from the Moon as much radiation as we do from the Sun, and in addition seven times seven (forty-nine) times as much as the Earth does from the Sun, or fifty times in all. The light we receive from the Moon is a ten-thousandth of the light we receive from the Sun, so we can ignore that. With these data we can compute the temperature of Heaven. The radiation falling on Heaven will heat it to the point where the heat lost by radiation is just equal to the heat received by radiation. In other words, Heaven loses fifty times as much heat as the Earth by radiation. Using the Stefan-Boltzmann fourth-power law for radiation,

(H/E)4 = 50,

where E is the absolute temperature of the Earth, viz. 300 K. This gives H as 798 K (525o C).

The exact temperature of Hell cannot be computed, but it must be less than 444.6o C, the temperature at which brimstone or sulfur changes from a liquid to a gas. Revelations 21:8: “But the fearful, and unbelieving… shall have their part in the lake which burneth with fire and brimstone.” A lake of molten brimstone means that its temperature must be below the boiling point, which is 444.6o C. (Above this temperature it would be a vapor, not a lake.)

We have, then, temperature of Heaven, 525o C. Temperature of Hell, less than 445o C. Therefore, Heaven is hotter than Hell.

This appeared in Applied Optics, vol. 11, A14, 1972. Along these lines, you might consider the thermodynamics of Hell.

05.16.09

A stress analysis of a strapless evening gown

Filed under: Dirty, Science humor — Travis @

Effective as the strapless evening gown is in attracting attention, it presents tremendous engineering problems to the structual engineer. He is faced with the problem of designing a dress which appears as if it will fall at any moment and yet actyuall stays up some small factor of safety. Some of the problems faced by the engineer readily appear from the following structual analysis of strapless evening gowns.

If a small elemental strip of cloth from a strapless evening gown is isolated as a free body in the area of plane A in figure 1, it can be seen that the tangential force F is balanced by the equal and opposite tangential force F. The downward vertical force W (weight of the dress) is balanced by the force V acting vertically upward due to the stress in the cloth above plane A. Since the algebraic summation of vertical and horizontal forces is zero and no moments are acting, the elemental strip is in equilibrium.

Consider now an elemental strip of cloth isolated as a free body in the area of plane B of figure 1. The two tangible forces F1 and F2 are equal and opposite as before, but the force W (weight of the dress) is not balanced by an upward force V because there is no cloth above plane B to supply the force. Thus, the algebraic summation of horizontal forces is zero, but the sum of the vertical forces is not zero. Therefore, this elemental strip is not in equilibrium; but it is imperative, for social reasons, that this elemental strip be in equilibrium. If the female is naturally blessed with sufficient pectoral development, she can supply this very vital force and maintain the elemental strip at equilibrium. If she is not, the engineer has to supply this force by artificial methods.

In some instances, the engineer has made use of friction to supply this force. The friction force is expressed by F = f N, where F is the frictional force, f is the coefficient of friction, and N is the normal force acting perpendicularly to F. Since, for a given female and a given dress, f is constant, then to increase F, the normal force N must be increased. One obvious method of increasing the normal force is to make the diameter of the dress at c in figure 2 smaller than the diameter of the female at this point. This has, however, the disadvantage of causing the fibres along the line c to collapse, and, if too much force is applied, the wearer will experience discomfort.

As if the problem were not complex enough, some females require that the back of the gown be lowered to increase the exposure and correspondingly attract more attention. In this case, the horizontal forces F1 and F2 (figure 1) are no longer acting horizontally, but are replaced by forces T1 and T2 acting downward at an angle a. Therefore, there is a total downward force equal to the weight of the dress below B plus the vector summation of T1 and T2. This vector sum increases in magnitude as the back is lowered because R = 2 T sin(a), and the angle a increases as the back is lowered. Therefore, the vertical uplifting force which has to be supplied for equilibrium is increased for low-back gowns.

Since these evening gowns are worn to dances, an occasional horizontal force, shown in figure 2 as i, is accidentally delivered to the beam at the point c, causing impact loading, which compresses all the fibres of the beam. This compression tends to cancel the tension in the fibres between e and b, but it increases the compression between c and d. The critical area is a point d, as the fibres here are subject not only to compression due to moment and impact, but also to shear due to the force s; a combination of low, heavy dress with impact loading may bring the fibres at point d to the “danger point.”

There are several reasons why the properties discussed in this paper have never been determined. For one, there is a scarcity of these beams for experimental investigation. Many females have been asked to volunteer for experiments along these lines in the interest of science, but unfortunately, no cooperation was encountered. There is also the difficulty of the investigator having the strength of mind to ascertain purely scientific facts. Meanwhile, trial and error and shrewd guesses will have to be used by the engineer in the design of strapless evening gowns until thorough investigations can be made.

Condensed from A Stress Analysis of a Strapless Evening Gown and other essays, ed. Robert A. Baker (Prentice-Hall) 1963.

05.15.09

Snakes and Ladders for scientists

Filed under: Goofy graphs, Science humor — Travis @

– P. J. Duke

From Orbit, Journal of the Rutherford High Energy Laboratory, December 1963, p 10.

05.14.09

The secret to antigravity

Filed under: Science humor — Travis @

If you drop a buttered piece of bread, it will fall on the floor butter-side down. If a cat is dropped from a window or other high and towering place, it will land on its feet.

But what if you attach a buttered piece of bread, butter-side up to a cat’s back and toss them both out the window? Will the cat land on its feet? Or will the butter splat on the ground?

Even if you are too lazy to do the experiment yourself you should be able to deduce the obvious result. The laws of butterology demand that the butter must hit the ground, and the equally strict laws of feline aerodynamics demand that the cat can not smash its furry back. If the combined construct were to land, nature would have no way to resolve this paradox. Therefore it simply does not fall.

Thus has man discovered the secret of antigravity! A buttered cat will, when released, quickly move to a height where the forces of cat-twisting and butter repulsion are in equilibrium. This equilibrium point can be modified by scraping off some of the butter, providing lift, or removing some of the cat’s limbs, allowing descent.

Evidence suggests that most of the civilized species of the Universe already use this principle to drive their ships while within a planetary system. The loud humming heard by most sighters of UFOs is, in fact, the purring of several hundred tabbies.

The one obvious danger is, of course, if the cats manage to eat the bread off their backs they will instantly plummet. Of course the cats will land on their feet, but this usually doesn’t do them much good, since right after they make their graceful landing several tons of red-hot starship and extremely pissed-off aliens crash on top of them. No resolution to this problem is yet known.

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