# Hearing Hydrogen

The spectral lines characteristic of hydrogen are spaced according to the Rydberg formula, $$\displaystyle\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$.

The wavelengths $$\lambda$$ given by the formula can be given as frequencies $$\displaystyle f = \frac{c}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$ which can be rescaled into musical frequencies, which we will play. This has been done before, but I will give more attention to the science and musical perception and less attention to the programming.

## Play a pure tone

In [1]:
from IPython.display import Audio
from numpy import sin, pi

In [2]:
amplitude = 2**13
rate = 41000  # Hz
duration = 2.5  # seconds

time = np.linspace(0, duration, num=rate*duration)

def tone(freq):
return amplitude*sin(2*pi*freq*time)


As a test drive, let’s just play a single pitch, Concert A.

In [3]:
A = 440  # frequency of Concert A
Audio(tone(A), rate=rate)

Out[3]:

## Play the spectral lines

An "audible" Rydberg formula in Python:

In [4]:
scaling = 4*A  # rescale the frequencies into an audible range
def freq(n1, n2):
return scaling*(1./n1**2 - 1./n2**2)


Generate a spectrum of frequencies for $$n_1 = 1, 2, 3$$, corresponding to the yellow, black, and maroon lines in the illustration above. Verify that the lowest and highest frequencies are within the audible range of 20-20,000 Hz.

In [5]:
series = [1, 2, 3] #  Lyman, Balmer, Paschen series
spectrum = [freq(n1, n2) for n1 in series for n2 in range(n1 + 1, 9)]
min(spectrum), max(spectrum)

tones = [tone(f) for f in spectrum]
composite_tone = np.sum(tones, axis=0)


Listen.

In [6]:
Audio(composite_tone, rate=rate)

Out[6]:

The sound is eerie and disssonant, but more musical than one might expect. Why?

Most natural sounds, especially musical ones, consist of several frequencies (i.e., pitches) related to each other by the harmonic series: $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$$, etc. Our brains usually group tones related by the harmonic series together, interpreting them as part of the same sound. Further, we perceive the differences between these frequencies as beating, a pulsing sensation that is particularly obvious when two tones are almost but not quite in unison.

Each tone in the sound above corresponds exactly to one of beating patterns. The Rydberg formula takes the difference between two fractions from the harmonic series. (To be specific, the fractions are from the series $$\frac{1}{2}, \frac{1}{4}, \frac{1}{9}$$ …, a subset of the harmonic series.)

Although the spectrum of hydrogen is unrelated to music or acoustics, it happens to follow a pattern that also occurs is musical sound, and so it makes more sense to our ears than random tones or noise.

# Winter Is Coming

I coauthored an astrophysics paper, Winter is Coming, submitted to the physics arxiv on April 1, in which the irregular seasons of Game of Thrones are explained in astrophysical terms. Veselin Kostov did all the hard science; I am responsible for the plots and some of the writing.

We were Nerd Famous for a day. A roundup:

# The Audacity of Dispair

What else but the title of a blog by David Simon, former Baltimore Sun reporter?

The blog is old news — his first post was in April — but I only just discovered it and caught up. Two of my favorites:

# For Y’all Who Use Multiple Computers

I read a nifty suggestion for combining Dropbox with version control. Smart and very convenient.

# Turkey Time

Ways to estimate turkey cooking times:

• USDA guidelines
• the simple rule “18 minutes per pound”
• $\text{time} = \frac{\text{weight}^{2/3}}{1.5}$ (using weight in pounds)

The “18 minutes” rule works for some weights, but it doesn’t scale right for large turkeys. A wider range of accuracy is achieved by the simple formula, suggested by the late physicist and SLAC director Pief Panofsky. All of these assume the oven is set to 325 F.

The exponent in Panofsky’s formula comes from the ratio of the turkey’s surface area, through which heat flows, to its volume. The 1.5 is empirical, fitting the mathematical curve to data from actual cooked turkeys.

With a little more trouble, we can solve the problem without referring to actual cooking times, using only the basic material properties of turkey. We can use our solution to estimate cooking times at other temperatures, such as smoking a turkey on a 225 F grill.

What if we imagine that the turkey is a round ball of cold meat sitting in hot air? Simplifying its shape and ignoring the details of the cooking process, we have a straightforward physics problem. The properties we need (density $\rho$, conductivity $\kappa$, and diffusivity $\alpha$) were measured and published by the Canadian Food Research and Development Center.

How big should this imagined ball of turkey be? We could try mashing the turkey’s whole mass into one solid ball. But since that approach ignores the bones, which conduct heat faster than meat, we should expect it to overestimate the cooking time. (And it does.) Instead, we can try including only the meat, which comprises roughly half the mass. For a 6- to 22-pound turkey, we’ll be modeling a 13- to 21-cm meat ball. In comparison to a turkey breast, the thickest part of the meat, that seems about right.

Our turkey ball will start at 50 F, surrounded by 325 F air in the oven. I will solve the heat equation to compute when the center of the ball reaches 170 F, safe to eat.

A fine fit like that seems too good to be true. Perhaps our approximations balanced each other by chance. Now, extend the model to learn something new: If we set the surrounding air to 225 F, as on charcoal grill, the equation predicts longer cooking times that agree with experience.

The real physics of turkey cooking is explained in a nonmathematical post by Modernist Cuisine.

## Mathematical Appendix

Parameters:

• raw turkey temperature $T_{\text{raw}} = 50 \text{ F}$
• oven temperature $T_{\text{oven}} = 325 \text{ F}$
• thermal diffusivity $\alpha = 1.4\times10^{-7} \text{ m}^2\text{/s}$
• thermal conductivity $\kappa = 0.45 \text{ W/mK}$
• heat transfer coefficient of free air $h = 10 \text{ W/m}^2\text{s}$
• density $\rho = 1070 \text{ Kg/m}^3$

The temperature at the center of the turkey ball after time $t$ in the oven measured a distance $r$ from the center is given by

where $\zeta_n$ are given by the roots of the equation $f(x) = 1 - \zeta \cot(\zeta) - \frac{h R}{\kappa}$ and must be computed numerically.

We are mainly interested in the temperature at the center, the last part to cook. This is a slightly simpler expression.

When $T(t, 0) = 170 \text{ F}$, the turkey is done.

# Parting Shots

Quotations from Zlatko Tesanovic, brilliant physicist and colorful character. He was remembered today in a tribute held by our physics department.

Like beautiful people, beautiful functions are somewhat trivial.

It is OK to ask questions, but you must recognize the monumental clarity and precision of my answers.

In the past, when computers were not around, people had to think, and so they did.

One should not get emotional with methods of steepest descent, but somehow I do. It will be like a light to you in dark rooms in the middle of the night, when you are despairing and everything else has failed you… and you will realize, the Method of Steepest Descents is your only true friend.

This is our problem, not nature’s problem.

This is arbitrary… No. No. It really isn’t. Nothing I say in this course is arbitrary.

‘Ansatz’ means ‘guess,’ but when you say it in German it means ‘educated guess.’

This is where we separate real homo sapiens from other apes.

On Schrodinger, who lived in a house with his wife and her beautiful sister, who was also his mistress: > He knew absolutely everything about partial differential equations. So, you see, if you know everything about a particular method, it is like waiting on a platform for a train. And, sometimes the train does not come. but, for Schordinger of course, the train came Big Time.

Of Clebsch-Gordan coefficients: > The kindest thing anyone can say about them is that they are tabulated.

Of Bessel Inequality:

Rays of sunshine will come into your lonely room.

Now, just by enduring evidence, we come to the heart of its boring darkness: Special Functions.

To those who have more, everything will be given to them. Eventually, everything will be done by Gauss.

Physicists have played a great part in the economic collapse of the planet.

Dirac was a different sort of guy… in fact, people always suspected he was an alien. A real one, not just a foreigner.

The problem with liquids is that they are not a gas.

Ignorance doesn’t have to contain logic — truth — behind it.

Sources: My notes, Lynn Redding Carlson, and Jennifer Pursley

For the best written distillation of Zlatko’s humor, read his restaurant guide.

# What Shall I Call Thee When Thou Art a Man?

I downloaded Social Security Administration’s record of baby names. Here is what I learned about the name Kelly. I followed an example in this book on a handy set of tools for studying messy, real-life data called pandas.