If a blackbody has a temperature such that its peak frequency was well within our audible range, for example 1 KHz, what would that sound like if we used Planck's law to plot its spectral curve in the frequency domain and performed a transformation (like an inverse FFT) to obtain a waveform?

Planck's law tells us the peak wavelength and electromagnetic spectral emission curve of an ideal radiator an some temperature $T$.

If we know the peak frequency, $f$, then we can work backwards to figure its peak wavelength, $\lambda$. For our example, if $f$=1 KHz, then $\lambda$≈170,471 m, so $T$≈17 nK.

If we plot the power spectral density of a 17 nanokelvin blackbody as a function of frequency and performed an inverse FFT on that curve, what would the resulting waveform sound like?

According to this wikipedia article, blackbody radiation is just thermal noise (Johnson–Nyquist noise); if that's what I'm looking for, what does it sound like? Just to clarify, I'm looking for a waveform, maybe a WAV file, rather than a verbal description.

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    NB for anyone who attempts this: You can't just inverse Fourier transform the spectrum to get the time domain signal. You have to respect the fact that noisy time domain signals give noisy frequency domain signals. The amplitude and phases of your frequency domain points must be drawn from the proper probability distributions in order for the resulting time domain signal to have the right statistical properties. This is a commonly missed aspect of signal processing. – DanielSank Nov 3 '15 at 6:54
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    What kind of answer can possibly be given to this question? Suppose I were to construct the sound wave using my computer and a speaker. Can I post an answer describing the sound verbally? Can I post a link-only answer with a recording for everyone else to hear? I really like this question, but I'm wondering if/how it can be answered. – DanielSank Nov 3 '15 at 7:19
  • @DanielSank I think the answer should be objective rather than subjective (i.e. a description); perhaps you could upload a small wav file? I'm curious to analyze its spectrum using different window functions. – ayane Nov 3 '15 at 7:27
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    It's funny for me to inverse Fourier transform the Plank spectrum, and for you to then analyze it with a window function. Why not just use the windowed spectrum to produce the time domain signal in the first place? ^^ – DanielSank Nov 3 '15 at 7:28
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    Peak wavelength is not $c/f_{peak}$. – Rob Jeffries Nov 4 '15 at 22:22
up vote 38 down vote accepted
+200

This problem can be solved with noise-shaping. Since the shape of the spectrum is known, it can be used as a base for the power spectral density:

$$ P(f,T)=\frac{ 2 h f^3}{c^2} \frac{1}{e^\frac{h f}{k_\mathrm{B}T} - 1} $$

where $k_\mathrm{B}$ is the Boltzmann constant, $h$ is the Planck constant, and $c$ is the speed of light. This outputs the relative power of each band as a continuous function of frequency, $f$, and temperature, $T$. Since the output quantity must be expressed in decibels (dBr) to be meaningful for audio, we simply use a log scale and add an offset (a gain) to normalize the peak to 0. The equation of the EQ curve is:

$ E(f,T) = 10 \log{ P(f,T) } + G_{t}(T) $

where $G_{t}(T)$ is the gain required to normalize the peak to 0 dB. The required gain depends on the inverse cube of the temperature plus a constant, $G$ (187 dB); thus, $ G_{t}(T) = G - 10 \log T^3 $. The leading coefficient $10$ converts bels to decibels. Simplifying gives us:

$$ E(f,T) = 10 \log{ \left( \frac{ 2 h f^3}{c^2 T^3} \frac{1}{e^\frac{h f}{k_\mathrm{B}T} - 1} \right)} + G $$

tl;dr:

We obtain our waveform by applying an EQ to gaussian white noise from AudioCheck.net.


Examples:

  1. 17 nanokelvins is the temperature at which black noise has a peak frequency of 1 KHz. Its passband is limited to 1 Hz to 12 KHz.

  2. 30 nanokelvins is the lowest temperature at which black noise has a passband that spans the entire hearing range.

  3. 55 nanokelvins is the temperature at which black noise has a peak frequency of approximately 3 KHz, the most sensitve frequency of human ears.

  4. 340 nanokelvins is the temperature at which black noise has a peak frequency of just under 20 KHz, which is the limit of human hearing. Most of the audible spectrum is a linear upward ramp, which is very similar to violet noise. At higher temperatures, the frequency domain will be almost identical to violet noise.

All EQ filter parameters are in the descriptions of the tracks on SoundCloud.

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    This is probably my favorite Stack Exchange post ever. Great job! It would be really nice to post the code so that others can check it, but if you did it in a GUI application that may not be possible. It would also be really nice to post the original white noise for comparison, and perhaps a few more temperatures. – DanielSank Nov 4 '15 at 20:38
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    What is $B$ in the equation? It's good to define all symbols, even if they're defined in the external link. To clarify, it would be nice to upload the white noise to soundcloud so people don't have to download it etc. etc. – DanielSank Nov 4 '15 at 20:42
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    Sounds just like the tinnitus in my ears. :) – anna v Nov 5 '15 at 4:36
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    @DanielSank I added links for noise at two more temperatures. I also cleaned up the answer and refined the formula a bit. – ayane Nov 6 '15 at 5:24
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    @yuki96 I think this answer would be more clear if the first equation were to give just the power spectral density. Then, in a second equation, you can give the equalization curve, noting that audio processing is typically done with logarithmic scales. It's confusing (to me, and probably other physicists), to see the Plank law written as an equalizer curve expressed on a log scale. Please note that I'm only bugging you because I love this question and answer so much :D – DanielSank Nov 6 '15 at 7:33

If you are cooling your object that you wish to hear, then the exact sound will depend on the exact temperature (as given by yuki96's answer at 17nK).

However, any temperature above the nanoKelvin temperature scale will sound the same, but the volume will increase with temperature (according to the Stefan-Boltzmann law).

The sound of a warm blackbody (such as what you would get at room temperature) would sound like a violet or purple noise. You can listen to a sample of purple noise here.

What does a blackbody sound like?

It will sound like a musical note. Any spectrum looking like a bell is a musical note with a lot of harmonics. The narrower the bell the purer the musical note. If the bell is wide the ear perceives the sound more like a pop because the wider the spectrum the shorter its time image gets.

Planck's law

Planck's law

To be more precise, the inverse Fourier transform (the signal as a function of time) of a spectrum having the shape of a bell (like Plank's diagram) looks like in the image below, row no. 4 from the top.

Continuous and discrete spectra

Continuous and discrete spectra

Because I am seeing there are people who do not understand, the spectrum of a finite duration signal (a real musical note for instance) is continuous. Only repetitive signals have discrete spectra. If the musical note is played continuously its spectrum is discrete if not, and this is the case in practice, the spectrum is continuous.

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    Harmonics are multiples of a fundamental tone. How can harmonics give a continuous bell shape? – DanielSank Nov 3 '15 at 7:05
  • Harmonics would be discrete, not continuous, in the frequency domain. An example is a square wave. – ayane Nov 3 '15 at 7:10
  • If the number of harmonics of a musical note is big the spectrum of the note appears continuous. See the diagrams from here. – Energizer777 Nov 3 '15 at 7:20
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    @Energizer777 That has nothing to do with harmonics. The continuous spectrum there looks like it is due to the finite time duration of the measured sound wave causing spectral leakage in the DFT. It's a signal processing effect. – DanielSank Nov 3 '15 at 7:24
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    yuki96: "Harmonics would be discrete, not continuous, in the frequency domain. An example is a square wave." That square wave has to last indefinitely to have a discrete spectrum. If it last a finite number of periods its spectrum will be continuous. – Energizer777 Nov 3 '15 at 8:18

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