Here is the answer. Background: In the bottom surface of the kettle are many microscopically tiny pores, cracks and crevices. many of these retain extremely minute amounts of air after the kettle is filled with water. these air-charged pores will serve as nucleation sites, at which the boiling process will first be triggered.
as the water heats up, you will first notice air bubbles growing at the locations of the pores. this is because air dissolved in water is less soluble in hot water than cold, and the air already in the pores furnishes pre-existing sites at which the air can readily come out of solution. (As the water grows hotter, some of these bubbles will break free from the bottom of the kettle and float up to the surface, only to be replaced by fresh bubble growth at the vacated pores.) no sound is produced during this process.
Now note that although some convective heat transfer is occurring in the kettle, the bottom surface is where the heater element is and at this stage, the water next to it will be hotter than the rest of the bulk, and that this condition is actually assisted by the presence of the bubbles, which tend to insulate the bulk from the water closest to the kettle bottom.
Meanwhile, the air in the crevices is becoming enriched with water vapor at the same time that the temperature of the water is climbing, and eventually the air exsolvation process goes to completion and the pores are now charged with mostly water vapor and a little air, and most of the stuck air bubbles have detached and floated away.
each of the vapor-filled pores can be modeled as a crevice with an opening that contains a meniscus whose outward curvature, and hence its internal pressure, is given by the young-laplace equation. that pressure is balanced by the vapor pressure and as the surface grows still hotter, there comes a point where the water in the vicinity of that meniscus elects to vaporize and the meniscus suddenly everts from the mouth of the pore. because its radius of curvature decreases as its volume increases, the internal pressure required to inflate the bubble goes down as it grows, and a tiny vapor explosion occurs: the bubble undergoes very sudden growth, and it gives off a sound wave which propagates into the bulk of the water in the kettle.
That growth comes to a sudden halt once the bubble becomes big enough to protrude out of the hot environment next to the kettle bottom and contact the slightly cooler water above. the bubble is then suddenly quenched, and suddenly collapses. as the retreating meniscus collides with the kettle surface, it propagates a sound wave into the kettle bottom. almost immediately, this process repeats itself.
all across the kettle bottom, this mechanism of micro-explosions and cavitation collapses is occurring at each of the "activated" pores (those that are charged with vapor/air mixture and have menisci of appropriate radii) and the sounds thus produced blend into a hissing roar that you can hear. as more and more pores become active, the roar gets louder and very vigorous stirring of the water right next to the kettle bottom occurs. this mixes the hot water next to the bottom with the cooler water from above and soon the water far from the kettle bottom becomes just as hot as that near the bottom, and the boiling bubbles are no longer quenched by their own growth. the roar then diminishes.
Now what happens is that the boiling bubbles begin instead to detach and grow to macroscopic size, quickly bursting upward and away from the kettle bottom and the nature of the sound being produced shifts from the hissing roar of the microscopic vapor explosions to the familiar bubbledy-bubble noise, which grows in volume until it is the dominant noise you hear.
This mechanism accounts for the initial quiet phase, the gradual roar-and-hiss noise buildup, the brief minimum as the boiling mechanism switches modes, and the second bubbling noise growth phase that you observed.
Best Answer
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:
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.
30 nanokelvins is the lowest temperature at which black noise has a passband that spans the entire hearing range.
55 nanokelvins is the temperature at which black noise has a peak frequency of approximately 3 KHz, the most sensitve frequency of human ears.
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.