Why is the sound produced by a sonic boom low pitched?

Question

According to Doppler's effect

$$f'=\frac{v+v_0}{v-v_s}f$$

where $f'$ is the observed frequency, $f$ is the actual frequency, $v$ is the velocity of sound waves, $v_0$ is the velocity of the observer and $v_s$ is the velocity of the source.

Let us assume that I am at rest and I am observing a plane which is travelling at the speed of sound ( Mach 1 ). So the velocity of the observer would be zero and the velocity of the source would be equal to the speed of sound. Hence the denominator would be tending to zero and the observed frequency would be tending to infinite. Hence the pitch of the sonic boom should be high. But most of the times I hear a sonic boom it sounds like a low pitched thunder. Why is that?

Answer 1

First, "sonic booms" do not have to be low pitched. For example, the crack of a whip is created when the tip of the whip exceeds the speed of sound, and most would agree this crack is a high pitched sound (at least higher than the sonic boom of a plane).

Second, the Doppler's effect equation you give just applies to sounds emitted by the source of the sound.$^*$ However, a shock wave is not the same thing as just emitting a sound. To quote the wikipedia article on sonic booms from planes

There is a rise in pressure at the nose, decreasing steadily to a negative pressure at the tail, followed by a sudden return to normal pressure after the object passes. This "overpressure profile" is known as an N-wave because of its shape. The "boom" is experienced when there is a sudden change in pressure; therefore, an N-wave causes two booms – one when the initial pressure-rise reaches an observer, and another when the pressure returns to normal. This leads to a distinctive "double boom" from a supersonic aircraft. When the aircraft is maneuvering, the pressure distribution changes into different forms, with a characteristic U-wave shape.

And the pitch will depend on how much air is being pushed around:

The power, or volume, of the shock wave depends on the quantity of air that is being accelerated...Longer aircraft therefore "spread out" their booms more than smaller ones, which leads to a less powerful boom

and I am sure distance between observer and source also plays a role given how lower pitched sounds can propagate farther higher pitched ones.

So, it seems like the misunderstanding here is thinking when something emitting a sound exceeds the speed of sound, the Doppler shifted sound wave is the sonic boom. This is not the case. Sonic booms are much more complicated than this, and involve more physics than just the bunching up of sound waves due to a moving source.

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$^*$Not to mention it gives negative frequencies when $v_s>v$. Obviously, this equation breaks down at speeds equal to or larger than the speed of sound.

Answer 2

Hence the pitch of the sonic boom should be high.

As others have already pointed out, a sonic boom is not subject to the Doppler effect because it's just a nonlinear pressure pulse (i.e., a single or half period sound wave) due to something moving air faster than the speed of sound.

But most of the times I hear a sonic boom it sounds like a low pitched thunder. Why is that?

A sonic boom, when transformed to frequency space using a Fourier transform, has a very broadband response function (for a pure discontinuity or step function, the Fourier transform would have power at all frequencies). For the sake of simplicity, let's assume there is equal power at all frequencies (this isn't true but it's an easy way to understand the phenomena)

The frequency at the receiver (i.e., pitch heard by an individual) is dependent upon several things, one of which depends upon distance from the source (as previously noted). As you move further from the source, any broadband sound will shift to lower and lower frequency due to the higher frequencies being attenuated.

Another issue is that the human ear does not have a flat frequency response. Very strong sonic booms will sound different than weaker ones due to saturation at some (or all) frequencies within the range of human hearing. So in our overly idealized scenario of a short duration white noise pulse, if the amplitude of the white noise is large enough some ranges of frequency will saturate and cause the brain's interpretation of the "pitch" of the sound to be different than if the amplitude of the white noise was lower.

Note that the frequency response of a sonic boom depends upon the wavelength of the pressure pulse. The sharper the gradients the broader the range of sound frequencies.

Fun Side Note

I have heard sonic booms from whips, bullets passing close by, and fighter jets flying overhead. All of these sound like hollow cracks, not a high or low pitched noise. If you are farther from the source (e.g., the jet breaks the sound barrier at a very high altitude), then it can sound like a lower frequency rumble (the rumble versus crack would require another question and answer).

A few years ago we had an audification specialist working in our lab turning spacecraft data into sound files. At first glance it seemed like a superficial, subjective art project. After chatting with him, I started to realize they weren't superficial or subjective but rather had a very useful analysis technique. So he started listening to magnetic field data in the solar wind without knowing really anything about the data or phenomenon therein. He quickly found some very interesting sounds and we realized a few things. First, the ears can digest and differentiate a much larger bandwidth of information per unit time than the eyes. That is, he was able to search through and consistently find interesting time intervals orders of magnitude faster than anyone in our lab could do "by eye." He was able to categorize and itemize intervals for ~20 years of Wind magnetometer data in a matter of a few weeks. Trying to do the same thing "by eye" would have taken even the fastest of us several years.

Second, he was able to identify the unique sound of collisionless shock waves. They all had a similar auditory response, i.e., a kind of hollow crack and/or thud. When examined in his high end audio software, they all looked exactly as one would expect. They were an isolated band of power across the entire frequency range observed by the instrument.