# Is a sonic boom loudest when source is at exactly Mach 1?

Please imagine this cartoonish thing: a flying loudspeaker, playing a continuous sound, with zero air resistance (i.e its moving through the air doesn't generate any whooshing sounds) is coming towards you. Imagine it stops one meter before hitting you. How would its speed influence the amplitude of the sound you hear?

The way I think of it, the "shock wave effect", that is, the compression of the sound waves, would be biggest when at Mach 1, because the source of the sound would be always exactly on the crest of the sound wave.

After Mach 1, the sound would be heard in reverse: the faster the source, the more diluted the sound would get. Therefore, it's not as loud as Mach 1.

My questions are:

1. Is this thinking correct?
2. If so, does this mean that the amplitude heard at $v = 0.98c$ would be more or less the same as $v = 1.02 c$?
3. Could a flying airplane be modeled this way? (I guess this question would be the same as: is the main source of sound the plane's engines or air resistance?)

The amplitude/intensity of a sonic boom (in Earth's atmosphere) is dependent on the change in pressure across the shock wave. This should make sense, as the intensity of a sound wave is dependent upon its pressure relative to quiet periods. We also know that the ratio of the downstream to upstream pressure is proportional to the square of the Mach number. So the sonic boom intensity increases with Mach number (recent studies are working on changing the shape of the Mach cone by changing the nose shape of planes in an attempt to decrease the effective range of the sonic boom).

Except for speeds right near the M = 1 point, a sound produced by a source moving relative to you should not have an amplitude dependence on the speed of the source. Meaning, I see no reason why a jet would be louder because it moved towards you. The intensity of sound, so far as I know, only depends upon the inverse of the distance from the source squared (i.e., I $\propto$ $r^{-2}$).

Once you meet or exceed M = 1, then I imagine there would be complications introduced by the interaction between the source (presumably downstream of the Mach cone) and the shock wave itself (if the sound waves can even reach the shock wave, that is). For most purposes, I still think the intensity of the source would depend upon $r^{-2}$, not the speed of the source relative to you.

• I appreciate your whole answer, but I'm afraid you focused on the question title and failed to understand what I meant in the body, which relies completely on absence of air resistance (ergo, uh, lack of a proper sonic boom entirely?). I mean a travelling sound source at Mach 1. Like a drummer continuosly hitting, always on the wave crest. Doesn't it interfere constructively with the previous waves? Oct 29, 2014 at 2:21
• Are you thinking about a sort of "pile up" of waves? Again, I do not think it would be that simple. There might be a scenario where one could try and create conditions conducive for constructive interference, but that seems exceedingly unlikely. More importantly, if there is no air resistance, ergo, no air, then a source could not create sound. Oct 29, 2014 at 11:43
• About the "no air resistance -> no air" bit, I really don't mean to make this realistic. I just want to better understand if sonic booms have more to do with sound generated by air resistance or sound generated by turbines. Could you go back and try to address the three points I ask in the question body? I feel that what you said about "piling up waves" is the egg of an answer to point #1. Oct 29, 2014 at 18:02
• Point 3: The main source of noise from a jet flying at subsonic speeds would be due to the engines. This would still apply to supersonic speeds at the right distances, Mach numbers, temperature and pressure, etc. The pressure ratio increase across a normal shock (which controls the loudness of a sonic boom) depends upon the Mach number squared. The intensity drops off as r$^{-2}$, so you would need to balance to two to determine where there is overlap. Since the engine should have a set loudness, that should confine your results. Oct 29, 2014 at 19:00
• So, if I understand correctly, if the engines were completely silent, there would still be a shock wave/sonic boom, right? Oct 29, 2014 at 23:23

Pardon me if I substitute a different illustration — a car travelling in a straight line, such that it passes near to you. We are all familiar with the way the sound drops in pitch as it goes past. [In the case of your speaker illustration, the difference is that the sound would be of a high pitch as it approaches, and then it would suddenly drop to the actual pitch of the sound, when the speaker stopped moving.]

This is because, for each unitary sound wave, the point at which it is generated is moving… which makes each of {the resultant sound waves that travel through the air} shorter or longer — which equals higher or lower pitch (respectively). [This is “the Doppler Effect”.]

The other part of the picture is that Mach 1 is exactly the speed at which the size of the above phenomenon reaches 0. The idea is that, so to speak, it is the speed at which the sound source moves forward so fast that the peak of the next sound wave is generated at the point that the previous one moved to (at the speed of sound). Informally, the result of this is tremendous pressure in the air at the front of the moving object.

This brings us to part 2 of your question.

In the case of an object (an aircraft, or theoretically a car or a speaker) moving towards you at {approaching the speed of sound}, any sound it makes would conceptually be of an incredibly high pitch, and would arrive at about the same time as the object… and I imagine that the actual effect would be just one big push of air, like one drum beat — a sonic boom, I guess — as much at 0.98x as at 1x.

That is… this holds even if we imagine out the effect of the object itself physically pushing the air — which is also one big push. The combined effect, practically in the real world, is just one big powerful push of air.

The immediate answer to — particularly — the title of your question is that that is a sonic boom — it happens at the speed of sound, because the source is travelling at the speed of sound. Conversely, that does not address the intent and detail of the question.

The other aspect of all this is where you are standing. In the case of a car travelling nearby (assuming that it does not actually drive over you)… there is a point at which it is travelling perpendicularly to you; at that point, the sound is of the actual pitch of how it would sound stationary.

In the case of an aircraft that travels nearby, really really fast, the same is true, but the pitch of the sounds it makes would change from extremely high to extremely low in a very short period of time. If it is farther away, the change will be slower.

Thus… the sonic boom at Mach 1 does not exist in front of the aircraft (as in, say, 100m in front)… and you hear it from the side as a big boom as the aircraft passes by… and behind the aircraft all there is is turbulent air and low-frequency sound.

So what about this pressure boom at (say) Mach 2, then? Certainly there is still all that pressurised air happening!? Again, there is no sound in front of the aircraft, and behind it all there is is turbulent air and low-frequency sound. Why is there no sonic boom as it passes? At face value, it seems to me that there certainly should be a big boom… but the real-world fact is that there is not (I believe).

I am thinking that it is like the fact that air pressure reduces to the average combined effect of all the individual air molecules hitting a surface. In the same way, the speed of sound supervenes on individual air molecules moving around [at well above Mach 1, I think] and hitting each other, but it is “actually” a pressure wave passing through air — at the speed of sound — Mach 1.

A jet at Mach 2 will certainly disturb a lot of air, but the sonic boom at Mach 1 arises from the speed of the pressure wave, not of the individual air molecules. Either way, a lot of air is disturbed — arguably twice as much at Mach 2 — but it is only when the aircraft is moving at the same speed as the pressure wave that this pressure wave-related phenomenon is generated.

As for part 1 — the volume (loudness) of the sound… . The equation about less sound at less distance is just about distance; it does not deal with sound waves getting compressed or expanded because of the source moving. Given that sound waves reduce to air pushing and then retreating (as opposed to air molecules travelling long distances at the speed of sound)… it seems to me to be an obvious and direct fact that there is more energy in the same size of push happening more times per second… but conversely that the overall average air pressure does not change.

This is a frame-of-reference problem, I’ll wager. Yes, there is more energy in the same sound pitch-compressed by an object moving towards one… but no, the ear should hear the same volume [loudness] for the same size of wave. (That is what volume is — the size of the wave.)

The overpressure of sonic boom is practically indepedent of the noise of engines. It mostly depends on the geometry and size of supersonic object and conditions of atmosphere