Why you don't (often) hear the sonic boom of your rocket

[JLebow]

Last year, my L3 tap heard me commenting that my L3 rocket went Mach 1.5 but I didn't hear the sonic boom. He said that I was in the wrong location to hear it. I've tried to make sense of his comment.

If an object is stationary, and emitting a sound, the sound waves will radiate as expanding spheres. In this diagram (2D for simplicity) the circles show how far the sound radiates at time = 1, 2, and 3 seconds. I'm going to use 343 m/s as the speed of sound.

If the object is traveling to the right at a velocity of half the speed of sound, the sound still radiates out as a sphere from the point the object was located when the sound was emitted, but the center point of each sphere moves to the right. This diagram also illustrates the doppler shift. To an observer that sees the object approaching, the sound is shifted up in frequency, and if the object is moving away the frequency is shifted down. This is because the spacing between waves is affected by the motion of the object.

An interesting thing happens when the object is moving at the speed of sound. The object moves at the same rate that the wave propagates, and on one edge, all of the cycles of sound coincide. Each wave, lining up in phase causes a much bigger pressure wave - a sonic boom. And we can see that the wave occurs 90 degrees to the direction of motion. To hear the boom, you have to be located in a position where this high pressure wave crosses your location.


If the object is traveling faster than the speed of sound, the sum of the emitted waves no longer coincide at the location of the object. The object is out running the previous sound waves. The waves will coincide in the geometry of a cone trailing the object. It is important to note that only the shell of this trailing cone will have all of the sound waves coinciding in phase. To hear the shock wave, you must be located in a position that intersects the shell of the cone. Inside the cone you will hear sound, but not the boom, and outside the cone you will not hear anything.


As the object speeds up more cone gets more shallow.


Smart people have worked out that the cone angle follows Sin(alpha) = 1/Ma, where alpha is the cone half angle, and Ma is the Mach number.

So, in summary, anytime an object is traveling at or greater than the speed of sound, a sonic boom is being produced. The shock wave forms as a cone, heard when the shell of the cone crosses your observation point, and the shape of the cone depends on the speed of the object.

Here are some stats of a real flight, where I use the velocity and altitude of my rocket to determine the radius of the ring on the ground where the sonic boom might be heard. I say "might" because the sonic boom must also be energetic enough to propagate to the calculated location.



So, at 1.5 seconds into the flight, the rocket hits Mach 1. The shock wave is being radiated 90 degrees from the direction of travel, parallel to the ground, at an altitude of 228 meters. Half a second later, the rocket is now traveling Mach 1.2. The shock wave is in the shape of a cone, with an angle of 56 degrees. So if you were located more than 509 meters from the launch pad, the shock wave would have swept past your position, giving you a chance to hear the sonic boom. The flight line at our field is 150 meters away, so only somebody out retrieving a rocket would have a chance. At 3 seconds flight time, the rocket reaches max velocity, and the smaller cone angle is offset by the increase in altitude. The shock wave is now at a radius of 600 meters on the ground.

And here is the data from my L3 flight:

And the data is used to compute the radius of the sonic boom cone.


My tap was correct, and that is why I never heard the sonic boom.

 

[ebruce1361]

I would love to see a practical setup to test this math. I'm thinking five or six audio recording stations with identical equipment including recorder and datalogging sound level meter spaced 100 meters apart starting at 100 meters from the launch pad. The acoustic data from each recording station could then be used to form a "timeline" in a computer of the propagation of the soundwaves, both sonic booms and otherwise.

 

[Steve Shannon]

One other thing. The sonic boom propagates circularly, not spherically (sound emanates spherically). As such, unless you are perpendicular to the path while the rocket is traveling supersonic you won’t hear it. It’s not enough to just be farther away. The exception to that is when there is something the sonic boom can reflect from towards your ear.
Occasionally people at launches claim they heard a sonic crack. In my understanding it’s possible in three conditions, which all rely on the hearer being perpendicular to the path of the rocket while supersonic:

1. The rocket is flying at an angle, or
2. The rocket was launched from below their location so the rocket was traveling supersonic by the time its altitude matched the elevation of the person who heard the crack, or
3. The rocket was descending supersonically.

More frequently what people perceive as a sonic crack is the ejection of something from the nozzle, such as a piece of casting liner or propellant.

 

[dhbarr]

I think the usual explanation is that you can only really hear the motor unless you were somehow substantially elevated at the flight line.

I suppose something like a drone hovering in position then switching to an oversized chute might give enough dwell time to pick it up.

 

[jadebox]

So, at 1.5 seconds into the flight, the rocket hits Mach 1. The shock wave is being radiated 90 degrees from the direction of travel, parallel to the ground, at an altitude of 228 meters. Half a second later, the rocket is now traveling Mach 1.2. The shock wave is in the shape of a cone, with an angle of 56 degrees. So if you were located more than 509 meters from the launch pad, the shock wave would have swept past your position, giving you a chance to hear the sonic boom.

The shock wave doesn't form until the rocket reaches mach 1. So, as Steve pointed out above, you would have to be more than 228 meters above the ground for the shock wave to intersect with your location. The shock wave would not travel down to the ground.

An issue of Apogee's newsletter has a good article on the subject:

https://apogeerockets.com/education/downloads/Newsletter265.pdf

 

[Reinhard]

So, at 1.5 seconds into the flight, the rocket hits Mach 1. The shock wave is being radiated 90 degrees from the direction of travel, parallel to the ground, at an altitude of 228 meters. Half a second later, the rocket is now traveling Mach 1.2. The shock wave is in the shape of a cone, with an angle of 56 degrees. So if you were located more than 509 meters from the launch pad, the shock wave would have swept past your position, giving you a chance to hear the sonic boom. The flight line at our field is 150 meters away, so only somebody out retrieving a rocket would have a chance. At 3 seconds flight time, the rocket reaches max velocity, and the smaller cone angle is offset by the increase in altitude. The shock wave is now at a radius of 600 meters on the ground.

You're a bit off with the angle. At Mach 1 the shock wave travels forward, not parallel to the ground. Just repeat your circle diagram for multiple times while keeping the velocity at M1 and you see how it stays attached to the rocket. As the rocket becomes faster, a lateral component appears. At high Mach numbers, the shock wave travels mostly sidewards, but it still has a remaining forward component. If you draw the cone, the shock waves direction is always normal to the cone. It newer travels backward. The individual sound "spheres" travel backward too, but they can't overlap to a shock wave in this direction.

I've posted my attempt to visualize that previously, but I really need to convert it into a video, as it becomes much clearer this way:
https://www.rocketryforum.com/threads/sound-of-high-power-rocket-breaking-mach.155985/post-1935864
Reinhard

 

[billdz]

Does anyone have a drone recording of a sonic boom?

 

[JLebow]

One other thing. The sonic boom propagates circularly, not spherically (sound emanates spherically). As such, unless you are perpendicular to the path while the rocket is traveling supersonic you won’t hear it. It’s not enough to just be farther away.

Thank you for the insight. As many of you pointed out: I can see now that the cone does not extend below the point at which the object becomes supersonic. The sound still radiates spherically from the point of emission for the entire motor burn (we can hear the rocket motor the entire burn), but the stack up of many periods of of the sound waves responsible for the shock wave only occurs along the shell of the cone, which does not occur or propagate below the point the rocket reaches Mach. The cone does spread out normal to the velocity vector over time, and the shape of the cone (half angle) depends on the velocity.

This would explain why we don't hear sonic booms at Ball, where big rockets, traveling well in excess of mach 1, launch from 1600 meters (1 mile) out.

 

[Threemorewishes]

Forget that you are not in the boom carpet, the over pressure from a model rocket would be too small to hear over the background noise.