I Interval between double sonic booms

[sophiecentaur]

TL;DR Summary

Can you tell the length of the aircraft by the interval>

Yesterday afternoon I heard (and so did every bird in the district) a very loud, low frequency, double boom which was very vigorous and I suspected all sorts of things but then I thought "double boom!" and (engaging smartarse mode) I informed my wife that it was only a sonic boom. I was later proved right. Apparently, a Typhoon fighter had been ordered to race to the scene of a civilian aircraft and to accompany it back to Stanstead (UK) airport.
Now here's the thing. Yesterday's two booms were very close together and I remembered a few occasions in my teens when Concord was being tested in the West Country. Those booms were wider spaced iirc. I then thought that the interval between the booms would be strongly related to the length of the craft. A Typhoon is 16m long and Concord was 62m long. So the boom boom yesterday would have been about 1/4 the interval that I remember from the past. The actual speed of the aircraft would (I suspect) not be as important as that would only affect the angle of the shock wave and there would be some trigonometry involved. Does that sound reasonable or is my memory (50 years) just dodgy?

 

[.Scott]

OK. As I read through some articles on this, it seems as though the answer is "yes".

For the most part, the sonic booms are created where the cross-section of the aircraft (or spacecraft ) changes. And that means the forward end rearward ares of the craft. So the time between booms would be a measure of the time is takes for the vehicle to cross the distance between those two areas.

But I have also seen images of the shock waves coming off a supersonic projectile. Not all of the waves are traveling at the same speed - and some catch up to others and merge with them.
So I suspect there are cases where these waves diverge from each other after separating from the projectile.

 

[.Scott]

Here's a Schlieren photograph that shows the point I was making.
Notice that the shock waves do not run parallel to each other, so using the time between the double booms as a measure of how long it too the craft to travel it's own length will not be precise.

Depending on the shape of the projectile, it is also common for the trailing boom to shed from a locatoin behind the projectile - which would also make the boom interval longer.

 

[mfb]

You would have to study where the shock waves are coming from for the aircraft. The front is an obvious point, the wings will probably create at least one, the tail can also create one, and potentially other places as well. The relative strength depends on the aircraft.

Sonic booms can be quite characteristic. The Falcon 9 boosters create a triple-boom when they return to land, for example. "boom---boom-boom" - front, grid fins, tail

 

[sophiecentaur]

Thanks for the responses; interesting and it's not surprising that the answer is not simple. I now see that the position of the observer and the distance from the source could affect the spacing of the waves as well as the speed. Also the details of the structure.
I would suggest that there is a maximum interval between the booms, for any plane at a given range etc.. A long plane could generate waves with a larger interval. I am thinking in terms of ships and the bow waves they produce.
Despite the rather loose rule that I was trying to apply, I would say that rules as fuzzy as that one are often applied in astronomy when estimating the sizes and masses of distant structures.

Notice that the shock waves do not run parallel to each other,

As I recall, the actual shock wave (where a wave has higher velocity than sound) is limited to a region quite close to the plane. (We do not 'hear a shock wave'; just the resulting boom). The waves take up a sonic speed quite soon and then I would have to think they would more or less have the same 'virtual direction' for a distant observer (parallel plane waves) and an unchanging interval. I don't know whether that makes things easier or harder.

 

[hutchphd]

The waves take up a sonic speed quite soon

Is their final spacing then speed dependent? Can one use that to gauge mach number if you know the aircraft?

 

[sophiecentaur]

Is their final spacing then speed dependent? Can one use that to gauge mach number if you know the aircraft?

I guess so. It could be useful in planning an engagement with an enemy craft. They will all have 'footprints' recorded of each aircraft type.
Arm waving here but the angle would depend on the Mach number, I guess so the geometry suggests that the spacing would be a function of the speed.

Not all of the waves are traveling at the same speed

The speed of the launched shock wave would, I imagine, be related to the Energy imparted to the displaced pocket of air so the larger features (?) could produce faster shock waves.

 

[mfb]

If you can hear the sonic boom you can probably see and track the aircraft, too. Even better: You can see/get it on the radar before it passes you, while the sonic boom comes much later.

 

[sophiecentaur]

If you can hear the sonic boom you can probably see and track the aircraft, too.

The circs would need to be just right. You don't know its there until it has gone at least half way across your field of vision - or even further. You may have a chance of just being lucky and out in the open (observation post and a trained observer) but it's all over before most people could get to see it. At 1200mph, it would go from overhead to perhaps five miles (visual limit for a small plane, moving away from you) in 5/1200 hours = 15s. Put down the potato you were peeling in the kitchen, walk outside and scan the whole sky - what do you think?
I think "possibly" rather than "probably".
Radar would have a minute or more to do the job and it would easily identify a fast target.

 

[mfb]

You were talking about "planning an engagement with an enemy craft". If you peel potatoes in the kitchen you probably don't plan to attack a supersonic aircraft.

 

[sophiecentaur]

My original scenario was peeling potatoes.
The engagement scenario was supplementary. The problem would be in setting up a response in such a short time. Course and position are hard to estimate from just a short visual contact. But any info is good info in combat.
Apparently there was a similar incident an another sonic boom yesterday - somewhere else.

 

[cjl]

Now here's the thing. Yesterday's two booms were very close together and I remembered a few occasions in my teens when Concord was being tested in the West Country. Those booms were wider spaced iirc. I then thought that the interval between the booms would be strongly related to the length of the craft. A Typhoon is 16m long and Concord was 62m long. So the boom boom yesterday would have been about 1/4 the interval that I remember from the past. The actual speed of the aircraft would (I suspect) not be as important as that would only affect the angle of the shock wave and there would be some trigonometry involved. Does that sound reasonable or is my memory (50 years) just dodgy?

The speed of the craft does matter as well, because you're hearing one boom off the nose of the craft and one off the tail. Therefore, the spacing is going to be basically the time between when the nose passed overhead and when the tail did. Therefore, an identical length craft traveling mach 3 will have the booms spaced at half of the interval of one traveling mach 1.5. A craft twice as long will have twice the interval between booms.

 

[sophiecentaur]

I realize that the speed of the craft is a factor but that analysis is too simplified. Why should the craft be overhead when the boom is heard ? The craft could be well away from perpendicular to hearer. There is a propagation delay which means the craft will be forward of the originating region of the boom and the difference in arrival times will be due only to the effects in the actual region of the ultrasonic phase of the wave. Most of the wave path will be at sonic speed. A difference in time but not proportional to the Mach number. Also, the range of Mach numbers is no great in practice.

 

[cjl]

The craft is well away, but the booms you hear were generated when the flightpath was perpendicular to your line of observation (directly overhead, if it flew over your location). The difference in heard time will scale nearly perfectly with mach number.

It's true that the N-wave you hear will be propagating at sonic speed, but the two sharp booms you hear will be closer together (both physically as they travel through the air and in time) if the originating craft was traveling more quickly.

 

[sophiecentaur]

Doesn’t the propagating wave have a conical shape? I should have thought that, as with a ship’s bow wave, the wave generated overhead would propagate to intercept the ground forward of the craft. A ship can be well past before it’s bow wave hits the shore. I am assuming that the wave is at sonic speed long before it is heard by most people on the ground.

 

[hutchphd]

Doesn’t the propagating wave have a conical shape? I should have thought that, as with a ship’s bow wave, the wave generated overhead would propagate to intercept the ground forward of the craft. A ship can be well past before it’s bow wave hits the shore. I am assuming that the wave is at sonic speed long before it is heard by most people on the ground.

But a bow wave from a ship makes a fixed angle (19.5deg??if I recall) determined by dispersion of surface waves (v∝√λ) independent of source speed. For the airplane I believe @cjl is correct...the particular sound that hits you is part of spherical wavefront that started approximately overhead. Of course the source is the "shock wave envelope" and not the plane proper as you pointed out.

 

[Klystron]

Talk about 40-50 year old memories; during my years on electronic warfare ranges I heard (endured) numerous segmented supersonic shock waves on the ground. Prevalent theories for the multiple sounds included

• hearing the leading and trailing edges of the sonic envelope.
• reflection of the initial sound from the ground into the sky and back down due to atmospheric effects or nearby hills.
• multiple aircraft in a flight transitioning to supersonic mode.

The idea that different sections of the aircraft cause multiple sonic effects intrigues me. I can confirm that acoustics alerted us to aircraft in the area but unlike sonar in water even supersonic shock waves were too diffuse for tracking from the ground compared to radar, thermal and visual tracking data.

I always enjoyed simulated warfare against RAF Typhoon fighters but was seriously impressed by Tornados flown so low radar was all but useless while dust clouds obscured optical tracking. Good thing the bombs were cement dummies or I would not be here to enjoy Physics Forums.

 

[sophiecentaur]

The craft is well away, but the booms you hear were generated when the flightpath was perpendicular to your line of observation (directly overhead, if it flew over your location). The difference in heard time will scale nearly perfectly with mach number.

It's true that the N-wave you hear will be propagating at sonic speed, but the two sharp booms you hear will be closer together (both physically as they travel through the air and in time) if the originating craft was traveling more quickly.

Are you saying that the waves you hear are due to individual local 'explosions'? I understood that the booms are caused by a pair of conical waves going past your ears. Such a wave front can only be synthesised by a distributed source (i.e. from a long section of the plane's path). There was earlier mention of Huygens' construction and that seems relevant here.

the particular sound that hits you is part of spherical wavefront that started approximately overhead

A spherical wave front would be heard by people at any position on the sphere. Individual boom(s) are heard distinctly, and that has to be because they result from summation of waves along the plane's path.

 

[hutchphd]

A spherical wave front would be heard by people at any position on the sphere. Individual boom(s) are heard distinctly, and that has to be because they result from summation of waves along the plane's path.

You are correct It is a (coherent?) sum over a finite segment of the path which causes the sound to be loud. The entire path will not contribute equally however. Those portions closest to the observer will (because of 1/r²) contribute most to the sum I think. Maybe the propagation geometry from path segment to the cone where it passes the observer matters. It is not obvious to me how much of the path is very significant. The boat wake is quite different physics (which always surprises me)
If I have the energy I will play with the Huygens a bit and share. There is doubtless already la master's thesis somewhere about this!...

 

[marcusl]

Shock waves are conical, not spherical. The cone half-angle depends on speed—it’s 90 degrees about the moving surface at exactly Mach 1, becoming acute (less than 90) at higher speeds.

Because of the nature of aerodynamics, the speed of air flowing over different surfaces (nose, fairing, wing root, wing tip, etc.) can vary even though all parts of the plane are traveling at the same speed relative to the ground. As a result, shock waves from different surfaces can generate different cone angles. This is a likely explanation of why the lines aren't parallel in the photo in post #3. It also explains transonic flight, where certain parts of a plane go supersonic while the rest is subsonic. See photo here, e.g.
https://en.wikipedia.org/wiki/Vapor_cone

 

[sophiecentaur]

The entire path will not contribute equally however.

Right. It has to be true that the nearest that the craft passes will involve the shortest propagation time and the least loss and there will be lower level contributions from either side of the near point. There is one cool person of light here for me and it is the fact that we are not dealing with an array of coherent sources so Huygens won't apply in a simple way. That would imply to me that the directivity of the 'array' would be due to a relatively short region of coherence for the generated waves. The spectrum of the Boom is very low frequency and that could be due to this low coherence; the envelope of the pressure wave that arrives at the ground is wide (100ms or more?) and that could imply that there are contributions from several tens of metres along the plane's path. High frequency components of the pulse just get smoothed out to produce a 'very' infrasonic wave.

The speed of the craft does matter as well, because you're hearing one boom off the nose of the craft and one off the tail.

This, as I now realize, has to be the crux and so the gap between booms depends both on the speed and the length of the craft equally (t=v/s). The actual propagation mechanism won't affect this.

 

[.Scott]

Doesn’t the propagating wave have a conical shape? I should have thought that, as with a ship’s bow wave, the wave generated overhead would propagate to intercept the ground forward of the craft. A ship can be well past before it’s bow wave hits the shore. I am assuming that the wave is at sonic speed long before it is heard by most people on the ground.

To be clear, it will be forward of where the aircraft was when that part of the sonic boom reaches the ground. By that time, the aircraft would have moved forward. So, at Mach 1, the shock waves are running along side the aircraft, so you would see the aircraft nearby - but that would be a marginal sonic boom. At Mach 2, you need to look forward and up at a 45-degrees elevation angle to spot the plane. That angle gives you a crude estimate of how fast the aircraft is flying.
When I was a kid, the military was still doing super-sonic practice flights over parts of New England and over my area in Lowell, Mass (perhaps from Fort Devens). So kids of that time (who spent much more time outdoors that they do now) became well-practiced at spotting these jets. In 80% of the cases, there was not enough unobstructed sky to find them in time.

 

[sophiecentaur]

The boat wake is quite different physics (which always surprises me)

I was looking at one of the pictures of a duck swimming and producing a wake and realized an explanation for the 'different physics' is because the wave speed of sound is more or less the same for all frequencies but a gravity wave (surface water wave) has a varying speed, according to frequency.

is from this Wiki Link (others are available) So the details are different and in the case of a duck, swimming along at a leisurely pace it can be traveling faster than the tiny waves it sets up and that can generate a wake. It's different from the effect of a boat going faster than its 'design speed', which is more like the supersonic aircraft boom formation.

 

[hutchphd]

It's different from the effect of a boat going faster than its 'design speed', which is more like the supersonic aircraft boom formation.

And of course it is why the "hull speed"scales as √ length.

Right. It has to be true that the nearest that the craft passes will involve the shortest propagation time and the least loss and there will be lower level contributions from either side of the near point. There is one cool person of light here for me and it is the fact that we are not dealing with an array of coherent sources so Huygens won't apply in a simple way. That would imply to me that the directivity of the 'array' would be due to a relatively short region of coherence for the generated waves. The spectrum of the Boom is very low frequency and that could be due to this low coherence; the envelope of the pressure wave that arrives at the ground is wide (100ms or more?) and that could imply that there are contributions from several tens of metres along the plane's path. High frequency components of the pulse just get smoothed out to produce a 'very' infrasonic wave.

This, as I now realize, has to be the crux and so the gap between booms depends both on the speed and the length of the craft equally (t=v/s). The actual propagation mechanism won't affect this.

I agree completely although I could have said it using twice as many words...
I think I can use something like Huygens (a green's function I guess is correct term) assuming the source at each segment of the plane has a constant character. The phasing is tricky because that will happen where the true shock wave transitions to "regular sound" I guess. Maybe just assume they are emitted in phase at some cylindrical distance (a few meters?)from the plane's path as i passes? Then the length of the contributing part of the path will scale with wavelength and a nice low boom will result.

 

[sophiecentaur]

Maybe just assume they are emitted in phase at some cylindrical distance (a few meters?)

The notion of phase gives me a problem because it implies some natural frequencies of oscillations. Isn't the shock wave more random than that? There must be turbulence but are the modes well defined if it's possible to have a spectrum of wavelengths which can interfere in some way. Trouble is that the verbal handwaving that I have read in the popular discussions doesn't do any more than say 'this or that' happens.

 

[sophiecentaur]

The notion of phase gives me a problem because it implies some natural frequencies of oscillations. Isn't the shock wave more random than that? There must be turbulence but are the modes well defined if it's possible to have a spectrum of wavelengths which can interfere in some way. Trouble is that the verbal handwaving that I have read in the popular discussions doesn't do any more than say 'this or that' happens.

That is not a very good explanation. This is better, I think: If the initial wave is an impulse of finite duration and finite attack and decay then there will be a spectrum consisting of an infinite number of frequencies which rolls off at 1/t_transitions. The filtering (dispersion) effect of the propagation on the conical wave will suppress the high frequencies and only the very low frequencies will survive as far as the ground.

It would be interesting to know if there is an audio recording of the shock wave in the close vicinity of the plane.
Supersonic shock waves are frequent with high velocity rifles but the initial explosion could interfere with hearing just the shock wave

 

[hutchphd]

My motivation is that the shock wave looks like a step function and at the point it becomes regular (linear) sound we can look at the Fourier decomposition of that shape as a boundary condition. I think the step center defines a surface of constant phase for all component waves.. I'm being a little loose here but I think you follow. I also glanced at Cerenkov radiation as a model and think it may work to save effort at "reinventing the wheel."

Supersonic shock waves are frequent with high velocity rifles but the initial explosion could interfere with hearing just the shock wave

Does a supersonic rifle give different sound than .22? Let us not forget that a bullwhip (I guess that really is true...)

 

[averow45]

Summary: Can you tell the length of the aircraft by the interval>

Yesterday afternoon I heard (and so did every bird in the district) a very loud, low frequency, double boom which was very vigorous and I suspected all sorts of things but then I thought "double boom!" and (engaging smartarse mode) I informed my wife that it was only a sonic boom. I was later proved right. Apparently, a Typhoon fighter had been ordered to race to the scene of a civilian aircraft and to accompany it back to Stanstead (UK) airport.
Now here's the thing. Yesterday's two booms were very close together and I remembered a few occasions in my teens when Concord was being tested in the West Country. Those booms were wider spaced iirc. I then thought that the interval between the booms would be strongly related to the length of the craft. A Typhoon is 16m long and Concord was 62m long. So the boom boom yesterday would have been about 1/4 the interval that I remember from the past. The actual speed of the aircraft would (I suspect) not be as important as that would only affect the angle of the shock wave and there would be some trigonometry involved. Does that sound reasonable or is my memory (50 years) just dodgy?

Back in the Space Shuttle era double sonic booms were frequently discussed when the Orbiter was in sight of the landing zone. A good example of this is at elapsed time of 03:01 in this youtube video:

 

[hutchphd]

I'm assuming the "double boom" is a "plus" step followed by a "minus" step. Yes?

 

[sophiecentaur]

I'm assuming the "double boom" is a "plus" step followed by a "minus" step. Yes?

I don't think that's a good way to look at it; you are implying a 'blow' at the front and a 'suck' at the back. But the lowest suck pressure (highest negative) pressure possible is Zero. The (positive) pressure at the front is enormous and the effect at the back is similar.
As far as I can see. the transitions both from air to plane and plane to air must produce positive shockwave pressure. I would look upon it as the air coming together at the back with a clang and that would generate positive pressure. My experience has been that both booms are of equal amplitude (for what it's worth).

 

[sophiecentaur]

My motivation is that the shock wave looks like a step function and at the point it becomes regular (linear) sound we can look at the Fourier decomposition of that shape as a boundary condition. I think the step center defines a surface of constant phase for all component waves.. I'm being a little loose here but I think you follow. I also glanced at Cerenkov radiation as a model and think it may work to save effort at "reinventing the wheel."

Does a supersonic rifle give different sound than .22? Let us not forget that a bullwhip (I guess that really is true...)

It would be hard to make a control experiment.
Also, the power is so much less than a plane and the dispersion effects would not be heard at the same distance. It would not ‘scale’.

 

[hutchphd]

As far as I can see. the transitions both from air to plane and plane to air must produce positive shockwave pressure. I would look upon it as the air coming together at the back with a clang and that would generate positive pressure. My experience has been that both booms are of equal amplitude (for what it's worth).

I recommend a quick look at

https://www.af.mil/About-Us/Fact-Sheets/Display/Article/104540/sonic-boom/
Apparently the "N" wave is positive -negative slope -positive, hence its name. The Max over-pressure ever measured was 144 psf (1 psi... it seems too small, doesn't it?) so you could easily have a negative "gauge" pressure that low

 

[hutchphd]

And just for semantic clarity: by definition a shock wave is always moving supersonic speed I think?. The sonic front that describes the boom at the ground is then not really a shock wave even though its point of contact with the Earth moves along at the supersonic speed of the plane.

 

[sophiecentaur]

I agree. If there was an actual shock wave on the ground there would be a net movement of air.
The N wave appears to be a measurement on the ground and, by then, the shock wave has been dissipated and what remains is a low frequency sound pulse with fairly low peak to peak air pressure variation.
Nowhere can the absolute air pressure be less than zero. Descriptions of this need to use the term ‘gauge pressure’ to avoid confusion, I think.

 

[hutchphd]

I agree. If there was an actual shock wave on the ground there would be a net movement of air.
The N wave appears to be a measurement on the ground and, by then, the shock wave has been dissipated and what remains is a low frequency sound pulse with fairly low peak to peak air pressure variation.
Nowhere can the absolute air pressure be less than zero. Descriptions of this need to use the term ‘gauge pressure’ to avoid confusion, I think.

The term " shock wave" is used very carelessly in the literature. What I still don't understand at all is how far from the aircraft does the true shockwave persist. Centimeters ? hundreds of meters? You can see the "N' pressure profile in some of the color enhanced schlieren imagery;

 

[sophiecentaur]

What I still don't understand at all is how far from the aircraft does the true shockwave persist. Centimeters ? hundreds of meters?

I have the same problem. The only thing one can work on is those images which show a curved wavefront extending to a bit less than the size of the plane. Beyond that, the wavefront is 'straight' and that suggests that there is no change beyond the curved bit. The only alternative is to assume that the transition from shock wave to sound wave is way beyond what any of the photographs show. We can't be the only ones with this question so I have to conclude that the transition region is quite small.

I have some experience of ships' wakes and I can say that when I have been 'hit' by the wake of large ships, passing within several ship lengths (in deep water) I have not been aware of being pushed to one side; it's been largely up and down (scary at times) motion. So that implies to me that the 'shock wave' region is limited.

 

[cjl]

I'm assuming the "double boom" is a "plus" step followed by a "minus" step. Yes?

As has already been stated, but I feel like elaborating, it's actually usually a positive step, followed by a linear negative slope, then another positive step. This is called an "N-wave", for obvious reasons. The magnitude of the overpressure is very small, since it is basically just a sound wave at the point it reaches the ground.

Interestingly, the reason for this shape is because a positive step following another positive step will tend to catch up to the front one, while a negative pressure change will tend to spread out. As a result, all the smaller positive steps coming off of various parts of the aircraft will tend to coalesce, while the negative pressure gradients will tend to spread out and smooth out until you get that characteristic N shape. You can see this very well here.

 

[cjl]

The term " shock wave" is used very carelessly in the literature. What I still don't understand at all is how far from the aircraft does the true shockwave persist. Centimeters ? hundreds of meters? You can see the "N' pressure profile in some of the color enhanced schlieren imagery;

The actual shock only really persists as far from the plane as the flow itself is impacted. An oblique shock (by definition) causes the flow to change direction, so as soon as you're far enough away that the flow direction is basically unchanged through the wave, you're at the region where you have a sonic boom rather than a shock.

EDIT: Interestingly, this means that the shock will both be significantly stronger and persist farther from the plane when it is making more lift, such as when it is pulling a high-G turn, even if the airplane's shape and mach number are identical, simply because it has to affect more air in a larger volume around the plane.

 

[sophiecentaur]

The actual shock only really persists as far from the plane as the flow itself is impacted.

Great; there's the answer.

while the negative pressure gradients will tend to spread out

A brilliant observation. It's something I have seen (a classic demo) of the wind in front of a loudspeaker at high sound levels. The wind is in the positive direction - momentum transfer directly to the air molecules - but the return flow is due to the pressure from all around. A candle flame is constantly pushed away.
There must be similar circular motion in the shock wave - I guess that's the quoted turbulence idea.

the shock will both be significantly stronger and persist farther from the plane when it is making more lift,

Another good observation!

 

[russ_watters]

Are you saying that the waves you hear are due to individual local 'explosions'? I understood that the booms are caused by a pair of conical waves going past your ears.

It's not "explosions"; many concentric circles are just a simple way to describe it, and may help here. But think about the geometry of this. When the cone reaches you, the plane is past you. So where was the plane when the sound incorporated into that part of the cone was released?

It must be the shortest distance the sound can travel, which is the perpendicular/closest point of approach.

 

[sophiecentaur]

It's not "explosions"; many concentric circles are just a simple way to describe it, and may help here. But think about the geometry of this. When the cone reaches you, the plane is past you. So where was the plane when the sound incorporated into that part of the cone was released?

It must be the shortest distance the sound can travel, which is the perpendicular/closest point of approach.

Yes - I get it now. It's isolated pulses involved and the nearest point is the source of the main part of the energy. The conical wavefront is formed by many contributing spherical wavelets though. It's a diffraction mechanism. The period of the pulse received is very long and it is only the lowest frequency parts of the spectrum that make it to the ground. Almost more of a 'woomph" than a "bang".

 

[hutchphd]

It must be the shortest distance the sound can travel, which is the perpendicular/closest point of approach.

This is fundamentally true but oversimplified. Waves from parts of the path before and after the distance of closest approach also contribute to the front (they can do that only because the plane is supersonic). The ones that do it coherently are Gaussian distributed along the path with a sigma proportional to the wavelength centered at the point of closest approach. I have worked this out ( in Eikonal approximation) and will write it up when I can find a few hours. Obviously the long wavelengths then sample more of the path and so are more prominent in the "boom" front. The amplitude of the boom falls off like √ (distance of closest approach) because the wave is essentially a cylindrical wave.
This has been a great question!