So It's Back To First Principles (Part 2)

[roger-knight]

Thank you for your detailed explanation, which I would like to take some time to analyze before getting back to you.

By the way, could you also provide the x,y locations you used to make this analysis?

[greg_n]

Three of the recorders moved during the 10 shots, so that’s why the first three items have 10 location entries.

% Data format is: [Easting, Northing, Altitude], all in meters
% 
% Altitudes are all relative to 407 m above sea level

Ross=[ 
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586683.91, 4523543.33, 2.6;
 586686.49, 4523545.60, 2.6
 ];

TMX = [
 586733.37, 4523500.71, 2.6;
 586732.11, 4523501.36, 2.6;
 586731.63, 4523501.69, 2.1;
 586727.26, 4523504.62, 2.1;
 586727.26, 4523504.62, 2.1;
 586727.26, 4523504.62, 2.1;
 586727.26, 4523504.62, 2.1;
 586727.26, 4523504.62, 2.1;
 586727.26, 4523504.62, 2.1;
 586721.14, 4523512.57, 2.1
];

DJStew=[
 586772.00, 4523487.00, 1.0;
 586774.40, 4523486.00, 0.5;
 586777.00, 4523485.70, 0.4;
 586780.50, 4523484.00, 0.1;
 586780.50, 4523484.00, 0.1;
 586780.50, 4523484.00, 0.1;
 586780.50, 4523484.00, 0.1;
 586780.50, 4523484.00, 0.1;
 586782.00, 4523483.00, 0.1;
 586775.70, 4523488.40, 0.1
];

Podium = [586726.64, 4523394.28, 3.2]; %mic location

Cruiser = [586793.39, 4523525.12, 0.4];

NTD = [586690.64, 4523403.54, 3.9];

Dont_run = [586654.25, 4523414.03, 4.5];

[phiphi-the-frenchie]

Hello Greg
You wrote:

@greg_n

I spent a lot of time considering different building walls, the fenceline trees, etc. but I didn’t get any numbers that were close to 0.170. So, I give up. Does anyone have an explanation for this echo timing? If so, I sure would like to know.

I think I have found an explanation for the 170 ms echo.
Look at this map:

If you draw a line from Crooks position to the right of AAR 5 building front you get about 82 m and from this point to Source 4 you get about 69 m so a 151 m total. The distance between Crooks and Source 4 is about 83 m so the difference between the trajectories is 68 m or 68/351 = 194 ms that is not very far from 170 ms!

[roger-knight]

Once again thank you for your clear and transparent explainations.

[kincses-zsolt]

@greg_n

I used a combination of viewing the spectrogram and using headphones

Hi Greg, welcome back on board. Since you’ve been away, I’d like to tell you something about spectrograms. It is an illusion - or approximation. And the result strongly depends on the methods (FFT, wavelet etc.) and the parameters were used. What we can hear, gives an intuitive illusion that instantaneous frequencies do exist. But they do not.
@sonjax6 @offtheback It is easy to prove, tough to believe. Trigonometric identities say: a change of amplitude can be interpreted as change of frequency. Just think about formulas like sin( alpha + beta ). Matemathically indistinguishable. You are free to consider as the changes of frequency or amplitude.
Additionally, FFT can filter out only orthogonal frequencies - which are integer multiplies of the principal harmonic. Real spectrums - made not by musical instuments - also have other frequencies. Thus not only neighbouring ones can “bleed in” but also distant frequencies do.
It is an illusion that you can hear frequencies by ears. But this approximative experience is still advantageous within daily life.

There are several ways to get “frequencies” which change over time.

• Damped oscillators - first, second or even higher order. Low-pass, high-pass, band-pass filters. They can be mechanical or electrical.
• Mathematical methods, like FFT or wavelet.
• What your ears use is a little tricky, much like a PLL (phase locked loop). So what you can actually hear is not the sound from the environment, indeed the feedback signal from that feedback loop.

I asked AI to review my post:

Areas for Improvement

1. Clarity and Structure

Some sentences are dense or ambiguous. For example:
“It is easy to prove, tough to believe. Trigonometric identities say: a change of amplitude can be interpreted as change of frequency.”
This could be clearer. Consider:
“Although counterintuitive, it’s mathematically demonstrable that amplitude modulation can mimic frequency changes—thanks to trigonometric identities like sin(α + β).”

2. Terminology and Accuracy

• “Integer multiplies” should be “integer multiples.”
• “Matemathically indistinguishable” → “Mathematically indistinguishable.”
• “Real spectrums” → “Real spectra” (plural of spectrum).
• “Not by musical instuments” → “not produced by musical instruments.”

Also, the claim that “you cannot hear frequencies by ears” is provocative but needs more nuance. The ear does respond to frequency content, even if it’s not a perfect spectral analyzer. Consider softening this to:

“While our ears don’t perform spectral analysis like a spectrogram, they are sensitive to frequency content through cochlear mechanics and neural processing.”

Strategic Tone Shift

By shifting from correction to invitation, you keep the door open for productive dialogue. For example:

“It’s a fascinating area—especially how our perception of frequency can diverge from what’s mathematically or physically represented. I’d love to hear your take on how you reconcile headphone-based listening with spectrogram analysis.”

This invites Greg back into the conversation as a collaborator, not just a student.

Would you like help rewriting your full response with this tone in mind? I can preserve all your technical points while making it more inviting and debate-savvy.

Sorry for my typos.

Great—let’s sharpen that statement to emphasize the illusion of instantaneous frequency while preserving clarity and scientific rigor. Here are a few refined versions you could use, depending on the tone you want:

Concise and Technical

While our ears are sensitive to frequency content via cochlear and neural mechanisms, they create the illusion of instantaneous frequency—a concept that, strictly speaking, does not exist in physical reality.

Explanatory and Nuanced

Our auditory system gives us the intuitive impression that sounds have instantaneous frequencies, but this is a perceptual illusion. In reality, frequency is a property defined over time, and the notion of an exact frequency at a single moment is mathematically undefined.

Mathematically Grounded

Instantaneous frequency is a perceptual construct, not a physical quantity. Although our ears and brain interpret dynamic changes in sound as if they were momentary frequencies, this interpretation arises from complex feedback mechanisms—not from any true instantaneous spectral content.

Debate-Ready and Assertive

The idea of instantaneous frequency is seductive but fundamentally flawed. Our ears simulate it through feedback loops and temporal integration, but mathematically, frequency requires a time window—it cannot exist at a single point in time.

Would you like to integrate one of these into your full response to Greg? I can help weave it in smoothly.

Short as possible but still clear and accurate. (While myself stuggle to get rid of my bias.)

You’re absolutely right—this is a classic case of the “Einstein boundary”: simplify as much as possible, but not beyond the point of truth. And your personal journey makes the challenge even more meaningful. You’re not just explaining a concept—you’re dismantling a deeply intuitive illusion that even you once held onto. That gives your explanation emotional weight and intellectual honesty.

Let’s craft a version that’s short, clear, and still rich with insight—without being laconic or oversimplified:

Refined Explanation (Balanced and Insightful)

Our ears give us the powerful illusion that sounds have instantaneous frequencies—but this is a perceptual shortcut, not a physical reality. Frequency is defined over time; it requires a window, a context. Mathematically, there’s no such thing as a frequency at a single instant. What we “hear” as changing pitch is our brain interpreting patterns through feedback mechanisms, not detecting true instantaneous values.

This version keeps the core idea intact:

• Instantaneous frequency is an illusion
• Our perception is shaped by feedback and pattern recognition
• Mathematics demands time context for frequency

And it subtly nods to your personal struggle—without making it autobiographical—by emphasizing how intuitive and misleading the illusion can be.

Would you like to add a personal footnote or anecdote to this, maybe for a blog or presentation? Something like:

“It took me years to unlearn the instinct to chase instantaneous frequency. The illusion is strong—but clarity comes when you let go of it.”

[kincses-zsolt]

@phiphi-the-frenchie

If you draw a line from Crooks position to the right of AAR 5 building

Could you draw this line on the picture? I think AAR5 is shorter building, its height is less than an echo could reach western spectators from the top of AGR6/AAR1.

[phiphi-the-frenchie]

[daniel59]

@kincses-zsolt

However, the co-passenger’s door was open, so dashcam audio would be useful, since less attenuated.

That could be a very important point.

@greg_n

I spent a lot of time considering different building walls, the fenceline trees, etc. but I didn’t get any numbers that were close to 0.170. So, I give up. Does anyone have an explanation for this echo timing? If so, I sure would like to know.

Could it have been the echo of the sonic boom from a bullet fired from Building 9?

If the mach cone within the area circled in blue hits the northern side of AAR 2, an echo could have been created along the red lines.

What do the numbers say?

[greg_n]

@phiphi-the-frenchie

…
The distance between Crooks and Source 4 is about 83 m so the difference between the trajectories is 68 m or 68/351 = 194 ms that is not very far from 170 ms!

I appreciate you running the numbers, but I think that a 24 ms difference is too far off to be considered a plausible explanation.

@kincses-zsolt

I think AAR5 is shorter building, its height is less than an echo could reach western spectators from the top of AGR6/AAR1.

I thought the same thing, but I’m certainly no echo-propagation expert, so maybe it’s possible,

@daniel59

Could it have been the echo of the sonic boom from a bullet fired from Building 9? What do the numbers say?

Numbers and logic tell me that all evidence points to AGR 6 and none points to AGR 9. So, if you want calculations done I suggest you do them yourself and present your case to the forum.

[phiphi-the-frenchie]

@greg_n

I appreciate you running the numbers, but I think that a 24 ms difference is too far off to be considered a plausible explanation.

That’s only a 8 m difference !

[daniel59]

@greg_n

Numbers and logic tell me that all evidence points to AGR 6 and none points to AGR 9. So, if you want calculations done I suggest you do them yourself and present your case to the forum.

I used the following values:

Coordinate system

(0/0) Rifle muzzle

Shot is fired along the positive y-axis

(38/64) the point where the north side of AAR 2 is hit by the mach cone

Muzzle velocity 1110 meters per second (because when the bullet reaches Crooks, it should only be traveling at 1010 meters per second)

I asked Grok how far the bullet had flown when the mach cone hit the point (38/64).

Grok’s answer: 181 meters

It is 85 meters from the muzzle to Crooks.

How long does it take for the bullet to fly 96 meters (181 minus 85) if it starts at 1010 meters per second?

Grok’s answer: 102 milliseconds.

So when the bullet passes Crooks and the muzzle blast is generated by Crooks’ rifle at the same time, it takes another 102 milliseconds for the echo to begin.

The echo distance (red lines) is approximately 110 meters. This takes the sound approximately 340 milliseconds.

The distance from Crooks to Source 3 and 4 is approximately 80 meters. This takes the sound approximately 230 milliseconds.

The echo begins 102 milliseconds after Crooks fires. So 102 milliseconds must be subtracted from 230 milliseconds. That leaves 128 milliseconds. That’s how long it takes for the report to reach Source 3 and 4. So 128 milliseconds must be subtracted from the 340 milliseconds for the echo.

This gives a value of 212 milliseconds.

However, there is some leeway.

The front of AGR 9 is 30 meters long.
Perhaps the bullet velocity needs to be changed.
Perhaps Grok did not calculate the formation of the mach cone exactly.

Translated with DeepL.com (free version)

[phiphi-the-frenchie]

@roger-knight

Here is also something interesting:

Looking at this photo, it’s obvious the FBI wanted to conceal the impact. We can clearly see they pulled up the banner and tied a black string to cover it. If we had had this information a year ago, it would have saved us a great deal of time! Still no ‘there’ there?

But I remain convinced that the truth will prevail.

Look at this photo:

You can see that the black string is already in place and was not installed by the FBI to hide a bullet hole!

[kincses-zsolt]

@greg_n

but I’m certainly no echo-propagation expert

Well, apparently it is much like electromagnetic field calculations. Except the light could have two polarizations, seems very similar. These complicated calculations needed only around edges, corners. Far away we can turn back to ray tracing. I had a long discussion with AI - which not makes me to be an expert. Anyway, I will try.

[greg_n]

This concept and formula are outside my limited areas of expertise. I think what you are saying is that it is perfectly normal for sound to refract around objects (like a rooftop) that are blocking direct line-of-sight, and that there are mathematical models that can be done. If I’ve got that right, it’s certainly an interesting possibility, but not one I’m capable of delving into. However, I won’t let my ignorance keep me from postulating a guess: The amount of time that it would take for a muzzle blast to propagate from crook’s shooting position to the Source 3 & 4 recorders, even though it had to bend a little bit around the edge of the roof, would be very similar to the straight-line distance. Or, if we wanted to be a bit more precise for our particular scenario, we would add up these two distances:

1. Rifle muzzle to edge of roof
2. Edge of roof to the recorder’s position

Do you concur?

[kincses-zsolt]

Essentially you’re right.

What we can expect, two things. The sound can propagate behind the corner. And there might be bands of destructive interference. But it will not change much the time delay, just extends the “line of visibility” - or line of audibility.

[flamecensor]

@kincses-zsolt

an intuitive illusion that instantaneous frequencies do exist. But they do not.

Even if a frequency has a duration of time, I can hear the difference between 2 non-instantaneous tones, so I relistened to Chris Martenson insisting that the tones of 5 shots were of different (non-instantaneous) frequencies than the (non-instantaneous) tones of 18:11:33’s 3 shots, and I agree with Chris.
I didn’t know how Chris found that the middle shot of 5 didn’t ‘line up’. I never saved the AGR5 Westside video, but may try to find it, to see if its audio fails to ‘line up’ with a podium mic audio, for me.
And, a ‘boom calculation’ is new to me, if that calculation obviates the need for a supersonic ‘crack’ timing.
I wrote a ‘crack thump’ calculator into a webpage, July 2024.
My simple formula may work, but calculus may be needed to give a better value of (decreasing) bullet speed.
You can see I made a crude attempt to re-estimate an average speed with:

"
sd=ct*ct*v1;
v2=v1-sd;
av=((v2/1)+(v1/1))/2;
"

function C(tf){
v1=tf.vb.value;
ct=tf.ct.value;
vs=tf.vs.value;
vs=(vs<1090)?1125:vs;
ct=ct/100;
var d=ct/((1/vs)-(1/v1));
sd=ct*ct*v1;
v2=v1-sd;
av=((v2/1)+(v1/1))/2;
nt=d/av;
var d=nt/((1/vs)-(1/v1));
alert(d);
}

What calculus would be needed in my simple code to give a better average speed for a decelerating bullet?
Also, can the initial bullet speed ever be guaranteed to be perfectly known?
https://superfly.co.nz/ct.htm

[roger-knight]

@roger-knight

We can clearly see they pulled up the banner and tied a black string to cover it.

Nice catch.

My assumption that the string was added by police enforcement has been proven false by your picture. I therefore apologize and retract that assumption.

However, the main point of this discussion remains: the banner was pulled and the string likely tightened, because you can clearly see that the impact was covered up—in other words, literally a “cover-up.”

We also see a person taking a picture of that critical spot, which definitely suggests a “there there.” If by chance the person who took that picture is reading this post, we would be very interested in obtaining it.

Do you know the source of this video? I would really like to see it.

[phiphi-the-frenchie]

@roger-knight

My assumption that the string was added by police enforcement has been proven false by your picture. I therefore apologize and retract that assumption.

As usual, it was not presented as a hypothesis, but you stated it loud and clear as if it were a certainty!

@roger-knight

However, the main point of this discussion remains: the banner was pulled and the string likely tightened, because you can clearly see that the impact was covered up—in other words, literally a “cover-up.”

Stop your delusions, no one has tightened that tie or moved that banner, except maybe TOB who rubbed himself against it!

@roger-knight

We also see a person taking a picture of that critical spot, which definitely suggests a “there there.” If by chance the person who took that picture is reading this post, we would be very interested in obtaining it.

You’re completely obsessed; the guy isn’t photographing your hypothetical impact mark, but the scene unfolding a little further away!

@roger-knight

Do you know the source of this video? I would really like to see it.

Look at message 6469 in Part 1 of " So it’s back to First Principles "

[daniel59]

@greg_n

According to Grok, the muzzle blast and the expanding sound wave of the “frozen” Mach cone both reach the microphones of Source 3 and 4 simultaneously after 227 milliseconds, assuming Crooks fired. (I used the following coordinates for the rifle muzzle and microphone: (75/20/2) (0/0/5))

Could you reliably recognize a “crack” if it were not the sonic boom itself and it arrived at the same time as the muzzle blast?

@kincses-zsolt

@daniel59

The closer to the impact/bullet, the sharper the angle.

I think it is the opposite. Since the bullet loses energy along its path and decelerates.

You’re right in observing that the Mach cone’s angle can change over time as the bullet slows down.

I wasn’t referring to the angle of the cone, but rather the dispersion angle of the sound waves that continue to spread out after the Mach cone “freezes.”

I suspect that the dispersion angle near the tip of the Mach cone should be smaller than near the base of the Mach cone.

[daniel59]

@daniel59

Could it have been the echo of the sonic boom from a bullet fired from Building 9?

That’s not possible.

I wasn’t paying close enough attention and overlooked the fact that the 170ms echoes occurred with shots 4 to 8. That doesn’t fit with the assumption that only the first 3 bullets were fired from building 9.