SSS.2 - Methodology Introduction

Before there were silencers, there were guns. And guns were loud.

Let’s take a look at two regions of the unsuppressed firearm loudness spectrum. These two regions are not necessarily the true extreme ends of the spectrum, but practical regions of which many people familiar with firearms are aware: subsonic rimfire and supersonic centerfire, both fired out of bolt action rifles.

Below are two measured free-field pressure waveforms; both measured on the same day, at the same location (1.0 m left of the weapon muzzle, 1.6 m above ground level, over 10 m away from any reflecting surfaces). Two cases are presented; subsonic .22 Long Rifle (.22LR) rimfire from a 16 inch barrel (Fig 1) and supersonic .308 Winchester (.308WIN) centerfire from a 20 inch barrel (Fig 2). The horizontal axis in each figure displays units of time in milliseconds [ms], while the vertical axis displays units of pressure in pascals [Pa]. Both scales are linear. The time scale is kept constant between the figures, at a total of 25 ms. For reference, it takes a housefly approximately 3 ms to flap its wings, once.

If you click (or tap) on each figure, a full-resolution version will be displayed.

Fig 1. subsonic rimfire - CCI Standard Velocity out of a 16” barrel

Fig 2. supersonic centerfire - Federal XM80C out of a 20” Barrel

PEW-SOFT records data at a rate of 1,000,000 samples per second (1 MHz). The data is not filtered or averaged. Each of the waveforms shown in the figures is composed of 25,000 discrete points. Viewing the gross waveforms, you may notice some immediate differences between the subsonic .22LR and the supersonic .308WIN shots:

The subsonic .22LR waveform contains three major peaks. The first peak in Fig 1 is the pressure wave resulting from the compression of air as the subsonic bullet leaves the barrel; the bullet is traveling at close to the speed of sound in air, but not quite at the speed of sound in air. Because the distance between the air molecules is small enough such that the molecules can reach each other (the air can compress) faster than the bullet can push them out of the way, the molecules “pile up” in front of the bullet.
After the bullet exits the barrel, combustion products are released, forming the second peak in Fig 1. This second peak is called the “muzzle blast.” The third peak is the ground reflection of the muzzle blast, arriving to the microphone late in time after traveling to the ground and back.
The supersonic .308WIN waveform in Fig 2 contains two major peaks. Unlike the subsonic .22LR case, the .308WIN round travels out of the barrel so quickly that the air in front of the bullet is not “aware” of the bullet’s arrival; that is, the speed of sound in air is not fast enough to accelerate the air molecules to the speed at which they would be able to close the distance to their nearest neighbors. Therefore, the first peak is the muzzle blast.
The second peak in Fig 2 is the ground reflection of the muzzle blast, again arriving to the microphone late in time after traversing its path.

The shapes of the waveforms described above are somewhat difficult to discern at full timescale, despite that scale being only 25 ms. To better view the data integrity, we can take a closer look.

Let’s Take a Closer Look

The same waveforms previously presented in Fig 1 and Fig 2 are presented again, in Fig 3 and Fig 4. This time, only one millisecond of each waveform is shown (4.9 ms to 5.9 ms), omitting the late-time reflection peak(s).

Fig 3. Subsonic Rimfire - millisecond time window

Fig 4. Supersonic Centerfire - millisecond time window

You may recognize the muzzle blast portions of both waveforms to be shocks. An explanation of this phenomena is beyond the scope of this article, but I urge you to satisfy your curiosity, at your leisure, by starting here. Save that one for a rainy day.

The rise time (time from ambient pressure to peak pressure) of the shocks in Fig 3 and Fig 4 are different, as expected. However, both waveforms rise to their peaks quickly, and capturing that rise correctly, each time it is measured, is only possible by gathering data fast enough so you don’t miss the exact moment in time where the peak occurs. The raw data is plotted in Fig 5 and Fig 6. Again, note that you can click (or tap) on the figures to obtain full resolution images.

Fig 5. subsonic rimfire - 1,000 discrete points in one millisecond

Fig 6. supersonic centerfire - 1,000 discrete points in one millisecond

Can you spot the highest peaks? Want an even closer look? Let’s look at a 0.1 millisecond time window (100 microseconds):

Enhance!

Note that the peaks in Fig 7 and Fig 8 are plainly visible, and that they are labeled with their corresponding peak pressure magnitudes in pascals [Pa] and equivalent decibels [dB].

Fig 7. Subsonic Rimfire - 100 Discrete Points in 0.1 milliseconds (100 Microseconds)

Fig 8. Supersonic centerfire - 100 Discrete Points in 0.1 milliseconds (100 Microseconds)

By this time, you have probably figured out at least one reason why firearm sound testing data is often irregular and inconsistent. Almost all publicly available sound testing data for impulsive gunshot noise, unsuppressed and suppressed, has been obtained using slower sample rates. At lower sample rates, gunshot waveform peaks are missed. In those cases, reported “peak” values may actually be values occurring elsewhere in the waveform, away from the real peak altogether.

Pascals? Oh, I’m sorry, I thought this was America!

I would have used pounds per square inch (psi), but we were going to end up with pascals eventually, anyway. A long time ago, we established that 20 micropascals of sound pressure represents approximately the quietest sound a young human with undamaged hearing can detect, at a frequency of 1,000 Hz. We call this the threshold of human hearing.

You can imagine how difficult it would be to express sound pressure in terms of Pa. The above .22LR waveform has a peak of 446 Pa, whereas the .308WIN waveform peaks at 6276 Pa. The difference between those two numbers is greater than an order of magnitude. Humans tend to be bad at estimating and interpreting data on large scales, but you’re in luck. We have a tool to help us with that: the logarithm.

In this case, our reference sound pressure is 20 micropascals (0.00002 Pa). A decibel is nothing more than a way of scaling exponential data by moving from a linear scale to a base-10 logarithmic scale.

We call 0.00002 Pa “0 dB.” This is the quietest perceptible sound, as described above.
We call 101,325 Pa = 14.7 psi = 1 ATM = 1 atmosphere of pressure “194 dB.” This is the loudest “sound” possible, when in Earth’s atmosphere at standard conditions. However, not all pressure waves are “sound waves,” and a pressure wave or shock wave can be much more intense than 194 dB.

A comprehensive discussion of the decibel is outside the scope of this Standard. If you want to convert any of the points on the above linear sound pressure plots to the logarithmic dB scale, use the equation in Fig 9 with the aforementioned reference sound pressure value.

Pressure? What about Duration?

In the main introduction to the Standard on this website, I mentioned that sound pressure duration also influences loudness. Over the years, different methods for computing transient sound pressure waveform durations have been used. In an effort to simplify the way the silencer community presents and consumes data (and to investigate the influence and correlation of the addition of a simplified metric to our end goal of “loudness”), PEW Science computed the following:

The application of a suddenly applied overpressure waveform, at the point at which it is measured, occurs over a finite period of time. By computing the area under the pressure-time curve in a certain time window, one can ascertain the magnitude of momentum transfer potential possessed by the overpressure wave to an object (your inner ear, for example). This change in momentum is called impulse. The initial peak positive-phase impulse contained within an overpressure waveform is the impulse per unit of projected area.

The peak pressure and impulse for our .22LR subsonic rimfire and .308WIN supersonic centerfire waveforms are shown in Fig 10 and Fig 11, respectively. Note that the units of impulse are [Pressure x Time]. In our case, the units are [Pa-ms], or on the decibel pascal sound pressure reference scale, [dB-ms]. You can read the blue impulse waveforms using the scale on the secondary (right) vertical axis.

Fig 10. Unsuppressed Subsonic .22 Rimfire Pressure and Impulse

Fig 11. Unsuppressed Centerfire .308 Pressure and Impulse

We now have two characterization metrics:

Peak sound pressure [dB]
Peak positive phase sound impulse [dB-ms]

The peak positive phase impulse of a pressure waveform isn’t necessarily the sole quantity to concern oneself with, regarding loudness. For that matter, neither is the peak pressure... necessarily. However, we know that both influence loudness. Damage estimates to blast overpressure almost always depend on both pressure and impulse, and damage to your ear is no different.

Hold that thought. We’re not finished yet, but let’s first go ahead and see what we have so far:

Putting this into Perspective

The two unsuppressed gunshot waveforms presented in the above figures were shown to give you some confidence and understanding of what sound pressure waves look like, analytically, when measured at an adequate sample rate. We now have two key quantities we can report for each gunshot: peak sound pressure and peak sound impulse.

Now, let’s plot our data!

Fig 12 shows unsuppressed subsonic .22LR and supersonic .308WIN on a logarithmic scale, using linear units [Pa] and [Pa-ms].
Fig 13 shows unsuppressed subsonic .22LR and supersonic .308WIN on a linear scale, using logarithmic units [dB] and [dB-ms].
Both figures show the same data. This is important for you to understand, because the difference in sound pressure, for example, between 140 dB and 170 dB is 30 dB, which is equal to 6125 Pa. However, while the difference between 110 dB and 140 dB is also 30 dB, that difference is only 194 Pa. The 30 dB difference, in each case, is on a different region of the dB scale.
Remember: Pa is a linear unit that we can choose to view on a linear scale (the waveform plots, above) or a logarithmic scale (Fig 12), while dB is a logarithmic unit that we read on a linear scale (Fig 13).