# Finger snapping: Maximum fingertip speed and generated pressure change

There are three components to the snapping finger sound: (1) The "friction" or "sliding" sound between the second (middle) finger and the thumb (2) The "impact" sound from the second finger colliding with a groove created by contacting the third (ring) finger with the palm and (3) The "pop" sound from the rapid compression and subsequent decompression of air. The third "pop" sound is the most audible of the three components and because it is caused by a compression of air between the fast moving second finger and the groove between the palm and third finger, the second finger must hit both the palm and a small portion of the top of the third finger in order to get the full "snap" sound.

Is it possible calculate approximations for:

• The maximum speed with which the the tip of the finger moves?

• The maximum pressure (change relative to standard atmospheric pressure) generated as the 2nd finger impacts the groove created by the third finger against palm?

# Background and Definitions

It turns out there is a lot more to this than I had originally anticipated. I originally thought I need only look up an effective spring constant for human tendons and then calculate potential energy, convert that into kinetic, then walla, things would follow.

There is a very informative review/discussion by Lanir  that goes into the gory details of tendon deformation mechanics. Although tendons obey Hooke's law for the most part, they are considered hyperelastic. Rather than delve into the pile of derivations etc. that I ran through this afternoon after diving down the rabbit hole, I will try to summarize things in an overly simplified way.

Let the total force magnitude applied to a tendon be defined as: $$F = \langle \kappa \rangle \ D \tag{0}$$ where $$\langle \kappa \rangle$$ is an average effective spring constant (effective because it has units of force, not force per unit length) and $$D$$ is the total strain (unitless). The $$\langle \kappa \rangle$$ term is composed of a terms due to collagen fibers and elastin fibers that make up mammalian tendons (didn't look into whether non-mammalian species have different proteins in their tendons so I'm just being careful specifying mammalian here). The $$D$$ term is also composed of contributions from the strains on the collagen and elastin fibers but also geometrical effects. However, under most circumstances, the following is satisfied $$D$$ ~ 1-2%.

Since the problem seems to be concerned with the extreme limits, in the high strain limit one finds that $$\langle \kappa \rangle$$ ~ $$\kappa_{c} + \kappa_{e}$$, where $$\kappa_{c}$$($$\kappa_{e}$$) is the effective spring constant of the collagen(elastin) fibers. In humans, the typical ratio of these two satisfies $$\kappa_{e} / \kappa_{c} \ll 1$$.

Finally, we know that the total elastic energy in such a system can be expressed as: $$U_{o} \sim \frac{ 1 }{ 2 } l_{o} \ \kappa_{c} \ D^{2} \tag{1}$$ where $$l_{o}$$ is the initial (or equilibrium) displacement of the elastic material and $$\kappa_{c}$$ enters here instead of $$\langle \kappa \rangle$$ because it's the constant associated with Young's modulus. It doesn't really matter since the typical values for these constants are $$\kappa_{c}$$ ~ $$3 \times 10^{5}$$ N and $$\kappa_{e}$$ ~ $$1 \times 10^{2}$$ N [Lanir, 1978]1.

The maximum speed with which the the tip of the finger moves?

Let's take things to 11 here and assume $$D$$ of ~ 4%, $$\langle \kappa \rangle$$ ~ $$\kappa_{c}$$ ~ $$3 \times 10^{5}$$ N, and $$l_{o}$$ ~ 1-2 cm2. This would correspond to a total stored energy of ~0.0012--0.0060 J. The typical human finger is on the order of ~100 g in mass, so if we just directly convert this potential energy into kinetic energy and solve for speed, we get ~3.5--4.9 m/s or ~350--490 cm/s. Given that the fastest punch is upwards of ~20 m/s, these numbers seem okay.

A quick check of GoogleTM seems to suggest a typical finger snap comes in around ~20 mph or ~11 m/s, a little over twice my estimated speeds so either my $$l_{o}$$ or $$D$$ were too low.

The maximum pressure (change relative to standard atmospheric pressure) generated as the 2nd finger impacts the groove created by the third finger against palm?

Suppose we could travel back in time and measure Bruce Lee's finger snap speed. I imagine it would be faster than the average person, so let's assume his finger reaches ~30 mph or ~13 m/s. The corresponding dynamic pressure of the incident air just ahead of the finger would be ~108 Pa. Technically, the speed of the air escaping transverse to the motion of the finger just before impact would be significantly faster than this3. Suppose this reaches ~100 m/s, then the corresponding dynamic pressure would be ~6000 Pa.

Note that the sound pressure levels for these two dynamic pressures, assuming they are short duration pulses, would be on the order of ~130 dB for the ~108 Pa and ~170 dB for the 6000 Pa (at zero distance from source). People snapping their fingers obviously are not rupturing anyone's ear drums or causing severe pain, so the pressures are not directly converting to a sound (i.e., it's mostly just a flow, not an over pressure or pressure pulse).

# Footnotes

1. Note that Lanir  did not actually state the units here, so I am assuming newtons since a $$D$$ of ~1-2% would corresponding to forces of ~3000-6000 N or ~675-1350 lbs. Given that the current world records for deadlift is roughly 1100 lbs, these numbers seem reasonable.
2. It's not clear what the correct value should be here, but Lanir  uses the symbol $$\lambda$$ and gives an example value of 200 nm in the intro under a context of limitations of measurements. Typically the equilibrium displacement and stretch are normalized into the strain term and the displacement of the tendon from equilibrium does not seem to be well defined (at least from my cursory search). However, here I assumed this would average out to the typical length of a 1-2 cm for the tendon length and I am being conservative here.
3. If we assume continuity and that the finger-palm system is like a constant velocity cylinder colliding (i.e., $$A_{in} v_{z} \sim A_{out} v_{r}$$), with finger radius ~ 1 cm and altitude from palm where output starts to matter of ~600 $$\mu$$m, then the radial outflow speed can exceed 100 m/s for an incident finger speed of ~13 m/s. The outflow speed is inversely proportional to the altitude at which air flow radially starts to dominate over vertical. If this altitude drops to even ~100 $$\mu$$m, the radial outflow speed can reach over 650 m/s (i.e., supersonic). There are several assumptions here that are not tremendously valid so it's not surprising that numbers seem unrealistically large.

# References

• Y. Lanir "Structure-strength relations in mammalian tendon," Biophys. J. 24(2), pp. 541-554, doi:10.1016/S0006-3495(78)85400-9, 1978.
• I hardly remember posting this question - Thanks for digging this up and providing such detailed answer! – Digital Trauma Jan 6 at 17:19