Displacement amplitude of the faintest sound detectable by a human ear

[FranzDiCoccio]

Homework Statement

the pressure amplitude for the faintest detectable sound at 1kHz is 2.8E-5 Pa. FInd the displacement amplitude for such a sound of speed 343 m/s in air of density 1.21 kg/m^3

Relevant Equations

$\Delta p_m = (v \rho \omega) s_m$

The problem per se is pretty straightforward:
$s_m = \frac{\Delta p_m}{v \rho \omega} \approx 1.1\cdot 10^{-11} \, \textrm{m}$

I found an exercise similar to this in a translation of Cutnell and Johnson's "Physics" 9th edition. I could not find the problem in the original version of the book, but it seems to me that it draws inspiration from "Sample problem 17.01" "Physics 10th edition" by Resnick, Halliday et al.
The discussion there remarks that this displacement is one tenth of the radius of a typical atom, and that human ears are indeed sensitive sound detectors.

I am a bit puzzled by this. It seems a really small displacement.
In what way does this make sense?
Of course the relation we used has a range of validity. Is this result within that range?

 

[256bits]

A simple search, of one of your own phrases, you can easily do, gives research and insight.

Search "human ears are sensitive sound detectors"
http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/earsens.html
https://www.uclahealth.org/news/rel...ns-more-sensitive-than-those-of-other-mammals
https://news.mit.edu/2019/mechanism-explains-ear-sensitivity-0116

A single auditory neuron from humans showed an amazing ability to distinguish between very subtle frequency differences, down to a tenth of an octave. This, compared to a sensitivity of about one octave in the cat, about a third of an octave in rats and a half to a full octave in the macaque.

 

[FranzDiCoccio]

Hi,
thanks for your answer. I am not questioning the sensitivity of human ear, though.
We can hear intensities from 1e-12 to 10, which is 13 orders of magnitude. I am not referring to the ability of distinguish between frequencies. Here we have one frequency.

The point is that the we have an estimate for the displacement of a gas particle, and that displacement is less than an atomic radius.
The comment about human hearing capability makes me think that the human ear can detect a displacement of 0.1 angstrom. I believe this is kind of misleading.

By the way, particles in a gas at room temperature travel at speeds of the order of 100 m/s.

Do you see what I mean?

 

[haruspex]

Bear in mind that the wave is acting over the area of the eardrum, more or less synchronously. What displacement might it produce in the surface layer? What relative displacement between the surface and subsurface might be detectable electrically?

 

[FranzDiCoccio]

Hi,
I am not sure I follow.
Are you suggesting that $s_m$ is very small, but after all not so small if compared to the distance between two layers in the atomic structure of the eardrum? Like 0.1 angstrom vs more or less 1.5 angstrom?

 

[haruspex]

Hi,
I am not sure I follow.
Are you suggesting that $s_m$ is very small, but after all not so small if compared to the distance between two layers in the atomic structure of the eardrum? Like 0.1 angstrom vs more or less 1.5 angstrom?

That sort of thing, yes. Not saying it strikes me as likely, just not impossible.

 

[DaveC426913]

Hi,
I am not sure I follow.
Are you suggesting that $s_m$ is very small, but after all not so small if compared to the distance between two layers in the atomic structure of the eardrum? Like 0.1 angstrom vs more or less 1.5 angstrom?

Not distance: area.

A displacement of .1 angstrom might be small but there's 7 trillion* atoms getting displaced.

*Number of atoms on the face of a 50mm^2 eardrum.

 

[haruspex]

… and the ossicles amplify the displacement.

 

[FranzDiCoccio]

Not distance: area.

A displacement of .1 angstrom might be small but there's 7 trillion* atoms getting displaced.

*Number of atoms on the face of a 50mm^2 eardrum.

Hmm no wait.
I might be mistaken, but the way I figure it is that a wavefront is a plane the same size as the eardrum. As the wave hits the surface of the eardrum each of the atoms in the first layer might very well be displaced by $s_m$, or even $2 s_m$ considering a complete oscillation. So, as you say, trillions of atoms are displaced by that amount in the first atomic layer of the eardrum.

This still seems definitely too small for the hearing mechanism to pick that up. I mean, we are talking of definitely macroscopic structures, some of which have sizes of the order of $10^{-2} - 10^{-3}\,\textrm{m}$.

I am no expert but, from what I remember, hearing has an important mechanical component.
I read that vibrations hit the eardrum, the ossicles amplify them and transmit them into the cochlea, where they propagate inside a fluid and finally reach the stereocilia. These are the structures converting vibration into an electrical signal, and have sizes of the order of 10 micrometers. This is only one order of magnitude less than the smallest feature an human eye can see... Still kind of macroscopic.

The fact that all this is triggered by a 0.1 angstrom displacement does not sound right to me (no pun intended )

Are you suggesting that there is another mechanism registering the displacement of one atomic layer wrt the next, and converting that into an electrical signal for the brain?

Anyway, I have the feeling that there is not even a need to consider the details of the hearing mechanism.
It seems to me that the model proposed in "Physics 10 ed" is very simple. Air is treated like a continuous medium. Does this model really hold all the way down to the subatomic scale?

I am not saying that human do not hear sounds with an intensity of $10^{-12} \mathrm{W/m^2}$ (not this particular human, though).

It seems to me that the calculations in this example are "stretching" the model a bit too far. Assuming I'm right, the book might say that the the model they are using holds only down to a certain scale, below which the explanation of the phenomenon is too complex to be discussed there. Still, the fact that humans hear sounds so faint is "as if" their ear could register a displacement of 0.1 angstrom.

 

[Tom.G]

Movement information is difficult to find! This finally popped up 45 minutes into the search.

from: https://pmc.ncbi.nlm.nih.gov/articles/PMC3219815/#F1

"The stimulus sound pressures varied between 80 and 110 dB SPL. The largest displacements are on the order of 500 nm, the smallest measurable displacement is about 20 nm."

Note the very high sound level used - not a problem for the subjects though, they were cadavers (corpses).

(above found with:
http://www.google.com/search?hl=en&q=Tympanic+membrane+movement&WIw86pc)

Cheers,
Tom

 

[Steve4Physics]

For comparison purposes (and hopefully of interest) LIGO (gravitational wave observatory) will measure length-changes of “1/10,000th the width of a proton” (https://www.ligo.caltech.edu/page/facts).

That’s length-changes of $10^{-19}$m ($10^{-9}$ angstrom)!

 

[Tom.G]

I think we have a bit of evolving to do before we can match LIGO.
It will be a while before we can get a 1200 kilometer pathway between our ears.

https://www.ligo.caltech.edu/page/ligos-ifo

Cheers,
Tom

 

[FranzDiCoccio]

For comparison purposes (and hopefully of interest) LIGO (gravitational wave observatory) will measure length-changes of “1/10,000th the width of a proton” (https://www.ligo.caltech.edu/page/facts).

That’s length-changes of $10^{-19}$m ($10^{-9}$ angstrom)!

I think we have a bit of evolving to do before we can match LIGO.
It will be a while before we can get a 1200 kilometer pathway between our ears.

https://www.ligo.caltech.edu/page/ligos-ifo

Cheers,
Tom

I also thought of that. But in that case we are trying to measure a distortion in spacetime, and using laser interferometry.
I'd say that the underlying physical model (General Relativity) treats spacetime as a continuum, which makes sense since everything else is "inside" spacetime.

I guess that I'm sort of confused about the mathematical model resulting in the equations adopted for deriving the relation usend in the exercise: $\Delta p_m = (v \rho \omega) s_m$.

It seems to me that air (that is a large number of individual particles moving at relatively high speeds) is treated like a continuous elastic medium. I expect that model to work well within a certain range of parameters, and to break down outside such range.

For instance, if my calculations are right, for $p \approx 10^5\, \mathrm{Pa}$ and. $T=293\, \mathrm{K} = 20 ^\circ \,\mathrm{C}$ we have $N \approx 2.5\times 10^7$ molecules in a cube of side $1 \mu \mathrm{m}$, which is still a lot.
The smallest cubic cell where I'm sure to find at least one particle on average is about $3.41\, \mathrm{nm}$ wide, more than 2 orders of magnitude wider than the displacement in the elastic medium.

Wouldn't the model need to have cells smaller than $10^{-11} \mathrm{m}$ for a displacement of that order of magnitude to make sense?

It's just I'm trying to figure out what happens, and I picture mostly void at that length scale, with the occasional very fast particle zooming by.

I'm not arguing that the ear cannot detect pressure variations of the order of $2.8\cdot 10^{-5} \,\mathrm{Pa}$. I can picture this in terms of the variation of the rate of particles knocking on the eardrum.
It's the displacement by $10^{-11} \, \mathrm{m}$ that feels somehow "off".

 

[haruspex]

.
It's the displacement by $10^{-11} \, \mathrm{m}$ that feels somehow "off".

Have you calculated the amplitude magnification produced by the ossicles?

 

[Steve4Physics]

I also thought of that. But in that case we are trying to measure a distortion in spacetime, and using laser interferometry.
I'd say that the underlying physical model (General Relativity) treats spacetime as a continuum, which makes sense since everything else is "inside" spacetime.

Yes. I noted the LIGO sensitivity (~$10^{-19}~m$) because I thought it might be the smallness of the displacement which bothered you. I find it remarkable that it is possible to make a device (from atoms!) capable of this sensitivity.

I'm not arguing that the ear cannot detect pressure variations of the order of $2.8\cdot 10^{-5} \,\mathrm{Pa}$. I can picture this in terms of the variation of the rate of particles knocking on the eardrum.
It's the displacement by $10^{-11} \, \mathrm{m}$ that feels somehow "off".

Maybe the process is assisisted by resonance (of some cochlear hair cells) effectively 'integrating' the signal over a number of cycles.

 

[hutchphd]

Maybe the process is assisisted by resonance (of some cochlear hair cells) effectively 'integrating' the signal over a number of cycles.

My understanding is that the ossicles are impedance matching (tympanic membrane in air to cochlear fluid) and that the cochlea provides resonances whose peak displacements are distributed along its spiral according to frequency. I do not know if this has been effectively modelled....but would guess yes.
/

 

[Tom.G]

I discovered that ... It's complicated!

https://pmc.ncbi.nlm.nih.gov/articles/PMC5792030/
TL;DR

(above found with:
http://www.google.com/search?hl=en&q=length+of+cochlear+hair+cells)

Cheers,
Tom

 

[hutchphd]

guinea-pig cochlea ! Amazing stuff.

 

[FranzDiCoccio]

All this is very interesting, but I still have some doubts about the result of the original problem. As I say, my doubts are not about the ability of human ear of hearing very faint sounds. I am wondering about the validity of the model at such a small scale.

I found that the typical displacement of atoms due to thermal lattice vibrations in a solid at room temperature is of the order of $0.01 - 0.1 \,\mathrm{nm}$. That is the same order of magnitude as the displacement resulting from the problem.
True, these oscillations are not likely to occur "coherently", they are not "wave-like". But it seems to me that they could at least provide a lot of "noise" over the displacement associated to the "wave" carrying the very faint sound.

Maybe the process is assisisted by resonance (of some cochlear hair cells) effectively 'integrating' the signal over a number of cycles.

The idea of effective integration assisted by resonance is interesting.
Would this mean that someone actually able to hear such a faint pure tone could also be able to recognize its pitch?

 

[hutchphd]

Not necessarilly. The evolution of hearing was not usually driven by abilities relating to pure tones. So that would be an assumption (without real basis) on your part. Surely our notion of Fourier synthesis and analysis is relevant but not dispositive.

 

[Steve4Physics]

All this is very interesting, but I still have some doubts about the result of the original problem. As I say, my doubts are not about the ability of human ear of hearing very faint sounds. I am wondering about the validity of the model at such a small scale.

I found that the typical displacement of atoms due to thermal lattice vibrations in a solid at room temperature is of the order of $0.01 - 0.1 \,\mathrm{nm}$. That is the same order of magnitude as the displacement resulting from the problem.
True, these oscillations are not likely to occur "coherently", they are not "wave-like". But it seems to me that they could at least provide a lot of "noise" over the displacement associated to the "wave" carrying the very faint sound.

I remember my school physics teacher telling us that (good) hearing allows us to detect sounds just above the background noise (from the random motion of air molecules hitting the ear drum).

Also, it is possible to recover a repeating signal similar to, or smaller than, background noise.

I'd guess something like that is happening - nature is very good at finding near-optimal solutions.

The idea of effective integration assisted by resonance is interesting.
Would this mean that someone actually able to hear such a faint pure tone could also be able to recognize its pitch?

Probably not - I'd guess that pitch-recognition requires a stronger signal. But, for survival purposes, simple sound-detection (without frequency recognition) might increase your chances of survival, e.g. as a prehistoric tiger sneaks-up on you during the night!

 

[FranzDiCoccio]

Yes, I can see how pitch would be immaterial for primordial hearing. On the other hand many species signal via specific pitches. It could be useful to be able to hear the call of a potential mate from a very large distance.
Anyway my question was prompted by Steve4Physics' comment about resonance.
By the way, it appears that the frequency of $1\mathrm{kHz}$ is in the range where human hearing is most sensitive. Other frequencies should have larger intensities to be detected.

 

[Herman Trivilino]

As I say, my doubts are not about the ability of human ear of hearing very faint sounds. I am wondering about the validity of the model at such a small scale.

The model is well-tested. Keep in mind that the size of an atom being on the order of 0.1 nm is not the size of a solid object, it's the size of the electron cloud. More precisely, the atom doesn't have a size, it's just that when we attempt to measure the size this the result we get ala Heisenberg's Uncertainty Principle.

 

[FranzDiCoccio]

The model is well-tested. Keep in mind that the size of an atom being on the order of 0.1 nm is not the size of a solid object, it's the size of the electron cloud. More precisely, the atom doesn't have a size, it's just that when we attempt to measure the size this the result we get ala Heisenberg's Uncertainty Principle.

In what respect (and range) is the model well tested?
Again, I'm not saying that the physical phenomenon does not take place. I do believe that humans are able to hear those faint noises.

Assuming that any model of a physical phenomenon has a range of validity, I'm wondering whether this specific example really falls within the range of validity of the model we are adopting to describe it.
Even if it does, is there a "sound" so faint that we cannot describe it using this model?
I mean, irrespective of the human ear being able to hear it...

 

[Herman Trivilino]

Thermal oscillations are random, rather than organized, motions of the air molecules and will not cause the ear drum to fluctuate in a way that is perceived as sound. Fluctuations can be large enough to rupture the ear drum. These are outside the limits of validity.

 

[Steve4Physics]

Assuming that any model of a physical phenomenon has a range of validity, I'm wondering whether this specific example really falls within the range of validity of the model we are adopting to describe it.

One thing we can do is see if the result is consistent with a different model. At the risk of being hand-wavy, some insight can be gleaned by considering the ear-drum as a single object of mass $m$ subject to Brownian motion from randomly impacting air molecules.

Applying the equipartition theorem, the average (rms) speed, $v$, of the ear-drum’s random motion is given by $mv^2 \sim k_BT$ (ignoring factors of order 1).

Once $v$ is found, we can consider the distance moved in 1 ms (the period of 1 kHz sound). If this distance is $\sim 10^{-11}$ m or less, this supports the idea that displacements of order $10^{-11}$ m due to a 1 kHz signal can be detected (i.e. will not be lost in thermal noise).

For a human ear-drum:
area $\sim 10^{-4}~m^2$ (various sources give it as $1~ cm^2$).
thickness $\sim 100~\mu m$ (e.g. see here)
density $\sim 1000~kg m^{-3}$ (typical tissue density)

This gives $m \sim 10^{-5}$ kg

$v \sim \sqrt {\frac {kT} m} \sim 10^{-8}$ m/s

Distance covered in 1 ms $\sim 10^{-11}$ m (surprise!)

 

[hutchphd]

Just to cloud the picture more, the rms molecular separation in air is ~1nm. I dasn't attempt any handwavery......at my age it is likely to cause soft-tissue damage.

/

 

[Steve4Physics]

Just to cloud the picture more, the rms molecular separation in air is ~1nm. I dasn't attempt any handwavery......at my age it is likely to cause soft-tissue damage.

FWIW I suspect that coherent displacement of the ear-drum can be much less than the average separation of air molecules because of statistical averaging.

And please take care of your soft-tissue.

 

[hutchphd]

Then don't tempt me to start gesticulating! Sensory organs are all fascinating and I am easily tempted.