# B Audible sound range for beats

#### FranzDiCoccio

Hi all,
I recently came across an exercise observing that human cannot hear ultrasounds, but could hear the beat between two different ultrasounds, provided that its frequency falls in the audible range between 20 Hz and 20kHz. The problem gives one frequency and asks for the range of the other where audible beats are produced.

I have no problems with the solution, but I have some doubts about the physics (or the physiology) involved.

The point is that the beat is a periodic modulation of a sound, and I realized that there is no need for a beat to have a frequency larger than 20 Hz. I can easily produce and hear 1Hz or 0.5 Hz beats.

So my question is: what about ultrasound beats?
What if the frequency difference of the ultrasound is around 1Hz? Do I hear their beat?
I'm inclined to say no, because it would be a modulation of the volume of something I am not hearing to begin with.
I wonder what I'd hear or experience in that case.

Also, would I really hear a sound if the beat between the ultrasounds is in the audible window?

Thanks a lot for any input

#### scottdave

Homework Helper
Gold Member
What you are hearing when it is the 2 audible frequencies beating at 1 Hz is a sound getting louder then softer. I am not certain, but I think the ear would act as a low pass filter, because it cannot react fast enough for the ultrasonic frequencies. But I believe the pressure is changing at the eardrum at the lower frequency, so it may be heard. I'd like to find out more, though.

#### sophiecentaur

Gold Member
The point is that the beat is a periodic modulation of a sound, and I realized that there is no need for a beat to have a frequency larger than 20 Hz. I can easily produce and hear 1Hz or 0.5 Hz beats.
Are you actually producing audible beats with two ultrasound sources? I'm not quite clear, from what you have written.
You can get 'mixing' of two frequencies when there is a non linearity and the mixing products can be many different combinations of nf1+mf2, (m and n can each have multiple positive and negative values) depending on the law of the non linearity. For this mixing to work and for you to hear the product(s) you need a high level of sound and for the product(s) to be within the audible range.
Otoh, I think the 'beat' between two audible tones is merely your ear hearing a variation in resultant level with time. I don't think this needs to involve a non linearity - but human hearing is very complex and the beat produced in this way may well be a psychological artefact (i.e. too hard for me to explain).

#### scottdave

Homework Helper
Gold Member
It will be interesting to try. I don't know if it is "high tech" enough, but a couple of phones with the Dog Whistle app could be a decent test to see if audible beats can be heard. I know as people get older, the higher frequencies go. According to my dog whistle app, I can't hear anything above 11000 Hz, yet my daughter can hear up to around 16000. It may be that my phone isn't capable of producing above 16000.

#### Nik_2213

There's 'beats', and there's 'wow' & 'flutter', the bane of weary tape players and vinyl decks...

FWIW, couple of years ago, I mentioned to 'Customer Service' desk at my usual supermarket that there might be a fault developing on the BIG illuminated sign on their foyer, as it had begun whistling shrilly...
They fetched the manager, who explained it was an experimental 'Teen Scarer' system.
Happens I was easily thrice the age of their usual loiterers and, as I said, most of them were probably hearing-impaired from raves & discos...

#### FranzDiCoccio

Are you actually producing audible beats with two ultrasound sources? I'm not quite clear, from what you have written.
Hi,
no, I'm not doing anything experimentally.
As I mention, this suggestion comes from a textbook exercise. The algebra for solving it is not hard, but the situation got me curious.
You can get 'mixing' of two frequencies when there is a non linearity and the mixing products can be many different combinations of nf1+mf2, (m and n can each have multiple positive and negative values) depending on the law of the non linearity. For this mixing to work and for you to hear the product(s) you need a high level of sound and for the product(s) to be within the audible range.
Otoh, I think the 'beat' between two audible tones is merely your ear hearing a variation in resultant level with time. I don't think this needs to involve a non linearity - but human hearing is very complex and the beat produced in this way may well be a psychological artefact (i.e. too hard for me to explain).
So what you're saying is that the proposed situation is much more complex than envisaged by the exercise, due e.g to nonlinear effects and the subtleties of human hearing.

#### FranzDiCoccio

Are you actually producing audible beats with two ultrasound sources? I'm not quite clear, from what you have written.
Hi,
no, I'm not doing anything experimentally. The suggestion came from a textbook exercise. The solution is not hard, but the situation got me curious.

You can get 'mixing' of two frequencies when there is a non linearity and the mixing products can be many different combinations of nf1+mf2, (m and n can each have multiple positive and negative values) depending on the law of the non linearity. For this mixing to work and for you to hear the product(s) you need a high level of sound and for the product(s) to be within the audible range.
Otoh, I think the 'beat' between two audible tones is merely your ear hearing a variation in resultant level with time. I don't think this needs to involve a non linearity - but human hearing is very complex and the beat produced in this way may well be a psychological artefact (i.e. too hard for me to explain).
So what you're saying is that reality is much more complex than the situation envisaged by the exercise, due e.g. to nonlinear effects and the subtleties of human hearing.

#### FranzDiCoccio

It will be interesting to try. I don't know if it is "high tech" enough, but a couple of phones with the Dog Whistle app could be a decent test to see if audible beats can be heard. I know as people get older, the higher frequencies go. According to my dog whistle app, I can't hear anything above 11000 Hz, yet my daughter can hear up to around 16000. It may be that my phone isn't capable of producing above 16000.
This is interesting. Beats with audible sound can be produced easily using two phones.
I think that the nontrivial thing is a fine control on the ultrasound frequency. Is the phone really able to make two different sounds at eg 50000 and 50030 Hz?
Then again, probably there would be some difference, so maybe a beat is produced but with an unknown frequency.

I'll look into that.

#### sophiecentaur

Gold Member
The algebra for solving it is not hard,
Oh yes - definitely the easiest bit.
I think you would need two proper ultrasound sources with fair output levels. The intermodulation between the two tones will need a non-linearity somewhere in the ear that the tones can actually reach and the ear canal may well not conduct too well at that frequency. I tried a half hearted version of the experiment with two 'school' sources but they were low power and I couldn't measure the frequencies. Unfortunately, I think you would need to spend money which you could save if you do a really good Google search. People are bound to have tried this already.

#### sophiecentaur

Gold Member
I found this link, which claims to be selling a product similar to what you are suggesting.

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#### FranzDiCoccio

Yes @sophiecentaur is right -do some extra Googling. I came across this on Britannica without much effort - https://www.britannica.com/science/beat-waves
Ok... right. But I think my question was more specific than that.
Maybe I did not explain myself clearly (sorry, I'm not a native English speaker).
I had no big problem with the fact that two ultrasounds can produce an audible beat.
Then I recalled that the beats I usually hear (e.g. when tuning a guitar) have a frequency that is definitely lower than the standard lower bound for audible sound (20Hz).
Of course that can be written off as a modulation in volume of a sound that a human can hear anyway.

I was simply wondering: what if the two ultrasound frequencies differ by, say, 2Hz? What would I hear, if anything?

I suspected that this could be a matter of physiology, rather than physics per se. And I guessed that I would not hear anything.
Probably I'm not the first asking this question, but honestly I'm not sure how to google it...
So I asked in PF anyway, just in case an expert with a specific answer for a specific question happened to browse the forum.
(BTW I just noticed that you were the first to reply. I somehow missed your first reply, and read only your second. Apologies about that).

#### tech99

Gold Member
I am wondering if the limitation in hearing is only due to the nerve hairs, and the ear drum and outer ear might respond mechanically to ultrasound. In this case the mixing could occur within the cochlea, say, so the nerves will pick up the beat frequency. In this way, is possible that the nerve hairs might respond below 30Hz, the traditional lower limit, as there is no lower limit imposed by the base resonance of the drum and ossicles.
Obviously it is not possible to hear a beat by the modulation of the ultrasound because that is inaudible anyway.

#### scottdave

Homework Helper
Gold Member
In my initial test with just a couple of laptops, I was able to confirm the computer was generating a tone of the desired frequency (within less than 1%). I used an app called Spectroid on my Android phone. It appears that my laptop can generate tones between 200 Hz and 21000 Hz.

I couldn't hear above about 11,000 Hz, but as I said, the phone app detected the sounds. I did not hear any beats when the "main sounds" were above my hearing range. I tried a few different frequencies. This is not enough to be conclusive.

It could be that the beat frequency is at a much lower volume to detect.
The wavelength of sound at these higher frequencies is less than 2 centimeters. Since my sound sources were not at the same location, the separation would cause a phase shift between the two signals. I did not try to measure the separation, since I don' think I would have been accurate enough to be meaningful.

Anyway, it was fun to try this.

#### sophiecentaur

Gold Member
I was simply wondering: what if the two ultrasound frequencies differ by, say, 2Hz? What would I hear, if anything?
I don't think so. The beats you hear between two audible tones is the result of your ear's perception of the two and their relationship. The fact is that the two tones have made the journey into your ear, across the ossicles and into the cochlea, where they would be detected.
If you try the same with ultrasonic tones, the ear cannot deal with those two so any detected 'sound' would be due to intermodulation due to some mechanism upstream of your ear's sound receptors. The ultrasound will be attenuated greatly and then not have any sensory hairs to excite in a direct way.
I have already suggested that you read more extensively around the topic before you can expect to 'extrapolate' on your knowledge in a valid way. There are a number of different mechanisms that you need to get a good grasp of before you can hope for much success in this unusual topic. Mixing frequencies is a common practice in Radio Frequency Engineering but translating this into psychoacoustics is a big step.

#### FranzDiCoccio

I don't think so. The beats you hear between two audible tones is the result of your ear's perception of the two and their relationship. The fact is that the two tones have made the journey into your ear, across the ossicles and into the cochlea, where they would be detected.
If you try the same with ultrasonic tones, the ear cannot deal with those two so any detected 'sound' would be due to intermodulation due to some mechanism upstream of your ear's sound receptors. The ultrasound will be attenuated greatly and then not have any sensory hairs to excite in a direct way.
I have already suggested that you read more extensively around the topic before you can expect to 'extrapolate' on your knowledge in a valid way. There are a number of different mechanisms that you need to get a good grasp of before you can hope for much success in this unusual topic. Mixing frequencies is a common practice in Radio Frequency Engineering but translating this into psychoacoustics is a big step.
Hi sophiecentaur
thanks for your answers. Psychoacoustics is surely a fascinating field, but I definitely have much less time than required to explore in depth all the interesting things I come across.
As I mention, my question arose from a textbook exercise. The result proposed by the book depends on the range of audibility of a beat. I easily solved it using the standard range of audibility provided by the book (20Hz - 20kHz). Then I started wondering whether the range provided by the book had to be extended on account of the fact that I could hear beats at less than 1Hz.

I was aware that the answer could be more about human physiology rather than the simple physics involved in the exercise. And, now that you mention the "phycho" part, I see how it must also be about how the mind processes the perceptions of its senses. Even more interesting, but even more complex.

So this was me "stretching" the question in a simple textbook exercise. The exercise asked for the condition so that the beat of the two ultrasounds is in the audible window, between 20Hz and 20kHz.
What kind of confused me is that I rephrased it in "personal" terms, so I asked myself "what would the second ultrasound frequency be for me to actually hear the beat".
But this question is not just about simple physical phenomena, like waves and beats. It involves a somewhat deep knowledge of how human hearing works.

Thanks again for your help.

I found this link, which claims to be selling a product similar to what you are suggesting.
BTW, I did not understand this comment of yours.

#### FranzDiCoccio

In my initial test with just a couple of laptops, [...] it was fun to try this.
This is really interesting. Thanks!

#### sophiecentaur

Gold Member
the fact that I could hear beats at less than 1Hz.
You do not have a 1Hz transducer in your ear and neither does anyone else (except for. perhaps an elephant or large whale) so you cannot 'hear 1Hz. You can still analyse two tones, one Hz apart and be aware of a 1Hz beat and so can everyone else. These are two different phenomena and there is no conflict in what you have read. Your brain is not able to distinguish between beats at audible frequencies and direct sounds. If you Single Sideband modulate a 1kHZ tone with 1Hz, the resulting perceived sound is very similar to the sound of 1kH and 1.001kHz tones, played together.
BTW, I did not understand this comment of yours.
The link is for a device which uses ultrasounds to produce audible frequencies. If you explore the whole website, they make it clear what they are doing.
Anyway, it was fun to try this.
Oh yes, definitely. But laptop DACs do not produce ultrasound so the OP's scenario is not possible to try. Also, you may need higher sound levels than are available from a laptop in order to get a good beat sensation. Playing the outputs at high level through an amp could improve the experience.
PS the different locations can result in an interference pattern but moving your head about should allow you to find a 'best' condition.

#### FranzDiCoccio

You do not have a 1Hz transducer in your ear and neither does anyone else (except for. perhaps an elephant or large whale) so you cannot 'hear 1Hz.
Sure, this is clear to me.

You can still analyse two tones, one Hz apart and be aware of a 1Hz beat and so can everyone else. These are two different phenomena and there is no conflict in what you have read. Your brain is not able to distinguish between beats at audible frequencies and direct sounds.
Ok, I am going to hear the beat as a modulation in the volume of a sound I was hearing to begin with, right?

If you Single Sideband modulate a 1kHZ tone with 1Hz, the resulting perceived sound is very similar to the sound of 1kH and 1.001kHz tones, played together.
Ok, this I sort of understand, but I am not familiar with the technical jargon. I'll look into that.
You mean that you can build a single "complex" waveform (from a single speaker) that sounds exactly as the beat from two simple "monochromatic" tones?

The link is for a device which uses ultrasounds to produce audible frequencies. If you explore the whole website, they make it clear what they are doing.
Ok, I looked the landing page and I completely missed the ultrasound bit. Apologies.

#### sophiecentaur

Gold Member
You mean that you can build a single "complex" waveform (from a single speaker) that sounds exactly as the beat from two simple "monochromatic" tones?
That's taken to the extreme with recorded music; many beats and intermodulation tones. The ear does its best to make sense of what it hears (always).

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