Beats - acoustics question

This is a bit of an old chestnut, but I can't find any definitive answer.

Beats (for simplicity between two sine waves near the unison)
constitute a clear periodic variation in loudness - a *subjective*
measure. However, I want to know what is the variation in *sound
power* - that is in physical energy transmitted by the sound wave.

The complication which makes this potentially interesting is that
sound power is not a function of pressure oscillation alone. Rather it
is the average of pressure times particle velocity over a cycle. For a
single sine wave it turns out to be simply (proportional to) the
square of the pressure oscillation.

So does the physical sound power also periodically vary for the sum of
two sine waves? My algebra suggests that it should do.

~~~T~~~

Hi Tom.

Not sure if this is what you are talking about, but the intensity variation
itself is also a sine period. For example, if you take a 100Hz sine wave and
add one of 120Hz to it, you get exactly the same as if you take a sine wave
of 110Hz and multiply (i.e. "ring-modulate") it by a sine of 10Hz. or, to be
more precise, it can be also a cosine wave if both the tones are in the same
phase at the begining. If they are totally out of phase at the beginning,
the imaginary amplitude modulator is a sine.

Examples:
1. Sine 100Hz + sine 120Hz = sine 110Hz * cosine 10Hz
2. Cosine 100Hz + cosine 120Hz = cosine 110Hz * cosine 10Hz
3. Sine 120Hz - sine 100Hz = cosine 110Hz * sine 10Hz
4. Cosine 100Hz - cosine 120Hz = sine 110Hz * sine 10Hz

Hope this helps.

Petr

OK, so if you stuck a microphone with a decibel meter there, the meter
would go up and down. That is my main point. (Though it would make
more sense here to take a slower variation e.g. beating of 100 Hz /
100.5 Hz so that the meter has time to respond.)

There are some acoustic phenomena (e.g. difference tones) that are due
to nonlinear response or other peculiarities of the ear or brain, and
some that are due to the physical nature of sound itself, and I'm
trying to work out which is which. It looks likely that beats are in
the latter category and that anyone or anything that can hear changes
in loudness can also hear beats in some range.

~~~T~~~

--- In tuning@yahoogroups.com, Petr Par�zek <p.parizek@c...> wrote:
>
> Hi Tom.
>
> Not sure if this is what you are talking about, but the intensity
variation
> itself is also a sine period. For example, if you take a 100Hz sine
wave and
> add one of 120Hz to it, you get exactly the same as if you take a
sine wave
> of 110Hz and multiply (i.e. "ring-modulate") it by a sine of 10Hz.
or, to be
> more precise, it can be also a cosine wave if both the tones are in
the same
> phase at the begining. If they are totally out of phase at the
beginning,
> the imaginary amplitude modulator is a sine.
>
> Examples:
> 1. Sine 100Hz + sine 120Hz = sine 110Hz * cosine 10Hz
> 2. Cosine 100Hz + cosine 120Hz = cosine 110Hz * cosine 10Hz
> 3. Sine 120Hz - sine 100Hz = cosine 110Hz * sine 10Hz
> 4. Cosine 100Hz - cosine 120Hz = sine 110Hz * sine 10Hz
>
> Hope this helps.
>
> Petr
>

Hi tom.
You wrote:

> There are some acoustic phenomena (e.g. difference tones) that are due
> to nonlinear response or other peculiarities of the ear or brain, and
> some that are due to the physical nature of sound itself, and I'm
> trying to work out which is which. It looks likely that beats are in
> the latter category and that anyone or anything that can hear changes
> in loudness can also hear beats in some range.

I'll tell you what's my view on this. The properties that can be clearly
shown by an "acoustical" example are the fundamental (i.e. the highest
common divisor in terms of frequencies) and the guide tone (i.e. the lowest
common multiple). The ones that are not so easily explainable and are more
or less resulting from some "processes in our brain" are the sum tone (X+Y),
the difference tone ()Y-X, the lower second difference tone (2X-Y), and the
higher second difference tone (2Y-X). BTW: I'm meaning the lower sounding
tone by X and the higher one by Y.

Petr

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
>
> This is a bit of an old chestnut, but I can't find any definitive
answer.
>
> Beats (for simplicity between two sine waves near the unison)
> constitute a clear periodic variation in loudness - a *subjective*
> measure. However, I want to know what is the variation in *sound
> power* - that is in physical energy transmitted by the sound wave.
>
> The complication which makes this potentially interesting is that
> sound power is not a function of pressure oscillation alone. Rather
it
> is the average of pressure times particle velocity over a cycle.
For a
> single sine wave it turns out to be simply (proportional to) the
> square of the pressure oscillation.
>
> So does the physical sound power also periodically vary for the sum
of
> two sine waves? My algebra suggests that it should do.

Be careful, Tom. You said "the average of pressure times particle
velocity over a cycle". This makes sense for a single sine wave,
where there's no problem determining what constituted a "cycle". In
reality, the physical formula involves an integral of this product
over time, divided by the length of time integrated over (that's the
definition of "average" used). For a periodic function, this will
clearly give the same answer as long as you're integrating over an
integer number of cycles. For the sum of two sine waves, though,
there isn't a clear choice of "cycle". You have to make a judgment,
based on some exogenous criteria, on what time scale it's most
appropriate to integrate over. These exogenous criteria in the case
of the human ear dictate a time scale much less than a second. So it
*is* meaningful, from the perspective of the human ear, to consider
the sum of two sine waves whose frequency differs by 1 Hz to be a
signal whose power varies from essentially zero to a maximum and back
every second. If the frequency of the two sine waves differs by many
times more than 1 Hz, you do not hear the beating, and the
appropriate integral for calculating sound power spans many cycles of
what would otherwise be 'beating'. It doesn't make sense to consider
two sine tones which don't audibly beat against one another (because
their frequencies are too far apart) to result in a varying level of
sound power, and indeed the appropriate calculation comes up with a
nearly constant power.

So, unfortunately for you perhaps, "sound power" isn't something that
can be pinned down quite as uniquely as certain other physical
quantities, such as energy . . . Even the most professional VU meters
must make some choice for the time scale over which to integrate; and
this time scale has to have some correspondence with the human ear
for the meter readings to be meaningful . . .

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
>
> OK, so if you stuck a microphone with a decibel meter there, the
meter
> would go up and down.

It might, depending on what time scale the decibel meter integrates
over.

> That is my main point. (Though it would make
> more sense here to take a slower variation e.g. beating of 100 Hz /
> 100.5 Hz so that the meter has time to respond.)

"Time to respond" is an extremely key part of this and it's this very
time scale which determines which definition of "loudness" the meter
is in effect using.

> There are some acoustic phenomena (e.g. difference tones) that are
due
> to nonlinear response or other peculiarities of the ear or brain,
and
> some that are due to the physical nature of sound itself, and I'm
> trying to work out which is which. It looks likely that beats are in
> the latter category

Not really -- what would be beats to us might not be beats, and vice
versa, for an elephant or mouse. I recently posted an explanation of
beating here -- let me know if you missed that and would like me to
find it in the archives . . .

> and that anyone or anything that can hear changes
> in loudness can also hear beats in some range.

That's the key -- some range (that is, some range of frequency).
Different beings are sensitive to changes in loudness over different
time scales, and thus to different rates of beating.

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@c...> wrote:
>
> Hi tom.
> You wrote:
>
> > There are some acoustic phenomena (e.g. difference tones) that
are due
> > to nonlinear response or other peculiarities of the ear or brain,
and
> > some that are due to the physical nature of sound itself, and I'm
> > trying to work out which is which. It looks likely that beats are
in
> > the latter category and that anyone or anything that can hear
changes
> > in loudness can also hear beats in some range.
>
> I'll tell you what's my view on this. The properties that can be
clearly
> shown by an "acoustical" example are the fundamental (i.e. the
highest
> common divisor in terms of frequencies) and the guide tone (i.e.
the lowest
> common multiple).

I don't understand. What do you mean by this? What if the two
frequencies are, say, 440 and 440*pi?

> The ones that are not so easily explainable and are more
> or less resulting from some "processes in our brain"

Mostly it's not the brain, but the ear, that's involved in the below,
as binaural experiments have shown.

> are the sum tone (X+Y),
> the difference tone ()Y-X, the lower second difference tone (2X-Y),
and the
> higher second difference tone (2Y-X). BTW: I'm meaning the lower
sounding
> tone by X and the higher one by Y.
>
> Petr

Around the same time I posted an explanation of beating, I posted an
explanation of first-order combinational tones here too. Let me know
if I should find that for you (it wasn't long ago).

Hi Paul.
You wrote:

> I don't understand. What do you mean by this? What if the two
> frequencies are, say, 440 and 440*pi?

OK, please let me clarify this with a simpler example which can be expressed
rationally. I'll choose frequencies of 200Hz and 301Hz. The third harmonic
of the lower tone is 600Hz, the second harmonic of the higher tone is 602Hz,
so you'll get beats of 2Hz. Such a slow beat rate makes the phase shifts
still not too rapid for you to think of an audible pseudo-fundamental (i.e.
slightly above 100Hz in this case).

> Around the same time I posted an explanation of beating, I posted an
> explanation of first-order combinational tones here too. Let me know
> if I should find that for you (it wasn't long ago).

If you could, it would be great. I'd definitely be interested in both of
these.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@c...> wrote:
>
> Hi Paul.
> You wrote:
>
> > I don't understand. What do you mean by this? What if the two
> > frequencies are, say, 440 and 440*pi?
>
> OK, please let me clarify this with a simpler example which can be
expressed
> rationally. I'll choose frequencies of 200Hz and 301Hz. The third
harmonic
> of the lower tone is 600Hz, the second harmonic of the higher tone
is 602Hz,
> so you'll get beats of 2Hz.

Right; so? How does this relate to what you wrote (and snipped) that
I was responding to above? What would irrational frequencies change
here?

> Such a slow beat rate makes the phase shifts
> still not too rapid for you to think of an audible pseudo-
fundamental (i.e.
> slightly above 100Hz in this case).

This is similar to virtual pitch, yet another acoustical phenomenon.

> > Around the same time I posted an explanation of beating, I posted
an
> > explanation of first-order combinational tones here too. Let me
know
> > if I should find that for you (it wasn't long ago).
>
> If you could, it would be great. I'd definitely be interested in
both of
> these.
>
> Petr

OK, I'll try, though the searches I've tried lately have seemed to
skip right over every one of my posts . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> OK, I'll try, though the searches I've tried lately have seemed to
> skip right over every one of my posts . . .

I think Yahoo might be trying an incredibly brilliant new system,
whereby searches always miss your own postings. Is there any way to
give feedback about these brainstorms?

Hi Paul.
You wrote:

> Right; so? How does this relate to what you wrote (and snipped) that
> I was responding to above? What would irrational frequencies change
> here?

Well, 440*pi is ~1382.3, which is close enough to 1320 (i.e. the third
harmonic of 440) to make beats of ~62.3Hz then. As far as I was trying it
out using regular sawtooth periods, this was exactly what I could hear.

> OK, I'll try, though the searches I've tried lately have seemed to
> skip right over every one of my posts . . .

I've got it, thanks a lot. I'll look at it in greater detail, if not sooner,
at latest on Monday after I finnish my new electronic experimental piece for
the upcoming composition exams at our school.

Petr

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > OK, I'll try, though the searches I've tried lately have seemed to
> > skip right over every one of my posts . . .
>
> I think Yahoo might be trying an incredibly brilliant new system,
> whereby searches always miss your own postings.

This time, it didn't. But only some of the keywords that should have
worked did . . .

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@c...> wrote:
>
> Hi Paul.
> You wrote:
>
> > Right; so? How does this relate to what you wrote (and snipped)
that
> > I was responding to above? What would irrational frequencies
change
> > here?
>
> Well, 440*pi is ~1382.3, which is close enough to 1320 (i.e. the
third
> harmonic of 440) to make beats of ~62.3Hz then. As far as I was
trying it
> out using regular sawtooth periods, this was exactly what I could
hear.

No one can hear beats this fast. But clearly, now that I know you're
making beats, a more appropriate example would have been 440 and
440+pi, or something.

> > OK, I'll try, though the searches I've tried lately have seemed to
> > skip right over every one of my posts . . .
>
> I've got it, thanks a lot. I'll look at it in greater detail, if
not sooner,
> at latest on Monday after I finnish my new electronic experimental
piece for
> the upcoming composition exams at our school.

Good luck and let us hear it!