[Dick Pierce[_2_]]

Edmund wrote:
On Wed, 31 Aug 2011 14:01:58 +0000, Arny Krueger wrote:
Not so much? All other things being equal, energy is proportional to
frequency.

Uh, no. If, "by all other things being equal," you mean
that for equal sound pressure level, your claim that
energy is proportional is provably wrong.

Sound pressure level is a measure of the amount
of acoustic power per unit area. Specifically,
0 dB SPL is defined as the equivalent of 10^-12,
and sound pressure level is defined as:

SPL = 20 log10 (Px/Pref)

where Pref = 10^-12 W

This same defining equation can be found in any number
of sources, such as Beranek, Blackenstock, Kinsler and
Frey, and many, many others. Not a single one of them
shows a frequency term.

Now, energy is power integrated over time. In the
simplest case, a speaker producing 1 acoustic watt
for 10 seconds produces 10 watt-seconds of energy
REGARDLESS of whether it's radiating 10 Hz, 100 Hz
or 100,000 Hz.

Now, it MAY, under some very narrowly constrained
circumstances, REQUIRE more input power to produce a
given acoustic power at some frequency than another,
but that's a matter of conversion efficiency, which
has no such instrinsic property of "energy is
proportional to frequency," as claimed.

Well I admit I am a bit rusty here, but are you
saying the ultrasonic sound of a bat requires more
energy to produce then the 7 Hz sound of an elephant?

That does appear to what he is saying and "all other
things being equal," which they simply can't be*, the
statement is wrong.

*All other things simply cannot be equal: for example,
the same surface radiating different frequencies has
different radiation patterns, different radiation
impedance, and so on. Two sources at different
frequencies with the same radiation impedance are
likely to have different radiating ares, moving
masses and the like.

Looking at instruments too I see the same phenomenon, low
frequencies require more air to be moved and much bigger
instruments en more power to drive these instruments.

No, they do not: they move a larger vlume of air, but,
for the same sound pressure level, they move it at a
substantially lower speed, indeed, the linear velocity
goes as the reciprocal of frequency. The net result is
that the volume ve,ocity of the source is constant with
frequency for a flat frequency response.

In loudspeakers too, the bass is bigger and need far more
energy then a tweeter.

Completely false. The amount of output acoustic power
compared to the input power is simnply a measure of
the efficiency of the system, and, for a flat frequency
response, that efficiency is, by definition, constant
over the pass band of the system. If your assertion was
correct, then the acoustic power radiated by a speaker
would would have a intrinsic direct dependence on
frequency for a constant input power, and this is
simply not the case. Conversely, if Mr. Krueger's
assertion were correct, the acoustic power radiated
by a speaker would have an intrinsic reciprocal
dependence on frequency, and this, as well, is not the
case.

The conditions required for a speaker to produce a
given sound pressure level is a specific volume
velocity. This is, essentially, the rate at which
a given volume of air can be moved. That means,
at low frequencies, the excursion of a diaphragm is
large, but the velocity is low, while at high
frequencies, the velocity is high, but the excursion
is low. It is the PRODUCT of these that determines
the sound pressure level, a measurement of power.
And that power, integrated over a given time interval,
is the energy radiated by the speaker (or whatever
happens to be producing the sound).

There simply is no intrinsic property that states
either that high frequencies requires more energy
or low frequencies require more energy. At least
not over a substantially broader bandwidth and at
much higher power than what we are talking about
for musci production. One can argue that the
basic gas laws go non-linear at extremely high
frequencies (orders of magnitude above the highest
imaginable musical sound, captured or otherwise)
or at sound levels where shock waves are forming.

Anyway it is not a problem to deliver the energy to drive
a tweeter for the very high frequencies.

No more than it is a problem to deliver the energy to
drive a woofer at very low frequencies. What's missing
is a definition of "very high" and "very low". Within
their pass band, a tweeter and a woofer of equal electro-
acoustic efficiency requires the same input power to
produce the same acoustic power REGARDLESS of the
frequency. Both drivers at "very high" and "very low"
frequencies, require substantially more power, if such
frequencies are outside the pass band of the drivers.


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+ Dick Pierce |
+ Professional Audio Development |
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