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[DPRG] Sonar 1/r (not 1/r squared)
Subject: [DPRG] Sonar 1/r (not 1/r squared)
From: John Swindle
swindle at compuserve.com
Date: Thu Jun 14 11:53:22 CDT 2007
Folks:
I gave a couple of presentations on sonar to DPRG in the past. I
made a big mistake in both of those presentations! (This is
really embarassing because I used to work for a company that
makes sound measurement equipment! That company's manuals are
correct, and so is every other document I've seen on the topic.
I'm the one who was wrong.)
For a spherical sound wave, the sound pressure is inversely
proportional to the distance from the sound source. Although the
sound intensity is inversely proportional to the square of the
distance, the pressure is inversely proportional to the distance
(not the square of the distance).
Does this matter? Microphones and ears are pressure (or velocity)
transducers, so the one over R rule applies. For sound, pressure
is directly proportional to displacement which in turn is
directly proportional to velocity. And all of these are directly
proportional to the voltage from a microphone. So, 1/r does
matter.
For sound, "intensity" is proportional to energy. Energy or work
is force through a distance, or sound pressure through a
displacement. Since both the sound pressure and displacement are
directly proportional, the energy is proportional to the square
of either one of them.
It's fairly intuitive that the inverse square law applies to
energy radiating from a point source. Since energy falls as
inverse square, and since energy is proportional to the square of
the pressure, the pressure falls as the inverse.
I wish I had an intuitive way of visualizing how the pressure
drops off. Aside from appealing to the energy derivation, I have
not come up with a visceral way of understanding why sound
pressure falls off as 1/r. It is not hydraulic, and it is not
static, so it's not as easy to visualize. The combination of the
pressure and the displacement carry the energy.
Who cares? This almost doesn't matter for sonar. The relative
strength of the echo is irrelevant to almost every hobbyist and
consumer sonar, as long as the echo is loud enough to trigger the
circuit. There may be a stepped or log amplifier to help with
that, and 1/r is used for that amplifier. But aside from that,
all that matters is the arrival time of the echo.
This email might matter if you're using sound to make a cheap
backyard beacon system using only two speakers. And, it might
matter if you are trying to digitize and process sonar echoes.
In a way, the inverse square law still applies because sound
pressure is often measured in decibels, and decibels are a
measure of power (unless specifically stated otherwise). So,
decibels of sound pressure are calculated using the square of the
pressure, thus making it a power measurement, not a pressure
measurement.
And on to other things: The next thing I want to do with sound is
to make a sound-driven weather station, to give temperature,
humidity, pressure, and velocity using nothing but three
transducers (no servos). Temperature and humidity affect the
speed of sound. Pressure affects the intensity. Velocity affects
the arrival time. What I haven't figured out is how to separate
the humidity measurement from the temperature measurement. And
what about measuring rainfall?
Later,
John Swindle
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