Academic Pursuits: The Scientific Essence Of Sound
Getting to the basis of everything we hear
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If you made it through the first installment of this series (Live
Sound, August 2003 issue), give yourself a high-five. Better
yet, bring those two hands together to create an impulsive sound
with a propagating omnidirectional wavefront! |
In other words, this is at the basis of everything we hear, the scientific
essence of sound.
Sound Particles. Sound requires a medium for the transmission of
vibrations, the most common being air. Understanding sound propagation
can be difficult because it’s not visible unless special opto-acoustical
instruments are used.
One way to “visualize” sound propagation is to imagine the vibrations
acting on invisible particles as the vibratory energy passes through a
given spatial region. The acoustic “particle” is a small volume unit of
air whose physical dimensions are smaller than the propagating sound wavelength.
The particles move about a fixed equilibrium position as a function of
time as the acoustical energy propagates through the medium.
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The collision of neighboring particles transmits energy through the medium. From the standpoint of elementary mechanics, the particles undergo displacement, velocity, and acceleration, just as any moving body does. Acoustic Pressure. Sound consists of a series of pressure maxima (compressions) and minima (rarefactions). The unit of acoustic pressure (p) is the pascal, abbreviated Pa. Acoustic pressure can be considered as the difference between the instantaneous pressure at a fixed point in a spatial region with the sound source present and with the source absent. The pressure maxima and minima oscillate above and below normal atmospheric pressure (po) in direct response to the acoustic particle motion. Figure 1 conceptually shows the fluid particle postions during one complete oscillatory cycle for a sinusoidal plane wave. |
A certain amount of acoustic pressure is necessary to evoke the sensation
of hearing. For individuals with “normal” hearing, and unfortunately this
may exclude some of our dear readers, the minimum acoustic pressure necessary
for the hearing sensation is 20 X 10-6 Pa (20 mPa). This value is referred
to as the reference acoustic pressure. We will see in a later installment
that this is equivalent to 0 dB sound pressure level (SPL).
The amplitude, and therefore the SPL, of a sound wave is directly proportional
to the acoustic pressure.
Propagation Medium Density. The density of a material is the mass
per unit volume, expressed in units of kg/m3. For air at normal atmospheric
conditions, the characteristic density (ro) is 1.2 kg/m3.
When the pressure in the propagating medium changes there will be a corresponding
density change in the medium. Pressure maxima result in a density increase
while pressure minima result in a density decrease. Because air is a compressible
fluid, there will be localized density changes as the acoustic energy
flows through the air.
Figure 2 shows conceptually what is happening for a longitudinal
sound wave in terms of the particle displacement and propagation medium
density change.
Particle Velocity. The particle velocity (u) is the velocity fluctuation
a particle undergoes in the acoustic medium about its equilibrium position
resulting from the passage of acoustic pressure. The units are m/s. Note
that the particle velocity should not be confused with the characteristic
propagation velocity of sound, 343 milliseconds (m/s), which describes
the rate at which sound travels through the medium.
One important characteristic of the particle velocity is that it is 90
degrees out-of-phase with the acoustic pressure when close to a physical
boundary or the radiating surface. This characteristic is of prime importance
when designing sound absorptive treatments such as “bass” traps. We will
examine sound absorption in a future article.
The amplitude of the particle velocity is directly proportional to the
acoustic pressure. Acoustic pressure and particle velocity are related
to each other by the following equation:
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where,
p = acoustic pressure, Pa
ro = air density, 1.2 kg/m3
c = velocity of sound, 343 m/s
u = particle velocity, m/s
The constant term in Equation 1, roc, is referred to as the characteristic
impedance of the acoustic medium. It is of prime importance in helping
us understand the interaction of a vibrating surface, such as a loudspeaker
cone, and the surrounding acoustic field that provides a “back pressure”
on the radiating surface. This radiation phenomenon will be examined in
a future article.
Acoustic Energy Relationships. A sound wave comprises both potential
energy and kinetic energy. Potential energy is energy that is stored and
is ready to do work. Kinetic energy is the energy of motion. The sum of
the potential and kinetic energies is always constant, assuming a lossless
transmission medium, even though each contributes a varying amount depending
on the position in the oscillatory cycle.
At the extremes of the oscillatory cycle (maximum positive or maximum
negative pressures) the energy is primarily potential with zero kinetic
energy. At the mid-point of the oscillatory cycle the energy is primarily
kinetic with zero potential energy. Between these limits there is a combination
of potential and kinetic energies. The total energy per unit volume is
called the sound energy density. Potential energy density, kinetic energy
density, and sound energy density are related to each other by the equations
below.
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where,
EP = potential energy density, Joules (J)
p = acoustic pressure, Pa
k = bulk modulus of acoustic fluid,
Pa (equal to roc3)
po = density of transmission medium, kg/m3
c = speed of sound, 343 m/s
The potential energy is directly proportional to the square of the acoustic pressure. The bulk modulus is used to specify the volume decrease of the acoustic medium occurring under uniform pressure. Think of it as the “compressibility” of the acoustic medium.
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where,
EK = kinetic energy density, J
ro = density of transmission medium, kg/m3
u = particle velocity, m/s
The kinetic energy is directly proportional to the square of the particle velocity.
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where,
E = sound energy density, J/m3
EP = potential energy density, J
EK = kinetic energy density, J
ro = density of transmission medium, kg/m3
u = particle velocity, m/s
The sound energy density in the acoustic medium is the sum of the potential
and kinetic energies.
Figure 3 illustrates the interaction of the potential and kinetic
energies in the form of the Heyser frequency spiral, here showing the
frequency response for a small loudspeaker separated into the real (potential
energy) and imaginary (kinetic energy) components. The frequency response
of the loudspeaker is shown in the center between the boxes. The real
(potential energy) response is shown at the bottom and the imaginary (kinetic
energy) response is shown to the right. The figure to the left is the
Nyquist (polar) response. This representation of energies is correct for
sound waves near a boundary surface or for standing waves, but not for
freely propagating waves where potential and kinetic energies are in phase.
Volume Velocity. The volume velocity (U) in its most simple sense
can be considered to be the amount of air that is moved by an acoustic
source, such as a loudspeaker, or the amount of air that causes a transducer
to move, such as a microphone diaphragm. The units of volume velocity
are m3/s. The “volume” here does not refer to level or loudness, but to
occupied space as measured in cubic units.
At the boundary of the vibrating object, the acoustic particle velocity
will be the same as the physical (moving) velocity of the object itself.
The vibrating air particles results in an acoustic medium flow perpendicular
to the vibrating object. The volume velocity is defined by the equation
below.
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where,
U = volume velocity, m3/s
u = particle velocity, m/s
S = surface area of vibrating object, m2
The volume velocity is directly proportional to particle velocity and
the surface area. Increasing either one will increase the volume velocity.
The volume velocity has the same sinusoid variation and phase as the particle
velocity.
Equation 5 readily shows us that if we want to move a given amount
of air, such as with a loudspeaker, we can either use a single cone (large
S) with a relatively low surface (particle) velocity or use a smaller
cone operating with a higher surface velocity. (Figure 4)
Of importance to volume velocity is its relationship to acoustic impedance
(Z). The units of acoustic impedance are Pa s/m3 and can be considered
analogous to acoustic ohms. The acoustic impedance is defined by the equation
below.
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where,
Z = acoustic impedance, Pa s/m3
p = acoustic pressure, Pa
U = volume velocity, m3/s
We will see in a future article the acoustic impedance is comprised of
acoustic resistance and acoustic reactance, similar to electrical impedance.
Examples. Let’s put some of the above into practice. A 100 mm (approximately
4 inch) diameter loudspeaker radiates 1 Pa acoustic pressure (94 dB).
The resulting volume velocity (U) can be determined by solving for the
particle velocity (u) using equation 1 and then solving for the volume
velocity using equation 5. Solving gives us a volume velocity of 1.9 x
10-5 m3/s, a very small quantity indeed.
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Assume we substitute a 400 mm (approximately 12 inch) diameter
loudspeaker that radiates the same acoustic pressure. Using the
calculation methods for the 100 mm loudspeaker example results in
a volume velocity approximately a factor of 10 larger, 3.1 x 10-4
m3/s. The acoustic impedance for the 100 mm loudspeaker radiating 1 Pa acoustic pressure is 5.3 x 104 Pa s/m3 and is 3.2 x 103 Pa s/m3 for the 400 mm loudspeaker. Note the lower acoustic impedance for the larger loudspeaker indicating more efficient acoustic radiation. |
The 100 and 400 mm loudspeakers, each radiating 1 Pa acoustic pressure, result in a sound energy density of 7 x 10-6 J/m3 for each as determined using equation 4. Again, like the particle velocity and volume velocity, the sound energy density is extremely small.
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You will notice that even though we are dealing with relatively
high SPL values, 94 and 120 dB, the physical acoustic variables,
excepting acoustic impedance, are fairly infinitesimal. This indicates
the extreme sensitivity of our hearing mechanism to the subjective
perception of loudness. |
Keep This In Mind
Here are some of Neil’s rules-of-thumb for calculating the basics of sound.
Try to remember them, or better yet, key them into your PDA or computer
“cheat sheet”:
• The reference acoustic pressure, 20 mPa, results in 0 dB SPL, the threshold of audibility.
• The motion of the ear drum due to a 0 dB SPL at 1,000 Hz is less than 1 angstrom (10-7 mm), the diameter of a hydrogen atom.
• A value of 1 Pa acoustic pressure will result in 94 dB SPL. This pressure value is commonly used in acoustic calibrators.
• Each successive doubling of acoustic pressure increases the SPL by 6 dB.
• Most physical acoustical parameters are very small quantities even though the SPL is relatively high.
Neil Thompson Shade has 22 years of experience in consulting and teaching acoustics, noise control and sound system design. He is president and principal consultant of Acoustical Design Collaborative, Ltd., located in Baltimore, Maryland, and for the past three years, he has also been teaching acoustic, sound system design, computer modeling and related topics at the Peabody Institute of Johns Hopkins University. He can be reached at neil@akustx.com
October 2003 Live Sound International