Scientific Essence Of Sound: Understanding Sound Propagation
Information that's truly at the heart of everything we do when working with sound...

March 31, 2014, by Neil Thompson Shade

sound

Let’s have a look at sound particles, acoustic pressure, propagation medium density, particle velocity, acoustic energy, volume velocity, and acoustic impedance.

Abstract concepts? Nope—far from it. All are related to the physics of sound propagation in an elastic medium that result in changes to the displacement, pressure, density, temperature, and velocity within the medium.

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.

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)

Figure 1: Fluid particle positiones through one complete oscillatory cycle. Y axis is acoustic pressure with 0 equal to atmospheric pressure, and X axis is distant (or time) corresponding to wave propagation. (Coutesy Pierce)

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 μPa).

This value is referred to as the reference acoustic pressure. 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 (ρo) 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.

Figure 2. Conceptualization of sinusoidal longitudinal wave motion. (Courtesy Houghton)

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 millieseconds (ms), 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:

Equation 1:
image
where,
p = acoustic pressure, Pa
ρo = air density, 1.2 kg/m3
c = velocity of sound, 343 ms
u = particle velocity, ms

The constant term in Equation 1, ρoc, 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.

Equation 2: 
image
where,
EP = potential energy density, Joules (J)
p = acoustic pressure, Pa
κ = bulk modulus of acoustic fluid, Pa (equal to poc2)
ρo = density of transmission medium, kg/m3
c = speed of sound, 343 ms

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.

Equation 3:
image
where,
EK = kinetic energy density, J
ρo = density of transmission medium, kg/m3
u = particle velocity, m/s

The kinetic energy is directly proportional to the square of the particle velocity.

Equation 4: 
image 
where,
E = sound energy density, J/m3
EP = potential energy density, J
EK = kinetic energy density, J
ρo = density of transmission medium, kg/m3
u = particle velocity, ms

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 spiral, a representation of the frequency response for a small loudspeaker separated into the real (kinetic energy) and imaginary (potential) energy relationships.

Figure 3. Real (kinetic energy) and imaginary (potential energy) relationships for a small loudspeaker. (Courtesy McNair)

The frequency response of the loudspeaker is shown in the center between the boxes. The real (kinetic energy) response is shown at the bottom and the imaginary (potential energy) response is shown to the right. The figure to the left is the Nyquist (polar) display of both responses.

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 magnitude of the acoustic medium flow will depend on the size of the vibrating object. The volume velocity is defined by the equation below.

Equation 5:
image 
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 amplitude surface (particle) velocity or use a smaller cone operating with a higher amplitude surface velocity. (Figure 4)

Figure 4: Concept of equal value velocity (U): loudspeaker at left has larger surface area (S) and smaller particle velocity (u), while loudspeaker at right has smaller surface area (S) and larger particle velocity (u).

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.

Equation 6: 
image 
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 m3s, a very small quantity indeed.

Assume we substitute a 400 mm (approximately 12 inch) diameter loudspeaker that radiates the same acoustic pressure. Using the calculations for the 100 mm loudspeaker example results in a volume velocity approximately a factor of 10 larger, 3.1 X 10-4 m3s.

When the guitarist in the band cranks his amp to “11” and the output increases to 120 dB, the volume velocity with the 12 inch loudspeaker is still surprisingly small at 6.1 X 10-3 m3/s, approximately a factor of 100 larger compared to the 1 Pa (94 dB) output.

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.

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.

I encourage you to take your time in digesting this information. No one becomes a scientific guru of sound overnight. However, your time will be well invested, because this information is truly at the heart of everything we do when working with sound.

Neil’s Rules Of Thumb For Calculating The Basics:

—The reference acoustic pressure, 20 μPa, 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.

Be sure to check out Neil’s related article, Scientific Essence Of Sound: Getting To The Basis Of What We Hear

Neil Thompson Shade has decades of experience in clouting 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 he has also been teaching acoustic, sound system design, computer modeling and related topics at the Peabody Institute of Johns Hopkins University.



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