Unit 1: BASIC ACOUSTIC PRINCIPLES - Subjective Questions

Sound is a mechanical disturbance that produces auditory sensation when it reaches the human ear. A sound wave is the propagation of pressure, density, and particle-velocity disturbances through an elastic medium.

• In fluids, sound generally propagates as a longitudinal wave.
• Fluid particles oscillate parallel to the direction of wave propagation.
• Alternating regions of compression and rarefaction are produced.
• The particles transfer energy without undergoing significant net displacement.
• Sound requires a material medium and cannot propagate through a perfect vacuum.

For a harmonic plane wave, acoustic pressure may be represented as

where is the pressure amplitude, is the angular frequency, and is the wave number.

• Frequency : Number of oscillation cycles completed per second, measured in hertz.
• Period : Time required for one complete cycle:
  
• Wavelength : Distance traveled by a wave during one period.
• Wave number : Spatial angular frequency:
  
• Phase: Angular quantity that specifies the position of a periodic disturbance within its cycle.
• Amplitude: Maximum departure of a fluctuating quantity from its equilibrium value.
• Speed of sound : Speed at which an acoustic disturbance propagates through a medium:
  

These quantities collectively describe the temporal and spatial behavior of a sound wave.

During one time period , a sound wave advances through one wavelength . Therefore,

Since ,

Angular frequency and wave number are defined by

Substitution gives

Hence, the complete relationship is

For a fixed sound speed, wavelength is inversely proportional to frequency. Thus, high-frequency sound has a short wavelength, while low-frequency sound has a long wavelength.

Let the harmonic particle displacement be

The particle velocity is the time derivative of displacement:

Thus, the velocity amplitude is

Particle acceleration is the derivative of velocity:

Its amplitude is

Therefore:

• Velocity leads displacement by in harmonic motion.
• Acceleration is out of phase with displacement.
• For a fixed displacement amplitude, velocity and acceleration increase with frequency.

Acoustic energy consists of kinetic energy due to particle motion and potential energy due to compression.

The instantaneous kinetic energy density is

The instantaneous potential energy density is

Hence, total energy density is

For a progressive plane wave, , so the time-averaged kinetic and potential energy densities are equal. For sinusoidal pressure with RMS value ,

Instantaneous acoustic intensity is

The time-averaged intensity of a progressive plane wave is

It also satisfies

Thus, intensity represents acoustic energy transmitted per unit area per unit time.

• Free field: A region where sound propagates without significant reflections, approximating an unbounded medium.
• Diffuse field: A field in which acoustic energy arrives from all directions with nearly equal probability and average intensity.
• Near field: Region close to a source where pressure and particle velocity may have complex phase relationships and the inverse-square law may not apply.
• Far field: Region sufficiently far from a source where wave fronts have stable directional characteristics and pressure generally decreases predictably with distance.
• Direct field: Portion of the field reaching a receiver directly from the source.
• Reverberant field: Portion formed by repeated reflections from surrounding surfaces.

In an enclosed room, the direct field dominates near the source, whereas the reverberant field may dominate farther away.

Sound is reflected when it encounters a boundary at which acoustic impedance changes. For normal incidence from medium 1 to medium 2, their characteristic impedances are

The pressure reflection coefficient is

The pressure transmission coefficient is

For lossless media, the intensity coefficients are

and satisfy .

• If , reflected pressure has no phase reversal.
• If , reflected pressure undergoes a phase reversal.
• A large impedance mismatch produces strong reflection.

Diffraction is the bending and spreading of sound waves around the edges of obstacles or through openings.

Its extent depends mainly on the ratio of wavelength to the characteristic obstacle or opening dimension :

• If , strong diffraction occurs and sound spreads widely.
• If , diffraction is significant.
• If , diffraction is weak and a distinct acoustic shadow may form.

Because , low-frequency sound generally diffracts more strongly than high-frequency sound. Diffraction explains why low-frequency noise can be heard behind barriers. According to the Huygens principle, each point on a wave front can be treated as a source of secondary wavelets, including points near an edge or within an aperture.

Sound pressure level:

For airborne sound,

Sound intensity level:

where

Sound power level:

where

Pressure uses a factor of because intensity is proportional to pressure squared. Decibel levels are logarithmic and dimensionless, although they are reported in decibels.

A typical acoustic measurement system contains:

• Microphone: Converts acoustic pressure into an electrical signal.
• Preamplifier and signal conditioner: Amplify and condition the microphone output.
• Sound-level meter or data-acquisition system: Measures and records sound pressure levels.
• Frequency analyzer: Provides octave, one-third-octave, or narrow-band spectra.
• Acoustic calibrator: Applies a known pressure level at a specified frequency to verify sensitivity.

Important practices include:

• Use the correct reference pressure, normally in air.
• Calibrate before and after measurements.
• Select suitable frequency weighting, such as A-, C-, or Z-weighting.
• Select appropriate time weighting, such as fast, slow, or impulse.
• Avoid microphone reflections, wind noise, overload, and background-noise contamination.
• Record environmental conditions because temperature and atmospheric properties affect propagation.
• Follow applicable IEC, ISO, or national standards and report measurement uncertainty.

Linear acoustic theory treats acoustic variables as small perturbations superimposed on equilibrium values:

The principal assumptions are:

• Small perturbations: and .
• Small particle velocity: .
• Initially stationary medium: Mean flow is absent unless specifically included.
• Homogeneous medium: Equilibrium properties are spatially uniform.
• Inviscid approximation: Viscous forces are neglected for ideal propagation.
• Adiabatic changes: Thermal exchange during rapid compression and expansion is negligible.
• Neglect of second-order terms: Products such as and are omitted.

These assumptions linearize the governing equations and permit superposition of acoustic waves.

The general continuity equation is

For a stationary equilibrium medium, write

Substitution gives

Because is constant and time-independent,

The term is second order and is neglected in linear acoustics. Therefore,

This equation states that local density changes are produced by convergence or divergence of acoustic particle velocity.

The nonlinear Euler momentum equation without body forces is

Introduce the acoustic variables

For uniform equilibrium pressure, . Neglecting the nonlinear convective term and products of perturbations gives

Therefore, the linearized Euler equation is

It states that a spatial pressure gradient accelerates fluid particles from high-pressure regions toward low-pressure regions. The equation represents conservation of linear momentum in an ideal acoustic medium.

For a perfect gas undergoing a reversible adiabatic process, Poisson's relation is

or

where is the ratio of specific heats. Differentiating about the equilibrium state gives

The adiabatic sound speed is defined as

Therefore,

For small acoustic perturbations, the linear pressure-density relation becomes

This constitutive relation closes the linear acoustic equations by relating pressure fluctuations to density fluctuations.

The linearized continuity equation is

Differentiate it with respect to time:

The linearized Euler equation is

Taking its divergence or substituting directly gives

Using the adiabatic relation , or , produces

Hence, the pressure wave equation is

Similar wave equations can be derived for density perturbation, velocity potential, and each component of particle velocity.

A plane harmonic wave traveling in the positive -direction can be written as

The one-dimensional linear Euler equation is

Differentiating the pressure and integrating with respect to time gives

because . Therefore,

The quantity

is the characteristic acoustic impedance of the medium. In a lossless progressive plane wave, pressure and particle velocity are in phase. For a wave traveling in the negative -direction, the velocity has the opposite sign relative to pressure.

For an ideal point source radiating uniformly in a free field, the acoustic power crosses a spherical surface of area

Conservation of energy gives the intensity

Therefore,

This is the inverse-square law. Since intensity is proportional to RMS pressure squared,

For two distances and ,

Thus, doubling the distance from an ideal point source reduces sound pressure level by approximately . The law applies in the source's far field under free-field conditions with negligible absorption and reflection.

• Transient response: Short-duration behavior occurring when a source starts, stops, or changes suddenly. It depends on initial conditions and decays with time in a damped system.
• Steady-state response: Persistent response after transient effects have diminished, usually at the excitation frequency for harmonic forcing.
• Standing waves: Produced by interference between waves traveling in opposite directions. They contain pressure nodes and antinodes.
• Resonance: Occurs when excitation frequency approaches a natural frequency, causing a large response if damping is low.
• Damping: Dissipates acoustic energy through viscous, thermal, radiation, or material losses and limits resonance amplitude.

For a one-dimensional cavity of length with rigid ends, typical natural frequencies are

The sharpness of resonance is often described by the quality factor

where is the resonant frequency and is the bandwidth between half-power points.

• Sound pressure : Local scalar fluctuation about ambient pressure, measured in pascals.
• Particle velocity : Oscillatory velocity of medium particles, measured in metres per second. It is distinct from the speed of sound.
• Acoustic intensity : Rate of acoustic-energy flow per unit area. Instantaneously,
  
  and it is measured in .
• Sound power : Total acoustic energy emitted or transmitted per unit time, measured in watts:
  

Pressure and particle velocity are local field variables, intensity is an energy-flux quantity, and power is a source or transmission quantity integrated over a surface. Unlike measured pressure, source sound power ideally does not depend on receiver distance.

The characteristic acoustic impedance is

RMS particle velocity:

Average intensity:

Sound pressure level:

Average acoustic energy density:

The consistency relation is also satisfied.