Unit 2: ACOUSTIC WAVE TRANSMISSION - Subjective Questions

Atmospheric properties influence both sound speed and attenuation:

• Temperature: For an ideal gas, the sound speed is
  
  where is the specific-heat ratio, is the specific gas constant, and is absolute temperature. Thus, sound travels faster in warmer air.
• Pressure and density: The general expression is
  
  At a fixed temperature, proportional changes in pressure and density largely cancel, so atmospheric pressure has little direct effect on sound speed.
• Humidity: Moist air has a lower effective molecular mass than dry air. Therefore, increasing humidity slightly increases sound speed and usually reduces molecular absorption at many frequencies.
• Density: Acoustic impedance is . Changes in density therefore affect particle velocity, intensity, reflection, and transmission.
• Frequency-dependent absorption: Viscosity, thermal conduction, and molecular relaxation convert acoustic energy into heat. High-frequency sound is generally attenuated more strongly than low-frequency sound.

Atmospheric refraction is the bending of sound rays caused by spatial variations in the effective propagation speed.

The effective sound speed along the direction of propagation is approximately

where is the local sound speed and is the wind component along the ray.

• Temperature gradient: During daytime, air near the ground is often warmer, so decreases with height. Rays bend upward, producing a shadow zone near the ground. During a temperature inversion, increases with height and rays bend downward, allowing sound to travel farther.
• Wind gradient: Wind speed generally increases with altitude. Downwind rays bend toward the ground, whereas upwind rays bend upward.
• Combined effect: Temperature and wind gradients jointly determine the vertical gradient of .
• Consequence: Refraction creates regions of enhanced sound, focusing, defocusing, and acoustic shadow zones.

Geometrical spreading distributes acoustic power over an increasing area. For a point source in a free field,

so pressure amplitude varies approximately as . The change in sound pressure level between distances and is

Thus, doubling the distance causes an approximate reduction.

Atmospheric absorption converts acoustic energy into heat through:

• viscous losses,
• thermal conduction,
• rotational and vibrational relaxation of atmospheric molecules.

If the absorption coefficient is in , the total level can be approximated by

Absorption increases strongly with frequency and also depends on temperature, humidity, and pressure.

Long-range outdoor sound propagation is controlled by several interacting mechanisms:

• Spherical spreading: Produces an approximately decrease per doubling of distance for a point source in a free field.
• Atmospheric absorption: Preferentially attenuates high-frequency components.
• Refraction: Temperature and wind gradients bend sound rays and may create focusing or shadow zones.
• Ground effect: Direct and ground-reflected waves interfere. The result depends on source height, receiver height, frequency, and ground impedance.
• Turbulence: Random temperature and velocity fluctuations scatter sound and cause fluctuating amplitude and phase.
• Obstacles and terrain: Buildings, barriers, and hills produce reflection, diffraction, and shielding.
• Atmospheric layering: Inversions and wind shear can form acoustic ducts that permit unusually long propagation.

A reliable prediction must therefore include source directivity, meteorology, terrain, ground impedance, and frequency-dependent losses.

Let a uniform fluid have mean density , sound speed , and constant velocity . Introduce small disturbances , , and .

The linearized continuity equation is

and the linearized momentum equation is

where the mean-flow material derivative is

For isentropic disturbances,

Apply to the continuity equation and take the divergence of the momentum equation. Eliminating and gives

Therefore,

This is the convected wave equation. It shows that acoustic disturbances propagate at speed relative to the fluid while being convected by the mean flow.

Consider a source of frequency moving with speed through a stationary medium of sound speed . During one source period , the source moves a distance .

For a source approaching a stationary observer, the wavelength ahead of the source is

Hence, the observed frequency is

For a receding source,

Using a sign convention in which the source velocity component toward the observer is positive,

The effect results from compression or expansion of successive wavefronts. It must be distinguished from convection: frequency is determined relative to the observer, whereas wave propagation speed is measured relative to the medium.

A moving source emits wavefronts centered at its successive emission positions.

• Subsonic motion, : Each wavefront expands faster than the source moves. Wavefronts are compressed in front and expanded behind, causing Doppler shift and directional amplification.
• Sonic motion, : The source travels with its own wavefronts, causing strong accumulation of disturbances.
• Supersonic motion, : The source outruns individual wavefronts. Their envelope forms a Mach cone.

From the wavefront geometry,

where is the Mach angle and .

The cone represents the locus at which disturbances emitted at different times arrive together. A pressure discontinuity or rapid pressure change may be perceived as a sonic boom. As Mach number increases, decreases and the cone becomes narrower.

Consider the inhomogeneous wave equation

Let the Green function satisfy

with the required boundary and causality conditions.

Multiply the field equation by , multiply the Green-function equation by , subtract, and integrate over a space-time domain. Green's second identity transforms the Laplacian terms into a boundary integral. The resulting representation is

with signs adjusted consistently to the chosen definition of and .

• The volume term represents distributed sources.
• The surface term represents boundary radiation or scattering.
• A retarded Green function ensures that only earlier source events contribute to the present field.

This formula generalizes the solution of acoustic radiation problems to include sources, initial data, and boundary effects.

For the three-dimensional wave operator, the causal free-space Green function can be written as

where

Its physical features are:

• The factor represents spherical spreading.
• The delta function enforces the propagation delay .
• The Green function is zero for events that cannot causally influence the observer.

For an equation of the form

the free-space solution is

The quantity is the retarded time. Therefore, the field measured now depends on the source strength at the earlier time required for sound to travel from the source point to the observer.

A Green function is selected to satisfy both the governing wave equation and the physical boundary conditions.

• Free-space Green function: It represents an unbounded medium and satisfies an outgoing radiation condition. No reflecting boundary is present.
• Rigid-boundary Green function: At a stationary rigid wall, normal particle velocity is zero. For harmonic acoustics this corresponds to the Neumann condition
  
  An image source of the same sign is commonly used for an infinite rigid plane.
• Pressure-release Green function: Acoustic pressure vanishes at the boundary, giving the Dirichlet condition
  
  An image source of opposite sign can represent an infinite pressure-release plane.

Choosing a Green function that already satisfies the boundary condition can eliminate part of the surface integral in Green's formula. This simplifies radiation and scattering calculations and gives the reflected field automatically.

Begin with exact conservation of mass and momentum:

Differentiate continuity with respect to time:

Take the divergence of momentum and use it to eliminate the mixed derivative. This yields

Add and subtract and define . The result is

where the Lighthill stress tensor is

The left side is a linear wave operator in a uniform medium, while the right side represents effective sound sources generated by the nonlinear flow.

The Lighthill tensor is

Its terms are:

• Reynolds-stress term, : Represents nonlinear momentum flux and is generally dominant in high-Reynolds-number turbulent flows.
• Entropy or nonlinear pressure term, : Accounts for departure from the reference isentropic pressure-density relation.
• Viscous stress term, : Represents molecular viscous effects and is often small in high-Reynolds-number free turbulence.

The source in Lighthill's equation is a double divergence:

Two spatial derivatives give the source the mathematical character of a quadrupole distribution. In free turbulence there is no solid surface to exert a net fluctuating force and no net fluctuating mass injection, so monopole and dipole contributions are usually absent or weaker. Solid boundaries, heat release, or mass addition can introduce stronger dipole or monopole components.

For a geometrically similar, compact, subsonic turbulent jet, dimensional reasoning applied to Lighthill's quadrupole source gives the approximate acoustic-power law

where is a characteristic jet velocity, is a characteristic jet diameter, and and are ambient sound speed and density.

Relative to a characteristic jet mechanical power , the acoustic efficiency scales as

where . This is commonly called the eighth-power law because under the assumed conditions.

Consequently, a modest increase in jet speed can cause a very large increase in noise. The law is an asymptotic scaling rather than a universal exact result; temperature differences, supersonic convection, shocks, nozzle geometry, and changes in turbulence structure can alter the exponent and directivity.

A standing wave is produced by the superposition of two waves of equal frequency and amplitude traveling in opposite directions. Let

Adding them gives

The spatial amplitude is .

• Pressure antinodes occur when :
  
• Pressure nodes occur when :
  

Adjacent pressure nodes, or adjacent pressure antinodes, are separated by . A pressure node and its nearest pressure antinode are separated by . Pressure and particle-velocity standing-wave patterns are spatially displaced by one-quarter wavelength.

A typical standing-wave apparatus consists of a long uniform tube, a loudspeaker or sound source at one end, and an adjustable termination or movable microphone/probe. The source is driven at a known frequency .

Procedure:

• Produce a steady sinusoidal sound in the tube.
• Move the microphone or probe along the tube and record pressure amplitude.
• Identify successive pressure maxima or successive pressure minima.
• Measure their separation .
• Since adjacent corresponding extrema are separated by half a wavelength,
  
• Calculate sound speed from
  

For improved accuracy, measure the distance across half-wavelength intervals:

The experiment should minimize frequency drift, background noise, tube leakage, and uncertainty in locating broad maxima or minima.

Open-open tube: Pressure nodes occur approximately at both open ends. The tube length contains an integer number of half-wavelengths:

Therefore,

All integer harmonics are permitted.

Closed-open tube: A pressure antinode occurs at the rigid closed end and a pressure node at the open end. Thus,

so

Only odd harmonics occur in the ideal model.

At an open end, the pressure node lies slightly outside the physical tube because the external air also oscillates. The effective length is therefore

For an unflanged circular opening, a commonly used approximation is per open end, where is the tube radius. Resonance formulas should use for improved accuracy.

Progressive wave:

• The waveform travels through the medium.
• Pressure and particle velocity are in phase for a plane wave traveling in the positive direction.
• The specific acoustic impedance is real:
  
• It transports nonzero time-averaged acoustic intensity:
  
• Amplitude is spatially uniform in an ideal lossless plane wave.

Standing wave:

• The spatial node-antinode pattern remains fixed.
• Pressure nodes coincide with velocity antinodes, and pressure antinodes coincide with velocity nodes.
• Pressure and particle velocity are in temporal quadrature at most positions.
• The local impedance is position-dependent and mainly reactive in an ideal lossless system.
• Equal counter-propagating components produce zero net time-averaged energy transport, although energy alternates locally between kinetic and compressional forms.

A partially standing field occurs when incident and reflected amplitudes are unequal.

Beam width specifies the angular extent of the main radiation lobe of a directional source. The most common measure is the half-power beam width, also called the beam width.

If the normalized intensity pattern is , the half-power angles satisfy

Since intensity is proportional to squared pressure, the corresponding pressure ratio is

The level difference is

The angular separation between the two half-power points around the principal axis is the half-power beam width. A smaller beam width indicates stronger concentration of acoustic energy and generally greater directivity. Beam width usually decreases when source size or frequency increases relative to wavelength.

The directivity factor is the ratio of intensity in a specified direction to the intensity that an omnidirectional source would produce at the same distance with the same total acoustic power:

In the direction of maximum radiation,

The directivity index is the logarithmic form:

Thus, corresponds to , while corresponds to approximately .

For a normalized intensity pattern , define the beam solid angle as

Since total power is ,

and therefore

A small beam solid angle consequently produces a large directivity index.

For a uniformly vibrating circular piston of radius mounted in an infinite rigid baffle, contributions from different surface elements have angle-dependent phase differences. Far-field integration gives the normalized pressure directivity

where is the first-order Bessel function and

At , the limiting value is .

The normalized intensity pattern is

The first null occurs when the argument reaches the first zero of :

Using piston diameter ,

• If , the piston is small compared with wavelength and radiates broadly.
• As increases, the main lobe narrows and directivity increases.
• Higher frequency or larger diameter therefore reduces beam width.
• At sufficiently large , sidelobes appear because of interference across the piston face.