Radiation from Simple Sources

Understanding how sound radiates from the voice requires first examining radiation from idealized simple sources. These theoretical models—particularly the monopole (point source) and dipole—provide essential insights into the directional patterns, efficiency, and frequency dependence of acoustic radiation that ultimately apply to the complex reality of speech and singing.

The Monopole (Simple Point Source)

The monopole represents the simplest theoretical sound source: a pulsating sphere of infinitesimal size that radiates uniformly in all directions.

Physical Model

Conceptual Description

Imaginary sphere expanding and contracting radially
Uniform surface velocity in all directions
No preferred direction of radiation
Creates spherical wave fronts propagating outward
Also called “simple source” or “breathing sphere”

Mathematical Characteristics

Volume velocity: rate of volume change (m³/s)
Strength parameter: Q = volume velocity amplitude
Radius assumed infinitesimally small
Only property: magnitude of oscillation
No directional characteristics in source itself

Radiation Pattern

Radiation patterns from simple sources Figure 9.2: Radiation patterns showing omnidirectional monopole (spherical) and figure-eight dipole patterns in free space.

Omnidirectional Radiation

Equal sound pressure at equal distances in all directions
Spherical wavefronts expand uniformly
Intensity decreases as 1/r² (inverse-square law)
Pressure decreases as 1/r (inverse distance)
No directivity: same output in all directions

Practical Implications

Provides baseline for comparing directional sources
Approximates radiation from small openings at low frequencies
Useful for understanding far-field behavior
Simplifies calculations of total radiated power

Acoustic Pressure from a Monopole

At distance r from a monopole source:

p(r,t) = (ρω²Q/4πrc) cos(ωt - kr)

Where:

p: acoustic pressure (Pa)
ρ: air density (kg/m³)
ω: angular frequency (rad/s)
Q: volume velocity amplitude (m³/s)
r: distance from source (m)
c: sound velocity (m/s)
k: wave number = ω/c (rad/m)

Key Observations

Pressure proportional to ω² (frequency squared)
Higher frequencies radiate more efficiently
Pressure inversely proportional to r
Phase lag kr accumulates with distance

Acoustic Intensity from a Monopole

The time-averaged intensity (power per unit area):

I = (ρω⁴Q²)/(32π²r²c)

Important Features

Intensity proportional to ω⁴ (frequency to fourth power)
Extremely strong frequency dependence
Intensity falls as 1/r² (inverse-square law)
Total power independent of distance (power conserved)

Frequency Dependence Implications

Doubling frequency quadruples pressure (12 dB increase)
Doubling frequency increases intensity 16-fold (12 dB + 12 dB = 24 dB)
Low frequencies radiate very inefficiently
High frequencies dominate radiated sound

The Dipole Source

A dipole consists of two monopole sources of equal strength but opposite phase placed infinitesimally close together.

Physical Model

Conceptual Description

Two point sources separated by small distance
Equal strength but 180° out of phase
One source expanding while other contracts
Creates pressure/rarefaction pattern with directivity
Also called “doublet” source

Physical Realization

Approximated by vibrating sphere (front/back motion)
Models sound from small vibrating surfaces
Represents oscillating force in fluid
Two sides of vibrating surface act as opposite-phase sources

Mathematical Representation

Characterized by source strength (dipole moment)
Dipole moment = source separation × monopole strength
Limiting case: separation approaches zero
Strength adjusted to maintain finite dipole moment

Radiation Pattern

Figure-Eight Pattern

Maximum radiation along axis of dipole
Zero radiation perpendicular to axis
“Figure-eight” or “dumbbell” shape in 3D
Front and back lobes of opposite phase
Strong directivity even at low frequencies

Directional Characteristics

Cosine directivity: pressure ∝ cos(θ)
θ = angle from dipole axis
At θ = 0° and 180°: maximum radiation
At θ = 90° and 270°: zero radiation (null)
Front/back lobes 180° out of phase

Acoustic Pressure from a Dipole

At distance r and angle θ from dipole axis:

p(r,θ,t) = (ρω²Qd cos(θ))/(4πrc) cos(ωt - kr)

Where:

d: effective dipole separation
θ: angle from dipole axis
cos(θ): directivity factor
Other parameters as defined for monopole

Comparison with Monopole

Additional cos(θ) term creates directivity
Same frequency dependence (ω²) for pressure
Same distance dependence (1/r) for pressure
But different total radiated power

Acoustic Intensity from a Dipole

The time-averaged intensity:

I = (ρω⁴Q²d² cos²(θ))/(32π²r²c)

Directional Features

Maximum on-axis (θ = 0°, cos²(θ) = 1)
Zero perpendicular to axis (θ = 90°, cos²(θ) = 0)
Same ω⁴ frequency dependence as monopole
But lower total radiated power (by factor related to (kd)²)

Comparison of Monopole and Dipole

Understanding the differences between these idealized sources illuminates voice radiation:

Directivity

Monopole

Omnidirectional: uniform in all directions
Spherically symmetric radiation
No preferred direction
Directivity index: 0 dB

Dipole

Highly directional: figure-eight pattern
Maximum on-axis, zero perpendicular
Strong directional preferences
Directivity varies with angle

Radiation Efficiency

Monopole

Total radiated power independent of frequency (at constant Q)
Actually, at constant amplitude: power ∝ ω⁴
High-frequency radiation efficient
Good radiator at all frequencies (relative to dipole)

Dipole

Total radiated power reduced compared to monopole
Cancellation between opposite-phase sources
Efficiency proportional to (kd)²
Very inefficient at low frequencies when kd << 1

Frequency Dependence

Common Features

Both show pressure ∝ ω²
Both show intensity ∝ ω⁴
Strong preference for high frequencies
Low-frequency radiation inherently difficult

Different Features

Monopole total power independent of frequency (constant Q)
Dipole total power additionally reduced at low frequencies
Dipole has extra factor (kd)² reducing low-frequency output

Relevance to Voice Production

Real voice radiation combines aspects of both monopole and dipole characteristics:

Glottal Source as Volume Velocity

Monopole-Like Behavior

Glottal airflow creates volume velocity into vocal tract
Acts as monopole source at vocal tract input
Flow pulses generate acoustic pressure waves
Drives resonances of vocal tract

Limitations

Source is not in free space (confined to tract)
Source is not point-like (distributed along glottis)
Flow is pulsatile (not sinusoidal)
Strong nonlinearities in flow

Mouth Opening as Complex Source

Low Frequencies (Long Wavelengths)

Wavelength >> mouth dimensions
Mouth acts approximately as monopole
Nearly omnidirectional radiation
Efficient radiation into hemisphere

High Frequencies (Short Wavelengths)

Wavelength comparable to mouth dimensions
Directional effects become important
Some dipole-like characteristics
Beaming in forward direction

Practical Consequences

For Listeners

Voice relatively omnidirectional at low frequencies
Directivity increases with frequency
Forward direction favored at high frequencies
Back of head attenuates high frequencies

For Speakers/Singers

Low-frequency energy radiates in all directions
High-frequency energy projects forward
Turning away from listener reduces intelligibility
Acoustic energy heard by speaker differs from listener’s experience

Near Field versus Far Field

The distinction between near and far fields affects radiation characteristics:

Near Field (Close to Source)

Definition

Distance r << λ (wavelength)
Typically within one wavelength of source
For speech (500 Hz): λ = 0.69 m, so near field < ~0.7 m
For high frequencies (4000 Hz): λ = 8.6 cm, near field < ~10 cm

Characteristics

Pressure and velocity not in phase
Reactive component dominates
Oscillating energy storage (like spring)
Intensity difficult to define
Pressure field complex and source-dependent

Clinical Relevance

Most voice assessment in near field
Microphone placement critical
Distance variations affect measurements
Standardization essential for comparison

Far Field (Distant from Source)

Definition

Distance r >> λ
Typically beyond several wavelengths
Pressure and velocity in phase
Simple progressive wave

Characteristics

Intensity well-defined
Inverse-square law applies cleanly
Directivity patterns stable
Source appears point-like

Practical Application

Most listeners in far field for speech
Simple acoustic relationships
SPL predictions reliable
Distance effects predictable

Acoustic Radiation Impedance

The relationship between volume velocity and resulting pressure defines radiation impedance:

For Monopole

Z_rad = R_rad + jX_rad

Where:

R_rad: radiation resistance (real part, dissipative)
X_rad: radiation reactance (imaginary part, reactive)
j: imaginary unit

Low Frequency (ka << 1, where a = source radius)

R_rad ≈ ρc(ka)² (very small)
X_rad ≈ ρcka (inertive reactance)
Reactance dominates
Poor radiation efficiency

High Frequency (ka >> 1)

R_rad approaches ρc (radiation resistance constant)
X_rad becomes negligible
Real impedance dominates
Good radiation efficiency

Physical Interpretation

Radiation Resistance

Represents power loss to acoustic radiation
Energy carried away by sound waves
Increases with frequency
Determines radiation efficiency

Radiation Reactance

Represents reactive energy storage
Energy oscillates near source
Does not propagate away
Inertive (mass-like) for monopole

Application to Mouth Radiation

The mouth opening can be approximated as a monopole at low frequencies:

Effective Radiation

At the mouth, the volume velocity from the vocal tract drives radiation:

Low frequencies: nearly omnidirectional (monopole-like)
Radiation impedance controls transfer from tract to free space
Resistance increases with frequency
Affects vowel spectra (boosts high frequencies)

Radiation Characteristic Transfer Function

The transfer from volume velocity to radiated pressure shows:

+6 dB per octave increase (pressure ∝ frequency)
High-pass filter characteristic
Emphasizes high-frequency components
Contributes to spectral slope of speech

Summary

Acoustic radiation from simple idealized sources provides fundamental understanding applicable to voice production. The monopole (point source) radiates omnidirectionally with pressure proportional to frequency squared and intensity to frequency to the fourth power, exhibiting strong high-frequency preference but no directional bias. The dipole consists of two opposite-phase monopoles creating a figure-eight radiation pattern with the same frequency dependence but reduced efficiency, especially at low frequencies.

Both source types demonstrate that radiation efficiency increases dramatically with frequency, explaining why high-frequency acoustic energy radiates more effectively than low-frequency energy. The distinction between near field (within a wavelength) and far field (beyond several wavelengths) affects measurement and perception, with most clinical voice assessment occurring in the near field where relationships are more complex. Radiation impedance, particularly its frequency-dependent resistance, determines the efficiency with which source volume velocity converts to radiated acoustic power.

For voice production, the glottal source generates volume velocity that acts as a monopole input to the vocal tract, while the mouth opening radiates this acoustic energy with characteristics intermediate between idealized monopole and dipole sources, becoming increasingly directional at high frequencies. Understanding simple source radiation patterns enables prediction of real voice radiation behavior and informs clinical measurement protocols and performance strategies.

Key Takeaways

✅ Monopole (point source) radiates omnidirectionally with spherical wavefronts and no directional preference
✅ Dipole creates figure-eight pattern with maximum on-axis radiation and nulls perpendicular to axis
✅ Both monopole and dipole show pressure ∝ ω² and intensity ∝ ω⁴, strongly favoring high frequencies
✅ Dipole is less efficient than monopole, especially at low frequencies, due to cancellation between opposite-phase sources
✅ Near field (r << λ) has complex pressure/velocity relationships; far field (r >> λ) follows simple inverse-square law
✅ Radiation impedance determines conversion efficiency from volume velocity to radiated pressure
✅ Radiation resistance (real part) increases with frequency, explaining high-frequency radiation advantage
✅ Mouth radiation combines monopole and dipole characteristics, becoming directional at high frequencies

Radiation from Simple Sources

Radiation from Simple Sources

The Monopole (Simple Point Source)

Physical Model

Radiation Pattern

Acoustic Pressure from a Monopole

Acoustic Intensity from a Monopole

The Dipole Source

Physical Model

Radiation Pattern

Acoustic Pressure from a Dipole

Acoustic Intensity from a Dipole

Comparison of Monopole and Dipole

Directivity

Radiation Efficiency

Frequency Dependence

Relevance to Voice Production

Glottal Source as Volume Velocity

Mouth Opening as Complex Source

Practical Consequences

Near Field versus Far Field

Near Field (Close to Source)

Far Field (Distant from Source)

Acoustic Radiation Impedance

For Monopole

Physical Interpretation

Application to Mouth Radiation

Effective Radiation

Radiation Characteristic Transfer Function

Summary

Key Takeaways

Related Topics

Further Reading