Radiation of Sound

The radiation of sound from the mouth represents the final stage in the voice production process, where acoustic energy generated at the glottis and shaped by the vocal tract propagates into the surrounding air. Understanding sound radiation principles is essential for interpreting vocal intensity measurements, comparing studies across different measurement conditions, and predicting how voice will carry in various environments.

The Mouth as a Sound Source

Radiation Characteristics

The mouth opening serves as the primary sound source for speech and singing, radiating acoustic energy into a hemisphere of space in front of the speaker. While some sound energy escapes through the nasal cavity during nasal consonants and nasalized vowels, and minimal energy transmits through the neck and chest walls, the vast majority of acoustic power exits through the oral aperture.

The mouth functions approximately as a simple source or monopole radiator at frequencies where the wavelength of sound is large compared to the dimensions of the mouth opening. For frequencies below approximately 500 Hz, this approximation holds well, and sound radiates nearly uniformly in all forward directions.

Directional Characteristics

At higher frequencies, where wavelength becomes comparable to or smaller than mouth dimensions, radiation becomes increasingly directional. Sound radiates more efficiently directly in front of the mouth than to the sides. This directional effect becomes more pronounced as frequency increases:

• Below 500 Hz: Nearly omnidirectional radiation into the forward hemisphere
• 500-2000 Hz: Moderate directionality, with 3-6 dB difference between frontal and lateral directions
• Above 2000 Hz: Strong directionality, with up to 10-15 dB difference between frontal and off-axis measurements

This frequency-dependent directionality has important implications for microphone placement in recording and measurement situations, and for understanding how voice projects in performance spaces.

The Inverse Square Law

Fundamental Principle

As sound radiates from a source, acoustic power spreads over an increasingly large area. For a simple source radiating into free space, the surface area of a sphere increases with the square of the radius:

A = 4πr²

where:

• A = surface area (m²)
• r = distance from source (m)

Since the total acoustic power (P) remains constant as it radiates outward (energy is conserved), the intensity (I) at any distance must equal the power divided by the area:

I = P / (4πr²)

This relationship demonstrates that intensity decreases as the inverse square of distance. Doubling the distance from the source reduces intensity to one-quarter of its original value.

Figure 9.2: Radiation of sound into the space in front of a speaker’s mouth, illustrating how acoustic power spreads over an increasingly large spherical surface as distance increases.

Practical Consequences

The inverse square law has several important practical implications:

Distance Specification: Any measurement of vocal intensity must specify the measurement distance. A sound pressure level of 85 dB at 1 meter represents four times the radiated power of 85 dB at 2 meters, even though the measured values are identical.

Standardization: Clinical voice assessments commonly use a standard distance of 30 cm (approximately 1 foot) or 1 meter to enable comparison across patients and studies. Without distance standardization, intensity measurements become difficult to interpret.

Environmental Adaptation: Speakers unconsciously adjust vocal intensity based on distance to listeners. In a small room with listeners at 1-2 meters, conversational speech at 60-65 dB SPL (at 1 meter) suffices. In a large auditorium with listeners at 10-20 meters, projected speech at 75-85 dB SPL (at 1 meter) becomes necessary to maintain the same perceived loudness.

Acoustic Power and Sound Pressure Level

Relationship Between Power and SPL

Sound pressure level (SPL) measured in decibels represents a logarithmic scale that compresses the enormous dynamic range of human hearing into a manageable numerical range. The relationship between acoustic power and sound pressure level depends on both the radiated power and the measurement distance.

For a simple source radiating into a hemisphere (appropriate for mouth radiation), the sound pressure level at distance r relates to acoustic power as:

SPL = PWL - 20 log₁₀(r) - 8 dB

where:

• SPL = sound pressure level (dB re 20 μPa)
• PWL = power level (dB re 10⁻¹² watts)
• r = distance in meters

The power level (PWL) is defined as:

PWL = 10 log₁₀(P / P₀)

where P₀ = 10⁻¹² watts (reference power).

Distance Effects on SPL

From the relationship above, we can derive that for each doubling of distance, sound pressure level decreases by approximately 6 dB:

20 log₁₀(2) ≈ 6 dB

This means:

• At 0.5 m: SPL = PWL - 8 dB
• At 1.0 m: SPL = PWL - 14 dB
• At 2.0 m: SPL = PWL - 20 dB
• At 4.0 m: SPL = PWL - 26 dB

A speaker producing 80 dB SPL at 1 meter radiates the same acoustic power as a speaker producing 74 dB SPL at 2 meters.

Typical Acoustic Power Values

Speech Production

Normal conversational speech produces remarkably small amounts of acoustic power:

Typical Values:

• Quiet speech: 10⁻⁶ to 10⁻⁵ watts (PWL = 60-70 dB)
• Conversational speech: 10⁻⁵ to 10⁻⁴ watts (PWL = 70-80 dB)
• Loud speech: 10⁻⁴ to 10⁻³ watts (PWL = 80-90 dB)
• Shouting: 10⁻³ to 10⁻² watts (PWL = 90-100 dB)

These values may seem surprisingly small compared to other sound sources, but they reflect the extraordinary sensitivity of the human auditory system, which can detect intensities as low as 10⁻¹² watts/m² at the threshold of hearing.

Singing

Trained singers, particularly in classical and operatic styles, develop the ability to produce greater acoustic power:

Typical Values:

• Soft singing (pianissimo): 10⁻⁵ to 10⁻⁴ watts
• Moderate singing (mezzo-forte): 10⁻⁴ to 10⁻³ watts
• Loud singing (fortissimo): 10⁻³ to 10⁻² watts
• Maximum trained voice: up to 0.1 watts (exceptional)

The maximum acoustic power produced by operatic sopranos and tenors at their highest intensities can reach 0.01 to 0.1 watts, representing a thousand-fold increase over conversational speech. This remarkable range enables singers to be heard over orchestral accompaniment in large performance halls.

Measurement Considerations

Microphone Placement

Standard practices for voice intensity measurement include:

Clinical Assessment:

• Distance: 30 cm or 1 meter from lips
• Angle: 0° (directly in front) or 45° (to avoid direct breath stream)
• Environment: Sound-treated room to minimize reflections
• Calibration: Regular SPL meter calibration with reference tones

Research Conditions:

• Distance: Most commonly 1 meter, clearly specified
• Angle: Typically 0° for maximum output or 45° for representative conditions
• Free-field conditions: Anechoic chamber or large room to minimize reflections
• Multiple positions: Sometimes measurements at several angles to characterize directivity

Converting SPL to Acoustic Power

To estimate acoustic power from a measured SPL value:

1. Measure SPL at known distance r and angle
2. Apply directivity correction if not measured at 0°
3. Calculate power level: PWL = SPL + 20 log₁₀(r) + 8 dB
4. Convert to watts: P = 10⁻¹² × 10^(PWL/10)

This conversion assumes hemispherical radiation and free-field conditions (no reflections). In real environments, room acoustics complicate this relationship, but the basic inverse square law still applies to the direct sound field.

Environmental and Contextual Factors

Room Acoustics

In enclosed spaces, sound reflects from walls, floor, and ceiling, creating a reverberant field that adds to the direct sound. The relationship between acoustic power and measured SPL becomes more complex:

• Near field (very close to mouth): Direct sound dominates, inverse square law applies
• Transition zone: Direct and reflected sound contribute comparably
• Far field (large distance): In small rooms, reflected sound dominates; SPL decreases less rapidly with distance

In typical rooms, the reverberant energy effectively raises the SPL at distant positions compared to free-field predictions, making voices more audible than they would be outdoors.

Atmospheric Absorption

Over long distances (tens of meters or more), air itself absorbs acoustic energy, particularly at high frequencies. This absorption increases with:

• Higher frequencies
• Lower humidity
• Higher temperature

For typical speech and singing distances (less than 20 meters), atmospheric absorption remains negligible. However, in large auditoriums or outdoor spaces, high-frequency energy attenuates more rapidly than low-frequency energy, affecting voice quality and intelligibility at distance.

Summary

Sound radiation from the mouth follows fundamental acoustic principles that govern the relationship between total acoustic power, intensity at specific locations, and measured sound pressure level. The mouth functions as an approximately simple source at low frequencies, with radiation becoming increasingly directional at higher frequencies.

The inverse square law dictates that intensity decreases with the square of distance from the source, requiring that all intensity measurements specify measurement distance for meaningful interpretation. Typical speech produces acoustic powers in the range of 10⁻⁵ to 10⁻³ watts, while trained singers can achieve up to 0.1 watts at maximum output.

Understanding these radiation principles enables accurate interpretation of voice measurements, appropriate standardization of clinical assessment protocols, and prediction of how vocal intensity will vary across different communication environments.

Key Takeaways

• ✅ The mouth radiates sound approximately as a simple source at low frequencies, becoming directional above 500 Hz
• ✅ Intensity decreases according to the inverse square law: doubling distance reduces intensity to one-quarter
• ✅ Sound pressure level decreases by 6 dB for each doubling of distance from the source
• ✅ Vocal intensity measurements must always specify measurement distance for meaningful interpretation
• ✅ Normal conversational speech produces 10⁻⁵ to 10⁻⁴ watts of acoustic power
• ✅ Trained singers can produce up to 0.1 watts at maximum intensity, about 1000 times conversational levels
• ✅ Converting between SPL and acoustic power requires knowing measurement distance and assuming radiation geometry

Related Topics

• Some Definitions of Terms
• Variation of Intensity with Adduction, Lung Pressure, and F₀
• Variation of Intensity with Vocal Tract Adjustments
• Acoustic Theory of Speech Production

Further Reading

1. Beranek, L. L. (1986). Acoustics. New York: American Institute of Physics.
2. Rossing, T. D., Moore, F. R., & Wheeler, P. A. (2002). The science of sound (3rd ed.). San Francisco: Addison Wesley.
3. Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J. V. (1982). Fundamentals of acoustics (3rd ed.). New York: John Wiley & Sons.
4. Sundberg, J. (1987). The science of the singing voice. DeKalb, IL: Northern Illinois University Press.