Acoustics

Acoustics is the science of sound, including its generation, transmission, and perception. Sound is a pressure wave that travels through a medium via particle oscillations, and therefore sound cannot travel in a vacuum. The term "sound" includes audible frequencies as well as ultrasound, infrasound, and other pressure waves, such as the seismic waves from earthquakes, which are felt as vibrations. For health care professionals, the main applications are audible sound and technologies such as sonography, in which ultrasound waves are sent into tissue and the reflected waves are detected to create an image.

The most important equations in acoustics that health care professionals may encounter are included in the following sections and illustrated with simple examples. For practical purposes, it is sufficient to know the basic formulas, which can be easily rearranged or used for substitution. If difficulties arise, see "Mathematics" for help. Only the physical properties of the auditory perception of sound are described here. Information on the auditory pathway is covered in "The ear."

Production of sound

• Vibrating source: Sound originates from the mechanical vibration of an object in a medium (e.g., solid, liquid, gas).
• Mechanical wave: The vibration transfers energy to the surrounding medium, creating a wave that propagates outward.
  • It requires a medium to travel and cannot propagate in a vacuum.
  • The particles of the medium oscillate around their equilibrium positions but do not travel with the wave.
• Longitudinal wave: The oscillations of the medium's particles are parallel to the direction of the wave's propagation.
• Pressure variations: The wave consists of alternating regions of high and low pressure.
  • Compressions: regions of higher density and pressure
  • Rarefactions: regions of lower density and pressure

Sound frequency

• Definition: the frequency of a sound wave
  • Indicates how quickly the oscillations repeat
  • Determines the pitch
  • Independent of the medium through which the sound propagates
• Formula: f = c/λ
  • Unit: hertz (Hz) or 1/s
  • f = sound frequency (Hz or 1/s), c = speed of propagation (m/s), λ = wavelength (m)
• Pitch: the subjective perception of sound frequency by the human ear
  • A high frequency is perceived as a high pitch.
  • A low frequency is perceived as a low pitch.
  • It is a psychoacoustic property, not a purely physical one.
• Period: the reciprocal of the frequency
  • Formula: T = 1 / f
    • Unit: seconds (s)
    • T = period (s), f = frequency (Hz)

The higher the frequency, the higher the pitch and the shorter the wavelength.

Speed of sound

• Definition: the speed at which sound propagates through a medium
• Dependence on medium properties: The speed of sound is determined by the resistance of the medium to compression, its temperature, and its density.
  • Temperature: increased temperature → higher kinetic energy → increased speed of sound
  • Bulk modulus (B): a measure of a substance's stiffness or resistance to compression; a higher bulk modulus allows for faster sound propagation
  • Density (ρ): a measure of mass per unit volume; a higher density slows down sound propagation
  • General trend: v_solid > v_liquid > v_gas
    • The speed of sound in air at room temperature is 343 m/s.
    • Sound travels much slower than light (approx. 300,000,000 m/s).
• Formula: c = √(B/ρ)
  • Unit: m/s
  • c = speed of sound (m/s), B = bulk modulus of the medium (Pa or N/m²), ρ = density of the medium (kg/m³)

Example calculation: speed of sound

Sound moves through a medium at 25 m/s. How does the wavelength change if the frequency increases from 50 Hz to 150 Hz?

• Find: wavelength (λ)
• Given: speed of sound (c) = 25 m/s, initial frequency (f₁) = 50 Hz, new frequency (f₂) = 150 Hz
  • f = c/λ → λ = c/f
  • λ₁ = 25 m/s / 50 Hz = 0.5 m
  • λ₂ = 25 m/s / 150 Hz = 0.17 m
• As the frequency increases from 50 Hz to 150 Hz, the wavelength decreases from 0.5 m to approximately 0.17 m. This demonstrates that when the frequency triples, the wavelength decreases to about one-third of its original value.

The speed of sound increases with stiffness and temperature and decreases with density.

The speed of sound is typically fastest in solids, slower in liquids, and slowest in gases.

Sound pressure

• Definition: the amplitude of the pressure variations caused by a sound wave, which is related to sound field quantities
  • The pressure fluctuation represents the force exerted by the sound source on a specific area.
• Formula: p = I/v
  • Unit: pascal (Pa) or N/m²
  • p = sound pressure (Pa or N/m²), I = sound intensity (W/m²), v = particle velocity (m/s)

Sound pressure decreases as the distance from the sound source increases.

Sound intensity (I)

• Definition: an objective measure of the energy a sound wave transmits per unit time through a unit area
• Formulas
  • General formula: I = P/A
    • Unit: W/m²
    • I = sound intensity (W/m²), P = power (W or J/s), A = area (m²)
  • Point source formula: I = P/4πr²
    • I = sound intensity (W/m²), P = power (W or J/s), r = distance from point source (m)
    • Shows how intensity decreases with distance from a point source
  • Intensity and pressure relationship: I = p²/2pv_w
    • I = sound intensity (W/m²), p = change in pressure or amplitude (Pa), p = density of the material the sound is traveling through (kg/m³), v_w = speed of sound in the medium (m/s)
  • Intensity and amplitude relationship: I ∝ A²
    • I = intensity (W/m²), A = amplitude of the wave (Pa)
    • Doubling amplitude quadruples intensity
  • Distance effect: I ∝ 1/r²
    • I = intensity (W/m²), r = distance from source (m)
    • Shows that intensity decreases with the square of the distance from the source

Sound intensity decreases as the distance from the sound source increases, as intensity is inversely proportional to the distance squared (I ∝ 1/r²). If the distance from the sound source is doubled, the intensity decreases by a factor of four (1/2²). If the distance is tripled, intensity decreases by a factor of nine (1/3²).

If the amplitude of a sound wave is doubled, its intensity is quadrupled.

Sound power

• Definition: the power a sound source produces to generate sound; it is a sound energy quantity
• Formula: P = I × A
  • Unit: watt (W)
  • P = sound power (W or J/s), I = sound intensity (W/m²), A = area (m²)

Sound power is independent of the distance to the sound source.

Sound attenuation (damping)

• Definition: the decrease in the amplitude and intensity of a sound wave as it travels through a medium due to loss of energy
• Mechanisms
  • Absorption: primarily caused by viscous friction and molecular interactions, leading to energy loss in the form of heat
  • Scattering: occurs when sound waves encounter obstacles in a medium, causing the waves to deviate from their original path in multiple directions, leading to a loss of signal strength
  • Reflection: occurs when sound waves encounter a boundary between different media, resulting in a portion of the wave being redirected back toward the source
• Formula: I(x) = I₀ × e^-μx
  • Unit: W/m²
  • I(x) = sound intensity after interaction (W/m²), I₀ = original sound intensity (W/m²), e = Euler’s number, approximately 2.718 (unitless), μ = absorption coefficient (unitless), x = distance traveled (m)

The intensity of ultrasound waves decreases as they penetrate deeper into the body due to attenuation, leading to weaker returning echoes and potentially lower image resolution.

• Loudness: a subjectively perceived quantity, usually specified in phons
  • Isophones: curves on a diagram of frequency (x-axis) vs. sound level (y-axis) in which all tones on a given curve are perceived as equally loud
• Audible sound: sound with a frequency (approx. 20–20,000 Hz) and pressure (loudness) that is above the hearing threshold and therefore perceptible to the human ear
  • The ear is most sensitive to sound frequencies of 2000–5000 Hz.
• Infrasound: sound with a frequency too low to be heard by humans (approx. < 20 Hz)
• Ultrasound: sound with a frequency too high to be heard by humans (approx. > 20,000 Hz)

Sound level measurement

A logarithmic scale (decibel scale) is used to define sound levels because human hearing perceives loudness logarithmically across a wide range of intensities.

• Sound pressure level (L_p)
  • Formula: L_p = 20 × log(p/p₀)
    • Unit: decibel (dB)
    • L_p = sound pressure level (dB), p = effective sound pressure (Pa), p₀ = reference sound pressure (typically 2 × 10^-5 Pa)
  • Special case: adding multiple sound sources of equal loudness
    • Formula: L_total = L_pi + 10 × log(n)
      • L_total = total sound pressure level (dB), L_pi = sound pressure level of individual sources (dB), n = number of equally loud sound sources
    • Example: Adding 10 identical sources increases the sound pressure level by 10 dB, 100 sources by 20 dB, and 1000 sources by 30 dB.
• Sound intensity level (L_I)
  • Formula: L_I = 10 × log(I/I₀)
    • Unit: decibel (dB)
    • L_I = sound intensity level (dB), I = perceived sound intensity (W/m²), I₀ = reference intensity, typically the threshold of human hearing (1 × 10^-12 W/m²)
  • Examples
    • An increase of 10 dB corresponds to a 10-fold increase in sound intensity.
    • An increase of 20 dB corresponds to a 100-fold increase in sound intensity.
• Sound power level
  • Formula: L_P = 10 × log(P/P₀)
    • Unit: decibel (dB)
    • L_P = sound power level (dB), P = perceived sound power (W), P₀ = reference sound power (1 × 10^-12 W)

Due to the logarithmic relationship, a 20 dB increase in sound pressure level corresponds to a 10-fold increase in pressure relative to the reference value (since log(10) = 1). In contrast, a 20 dB increase in sound intensity level corresponds to a 100-fold increase in intensity relative to the reference value (since log(100) = 2).

The subjectively perceived loudness is usually specified in phons, while objective measures such as sound pressure, power, and intensity are specified in decibels. By definition, the decibel scale and the phon scale are identical at a frequency of 1000 Hz.

Example calculation 1: sound level

A patient has a hearing loss of 20 dB in the right ear. By what factor is the sound intensity heard by the left ear greater than that heard by the right?

• Find: ratio of sound intensities (I_left/I_right)
• Given: difference in sound intensity levels between ears (ΔL_I)
  • ΔL_I = 10 × log(I_left/I_right)
  • Rearranged: log(I_left/I_right) = ΔL_I/10 = 20/10 = 2
  • I_left/I_right = 10² = 100
  • Sound intensity heard by the left ear is 100 times greater than that heard by the right ear.

Example calculation 2: sound level

A person is exposed to two sounds with levels of 14 dB and 5 dB. What is the total sound level exposure?

• Find: total sound level (L_total)
• Given: sound levels L1 and L2
  • Sound levels are additive, but because the decibel is a logarithmic unit, the intensities must be added, not the dB values directly.
  • L_total= 10_log(10^L1/10 + 10^L2/10)
  • Convert sound level to intensity ratio.
    • 10^L1/10= 10^14/10 = 10^1.4 ≈ 25.12
    • 10^L2/10= 10^5/10 = 10^0.5 ≈ 3.162
  • Add the intensities: 25.12 + 3.162 = 28.282.
  • Convert back to decibels.
    • L_total= 10_log(28.282)
    • log(28.282) ≈ 1.45
    • 10 × 1.45 = 14.5 dB
  • The quieter sound makes very little difference when added to the louder sound.

Sound propagates uniformly in waves. These waves can be reflected from surfaces (e.g., an echo), diffracted, scattered depending on frequency (dispersion), or absorbed. These phenomena, similar to those seen with light , are important because they can be utilized in medical applications such as ultrasound.

Impedance

Sound is transmitted via oscillations of particles in a medium; therefore, sound propagates at different speeds depending on the properties of that medium. Even the air temperature influences the speed of sound. Acoustic impedance describes this material-dependent resistance to sound wave propagation.

• Acoustic impedance: a material-dependent constant that represents the relationship between the speed of sound and the density of the material
  • Formula: Z = ρ × c
    • Unit: pascal-seconds per cubic meter (Pa·s/m³) or rayls (Rayl)
    • Z = impedance (Pa·s/m³), ρ = density of the medium (kg/m³), c = speed of propagation (m/s)
  • Acoustic impedance increases with both the density (ρ) of the material and the speed of sound (c) within it.
  • The frequency of sound does not change when passing between two media with different acoustic impedances.
  • The greater the difference in acoustic impedance between two media, the more sound is reflected at the interface between them.

In sonography, a transducer emits ultrasound waves and receives their echos, which are processed into images, such as B-mode (grayscale) images. A coupling gel is applied to bridge the air gap between the transducer and tissue, reducing acoustic impedance mismatch and enhancing wave transmission. Higher sound frequencies improve image resolution but decrease penetration depth due to increased tissue absorption. For different examinations, adjusting the ultrasound frequency may require changing the transducer to optimize imaging.

Resonance (acoustics)

A forced oscillation occurs when an external source, such as a sound source, excites a system. The amplitude of this oscillation increases as the frequency of the external source approaches the natural frequency of the oscillating medium. When a system is continuously driven at or near its natural frequency, resonance occurs, leading to significant amplification of oscillations. While resonance is desirable in musical instruments, it can also result in a resonance catastrophe, in which excessive amplification causes structural failure or collapse of the system.

• Definition: amplification of oscillations that occurs when an external force drives a system at a frequency close to its natural frequency
• Related terms
  • Forced oscillation: occurs when an external source, such as a sound source, excites a system
  • Natural frequency or resonant frequency: the frequency at which a system would oscillate on its own, without external excitation or damping
  • Amplification: The amplitude of oscillation increases as the frequency of the external source approaches the natural frequency of the oscillating medium.
  • Resonance catastrophe: a sharp increase in oscillation amplitude that occurs when a system is excited at its natural frequency
• Related formulas: V = f × λ or λ = V/f
  • Unit: meters per second (m/s)
  • V = speed of the wave (m/s), f = frequency (Hz or 1/s), λ = wavelength (m)
  • If wave speed (V) remains constant (e.g., in a medium such as air) and frequency increases, the wavelength (λ) must decrease to maintain the equality.

In string instruments, when a string is plucked, it vibrates at its natural frequency. If the player changes the tension or length of the string, the frequency changes, which in turn affects the wavelength and pitch of the sound produced.

Resonance in pipes and strings

• Standing waves: Resonance in strings and pipes creates stable wave patterns called standing waves.
  • Nodes: points on the wave that remain stationary (zero amplitude)
  • Antinodes: points of maximum amplitude
  • The length of the string or pipe determines which frequencies will resonate within them.
• Strings and open pipes
  • For strings fixed at both ends (such as those on a guitar) and open pipes (such as a flute), the following conditions apply:
    • The ends of the string must be nodes.
    • The ends of the open pipe must be antinodes.
  • Relationship for the length of a string or open pipe: L = n × (λ/2), where n = 1, 2, 3, ... (harmonic number)
    • L = length of the pipe (m), n = harmonic number (unitless), λ = wavelength (m)
  • Allowed wavelengths: λ = 2L/n
    • λ = wavelength (m), L = length of the pipe (m), n = harmonic number (unitless)
  • Resonant frequencies: f = nv/2L
    • f = frequency (Hz or 1/s), n = resonance node (unitless), v = wave speed (m/s), L = length of the string or pipe (m)
• Closed pipes
  • For closed pipes (one end open and the other end closed), the following conditions must be met:
    • The closed end must be a node.
    • The open end must be an antinode.
  • Relationship for the length of a closed pipe: L = n_odd·λ/4 or L = (2n−1)·λ/4, where n = 1, 3, 5, ... (only odd harmonics are possible)
    • Allowed wavelengths: λ = 4L/n
    • Resonant frequencies: f = nv/4L

Pipes create longitudinal waves, whereas strings generate transverse waves.

Different hair cells in the human inner ear are tuned to different resonance frequencies along the cochlea, producing a continuous tonotopic map that covers many thousands of distinct frequencies across the audible range.

Doppler effect

Normally, a sound is perceived by an observer at the same frequency at which the source emitted it. However, if the distance between the sound source (emitter) and the observer changes, the frequency of the received sound also changes. This phenomenon is called the Doppler effect.

• Doppler effect: f_o = f_s (v ± v_o / v ± v_s)
  • f_o = observed frequency (Hz), f_s = actual frequency emitted by source (Hz), v = speed of the sound wave in the medium (m/s), v_o = speed of the observer relative to the medium (positive if moving toward the source, negative if moving away) (m/s), v_s = speed of the source relative to the medium (positive if moving away from the receiver, negative if moving toward) (m/s)
• Sign convention
  • When the observer moves toward the source, use the + sign for f_o in the numerator.
  • When the source moves toward the observer, use the - sign for v_s in the denominator.
  • Use the opposite signs for movement away from each other.
• Movement toward each other
  • The effective path of the sound is shortened.
  • As the frequency of the sound increases, the pitch is perceived as higher than at standstill.
  • The wavelength of the sound becomes shorter.
• Movement away from each other
  • The effective path of the sound is lengthened.
  • As the frequency of the sound decreases, the pitch is perceived as lower than at standstill.
  • The wavelength of the sound becomes longer.

"Toward" motion always increases the perceived frequency. An observer moving toward the source increases the frequency, so the numerator becomes larger: "+" is used (v + v_o). A source moving toward the observer also increases the frequency, so the denominator becomes smaller: "-" is used (v - v_s).

Shock waves

• Formation: Shock waves form when a sound source moves at or faster than the speed of sound (v_s ≥ c).
• Mechanism: The sound waves emitted by the source cannot propagate forward faster than the source is moving.
  • Wavefronts pile up in front of the source.
  • Constructive interference between the wavefronts creates a single, highly compressed wave of large amplitude.
• Characteristics
  • The wave propagates outwards in a conical shape behind the object.
  • Associated with a massive and rapid change in pressure, temperature, and density
  • Sonic boom: the audible effect produced when the shock wave passes an observer
• Mach number: an object's speed in relation to the speed of sound
  • Formula: M = v/v_s
    • Unit: unitless
    • M = Mach number (unitless), v = velocity of the object (m/s), v_s = speed of sound in the medium (m/s)

If Mach is 1, the object is traveling at the speed of sound.

The siren of a moving ambulance is perceived at different pitches due to the Doppler effect. As the ambulance approaches the observer, the pitch sounds higher. This occurs because the ambulance is moving in the same direction as the sound waves, effectively "chasing" them and increasing their frequency. Conversely, as the ambulance moves away from the observer, the pitch sounds lower. In this case, the ambulance is moving in the opposite direction of the sound waves, effectively "outrunning" them and decreasing their frequency. This change in pitch illustrates how relative motion between a sound source and an observer affects the perceived frequency of sound.

Doppler ultrasound uses the Doppler effect to assess both the morphology of structures and the movement of blood. Depending on the direction and speed of blood flow, components in the blood cause a change in the frequency of the ultrasound waves sent by the device. From this frequency shift between the emitted and received sound, the device can calculate the blood's flow velocity. This procedure is used particularly in the diagnosis of vascular and heart diseases.