Physics of Sound, Wave Propagation & Decibel Scales (SPL, HL, SL)

Physics of Sound, Wave Propagation & Decibel Scales

Acoustic energy is the fundamental physical stimulus driving auditory perception and audiological diagnostics. For the clinical audiologist, a deep, quantitative understanding of wave physics, acoustic impedance, and decibel scales is not merely theoretical—it is essential for equipment calibration, differential diagnosis of middle ear pathology, and appropriate selection and verification of amplification devices.

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Simple Harmonic Motion & Wave Propagation Fundamentals

Sound is generated when a vibrating object perturbs an elastic medium. This mechanical disturbance is transmitted as a longitudinal wave through sequential compressions (regions of increased particle density and positive sound pressure) and rarefactions (regions of decreased particle density and negative sound pressure).

Simple Harmonic Motion (SHM)

The simplest acoustic vibration is simple harmonic motion (SHM), producing a sinusoidal waveform. SHM is governed by two opposing physical properties:

• Inertia (Mass): The tendency of a body in motion to remain in motion, or at rest to remain at rest (Newton's first law).
• Elasticity (Stiffness): The restoring force that drives displaced particles back toward their equilibrium position, described by Hooke's Law ($F = -kx$, where $k$ is the stiffness constant and $x$ is displacement).

The instantaneous displacement $d(t)$ of a particle undergoing SHM as a function of time $t$ is expressed by the wave equation: $d(t) = A \sin(2\pi f t + \theta)$ where:

• $A$ is peak displacement amplitude
• $f$ is frequency in Hertz ($\text{Hz}$)
• $t$ is time in seconds ($\text{s}$)
• $\theta$ is starting phase angle in radians or degrees ($0^\circ, 90^\circ, 180^\circ, 270^\circ$)

Frequency, Period, and Phase

• Frequency ($f$): The number of complete oscillatory cycles completed per second, expressed in $\text{Hz}$ ($1\,\text{Hz} = 1\,\text{cycle/s}$).
• Period ($T$): The time required to complete one full cycle of vibration, measured in seconds or milliseconds ($T = \frac{1}{f}$). For example, a $1000\,\text{Hz}$ tone has a period of $T = \frac{1}{1000} = 0.001\,\text{s} = 1.0\,\text{ms}$, whereas a $250\,\text{Hz}$ tone has a period of $4.0\,\text{ms}$.
• Phase ($\theta$): The fractional part of a cycle through which an oscillating particle has advanced relative to an arbitrary starting reference point ($0^\circ$ at equilibrium moving positive, $90^\circ$ at positive peak, $180^\circ$ at equilibrium moving negative, $270^\circ$ at negative peak, and $360^\circ$ returning to equilibrium). Two identical pure tones combined $180^\circ$ out of phase perfectly cancel each other out (destructive interference).

Velocity of Sound and Wavelength

The velocity of sound ($c$) is strictly a property of the transmitting medium, determined by the medium's volume elasticity (bulk modulus, $E$) and equilibrium density ($\rho$): $c = \sqrt{\frac{E}{\rho}}$

Sound travels faster through stiffer (higher elasticity), less compressible media because restoring forces rapidly accelerate displaced particles:

• Air at $20^\circ\text{C}$ (dry, sea level): $c \approx 343\,\text{m/s}$ (or $1125\,\text{ft/s}$). Sound velocity in air increases with temperature ($\approx 0.6\,\text{m/s}$ per $1^\circ\text{C}$ rise).
• Water ($20^\circ\text{C}$): $c \approx 1482\,\text{m/s}$ (more than 4 times faster than air due to high bulk modulus).
• Cranial Bone: $c \approx 3000 - 4000\,\text{m/s}$ (substantially faster, directly influencing bone-conduction transducer transmission and interaural time delays during transcranial bone conduction).

Wavelength ($\lambda$): The spatial physical distance spanned by one complete cycle of a sound wave in a given medium: $\lambda = \frac{c}{f}$

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Sound Pressure, Sound Intensity & The Decibel Scale

The human auditory system detects an extraordinary dynamic range of acoustic power, spanning more than $10^{14}$ to 1 in energy from threshold to the threshold of pain. A linear scale of sound measurement is mathematically unwieldy. The decibel ($\text{dB}$) provides a compact, logarithmic scale expressing the ratio of a measured physical acoustic quantity to a standardized reference.

Sound Pressure vs. Sound Intensity

• Sound Pressure ($P$): The force exerted perpendicular to a unit surface area by alternating acoustic compressions and rarefactions, measured in Newtons per square meter ($\text{N/m}^2$) or Pascals ($\text{Pa}$, where $1\,\text{Pa} = 1\,\text{N/m}^2$). The audiometric reference pressure ($P_0$) is: $P_0 = 20\,\mu\text{Pa} = 2 \times 10^{-5}\,\text{Pa} = 2 \times 10^{-4}\,\text{dyne/cm}^2$ This represents the nominal threshold of human hearing at $1000\,\text{Hz}$ under optimal listening conditions.
• Sound Intensity ($I$): The acoustic power transmitted per unit area perpendicular to the direction of wave propagation, measured in Watts per square meter ($\text{W/m}^2$). The audiometric reference intensity ($I_0$) is: $I_0 = 10^{-12}\,\text{W/m}^2 = 10^{-16}\,\text{W/cm}^2$
• Physical Coupling: Sound intensity is directly proportional to the square of sound pressure ($I \propto P^2$). In a free acoustic field with specific acoustic impedance $Z_0$: $I = \frac{P^2}{Z_0}$

Decibel Formulas: Intensity Level and Sound Pressure Level

Using the logarithmic decibel formula:

• Intensity Level ($\text{dB IL}$): $\text{dB IL} = 10 \log_{10}\left(\frac{I}{I_0}\right)$
• Sound Pressure Level ($\text{dB SPL}$): Because $I \propto P^2$, substituting $P^2$ yields: $\text{dB SPL} = 10 \log_{10}\left(\frac{P^2}{P_0^2}\right) = 20 \log_{10}\left(\frac{P}{P_0}\right)$

Decibel Mathematics & Summation Rules

Because decibels are logarithmic ratios, they cannot be added arithmetically.

• Doubling of Sound Power / Intensity: $10 \log_{10}(2) \approx +3.01\,\text{dB}$ Adding two independent (incoherent) sound sources of equal sound level increases total level by $3\,\text{dB}$ (e.g., $65\,\text{dB SPL} + 65\,\text{dB SPL} = 68\,\text{dB SPL}$).
• Doubling of Sound Pressure (Coherent / In-Phase): When two perfectly correlated, identical in-phase sound waves combine constructively, pressure doubles ($P_{\text{tot}} = 2P$): $20 \log_{10}(2) \approx +6.02\,\text{dB}$ For example, combining two identical in-phase $60\,\text{dB SPL}$ signals yields $66\,\text{dB SPL}$.
• Combining Uncorrelated Sources of Unequal Levels: When combining two sound sources differing in intensity, use the standard decibel addition rule:

Inverse Square Law

In a free, unbounded acoustic field (free from reflective surfaces), acoustic energy spreads spherically from a point source. The surface area of a sphere is $A = 4\pi r^2$. Consequently, sound intensity varies inversely with the square of the distance from the source ($I \propto \frac{1}{r^2}$).

• Sound Pressure Rule: Sound pressure drops in direct proportion to distance ($P \propto \frac{1}{r}$).
• The $6\,\text{dB}$ Rule: Every doubling of distance ($2r$) from a sound source in a free field reduces sound pressure level by $6\,\text{dB}$: $\Delta \text{dB SPL} = 20 \log_{10}\left(\frac{r_1}{r_2}\right) = 20 \log_{10}\left(\frac{1}{2}\right) = -6.02\,\text{dB}$ Clinical application: In classroom acoustics, a student seated $4\,\text{meters}$ from a teacher experiences a $12\,\text{dB}$ reduction in direct speech signal compared to a student seated $1\,\text{meter}$ away ($1\,\text{m} \to 2\,\text{m} = -6\,\text{dB}$; $2\,\text{m} \to 4\,\text{m} = -6\,\text{dB}$; total $-12\,\text{dB}$).

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Decibel Scales in Clinical Practice: dB SPL, dB HL, and dB SL

Clinical audiometry utilizes three distinct decibel references. Conflating these references leads to catastrophic errors in diagnostic interpretation, masking calculations, and hearing aid programming.

1. Decibel Sound Pressure Level (dB SPL)

• Reference: Absolute physical sound pressure of $20\,\mu\text{Pa}$ ($0\,\text{dB SPL} = 20\,\mu\text{Pa}$).
• Application: Physical sound measurement, sound level meters, hearing aid electroacoustic analysis (ANSI S3.22), probe-tube real-ear measurements (REAR, REUR), and otoacoustic emission measurements.
• Linearity: Independent of human hearing sensitivity. $80\,\text{dB SPL}$ represents the exact same physical pressure whether presented at $125\,\text{Hz}$ or $4000\,\text{Hz}$.

2. Decibel Hearing Level (dB HL)

• Reference: Human auditory threshold sensitivity across frequency, standardized by ANSI S3.6 and ISO 389.
• Minimum Audibility Curve: The human ear is not equally sensitive at all frequencies. It requires significantly more physical pressure to detect a $125\,\text{Hz}$ or $8000\,\text{Hz}$ tone than a $1000\,\text{Hz}$ or $3000\,\text{Hz}$ tone.
• RETSPL (Reference Equivalent Threshold Sound Pressure Level): The exact dB SPL required to reach the threshold of hearing ($0\,\text{dB HL}$) in a normal-hearing population for a specific transducer and acoustic coupler.
• An audiometer dial setting of $0\,\text{dB HL}$ outputs different dB SPL levels at each frequency to account for the human audibility curve. $\text{dB SPL} = \text{dB HL} + \text{RETSPL}$ $\text{dB HL} = \text{dB SPL} - \text{RETSPL}$

Clinical Calculation Example: If an audiometer outputs a $500\,\text{Hz}$ tone at $50\,\text{dB HL}$ through TDH-39 headphones (where RETSPL is $11.5\,\text{dB SPL}$), the physical output in the calibration coupler is $50 + 11.5 = 61.5\,\text{dB SPL}$. If using ER-3A insert earphones (where RETSPL is $5.5\,\text{dB SPL}$), $50\,\text{dB HL}$ corresponds to $50 + 5.5 = 55.5\,\text{dB SPL}$.

3. Decibel Sensation Level (dB SL)

• Reference: The individual patient's personal behavioral threshold at a specified frequency or for a specific stimulus.
• Formula: $\text{Presentation Level (dB SL)} = \text{Presentation Level (dB HL)} - \text{Patient Threshold (dB HL)}$
• Application: Suprathreshold diagnostic testing, including Word Recognition Score (WRS, routinely administered at $+30$ to $+40\,\text{dB SL}$ above the patient's SRT or pure-tone average), acoustic reflex threshold testing, and Tone Decay / PIPB rollover testing.
• Diagnostic Trap: Presenting speech at a fixed level of $40\,\text{dB HL}$ to a patient with a $40\,\text{dB HL}$ threshold provides $0\,\text{dB SL}$ (inaudible or barely audible), whereas presenting at $40\,\text{dB SL}$ requires an audiometer dial setting of $80\,\text{dB HL}$.

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Acoustic Impedance, Resonance & Acoustic Filtering

Acoustic Impedance ($Z$)

Acoustic impedance ($Z$) represents the total opposition encountered by an acoustic wave propagating through a mechanical or acoustic system (such as the middle ear). Measured in acoustic ohms ($\text{Pa}\cdot\text{s/m}^3$), impedance comprises resistive and reactive components: $Z = \sqrt{R^2 + (X_m - X_s)^2}$ where:

• Acoustic Resistance ($R$): Dissipative opposition caused by friction and viscosity (e.g., perilymph viscosity, middle ear mucosa, air resistance in the middle ear cavity). Resistance is largely independent of frequency.
• Mass Reactance ($X_m$): Energy storage opposition due to inertia and physical mass (e.g., mass of the tympanic membrane, ossicles, and perilymph): $X_m = 2\pi f M$ Mass reactance is directly proportional to frequency. As frequency increases, mass reactance increases linearly, impeding high-frequency energy transfer.
• Stiffness Reactance ($X_s$): Energy storage opposition due to elasticity and compliance (e.g., tension of the tympanic membrane, ossicular ligaments, tendon of the stapedius muscle, air volume elasticity of the tympanic cavity): $X_s = \frac{1}{2\pi f C} = \frac{K}{2\pi f}$ where $C$ is acoustic compliance and $K$ is stiffness. Stiffness reactance is inversely proportional to frequency. At low frequencies, stiffness reactance is exceptionally high, impeding low-frequency transmission.

Resonance

Natural resonance occurs at the specific frequency ($f_0$) where mass reactance equals stiffness reactance ($X_m = X_s$): $2\pi f_0 M = \frac{1}{2\pi f_0 C} \implies f_0 = \frac{1}{2\pi \sqrt{M C}}$ At resonance, $(X_m - X_s) = 0$, the reactive forces cancel out completely, and total impedance reaches its absolute minimum, equal solely to resistance ($Z = R$). Sound transmission through the system is maximized.

• Middle Ear Resonant Frequency: In normal human adults, the middle ear resonant frequency typically ranges from $800\,\text{Hz}$ to $1200\,\text{Hz}$ (mean $\approx 1000\,\text{Hz}$).
• Pathological Shifts:
  • Stiffness-dominated pathologies (e.g., otosclerosis, middle ear effusion, lateral ossicular fixation): Stiffness ($K$) increases, causing resonant frequency to shift upward ($> 1200\,\text{Hz}$). This selectively attenuates low-frequency transmission, producing a low-frequency conductive air-bone gap.
  • Mass-dominated pathologies (e.g., ossicular chain discontinuity, monomeric tympanic membrane): Stiffness drops and relative mass increases, shifting resonant frequency downward ($< 700\,\text{Hz}$).

Acoustic Filters

An acoustic filter is an acoustic system or electronic circuit that selectively transmits certain frequency components while attenuating others.

• Low-pass filter: Transmits all frequencies below a cutoff frequency ($f_c$) and attenuates frequencies above $f_c$.
• High-pass filter: Transmits all frequencies above $f_c$ and attenuates frequencies below $f_c$.
• Band-pass filter: Transmits a specific band of frequencies bounded by a lower cutoff ($f_1$) and upper cutoff ($f_2$). Characterized by center frequency ($f_0 = \sqrt{f_1 f_2}$), bandwidth ($BW = f_2 - f_1$), and filter sharpness/quality factor ($Q = \frac{f_0}{BW}$). The cochlea behaves as a bank of overlapping band-pass auditory filters.
• Band-reject (Notch) filter: Attenuates a narrow frequency band while passing all frequencies above and below (e.g., $60\,\text{Hz}$ electrical line-noise rejection filters in ABR equipment).
• Filter Roll-off (Slope): The steepness of attenuation beyond the cutoff frequency, measured in decibels per octave ($\text{dB/oct}$) or decibels per decade. A single-pole acoustic filter attenuates at $6\,\text{dB/oct}$; a two-pole filter rolls off at $12\,\text{dB/oct}$; steep multi-pole digital filters achieve $> 24-48\,\text{dB/oct}$.