Sound waves are all around us, shaping our auditory world. From the gentle whisper of leaves to the roar of a jet engine, these vibrations travel through the air, carrying energy and information. Understanding how sound intensity relates to amplitude and perception is key to grasping acoustic phenomena.

The decibel scale helps us make sense of the vast range of sound intensities we encounter daily. By using logarithms, it compresses this range into a more manageable form, allowing us to compare and quantify sounds from the barely audible to the painfully loud. This knowledge is crucial for everything from designing concert halls to protecting our hearing.

Sound Intensity and Perception

Amplitude, energy, and loudness relationships

• Sound waves cause particles to oscillate parallel to the direction of wave propagation (longitudinal pressure waves)
  • Amplitude represents the maximum displacement of particles from their equilibrium position
  • Higher amplitude waves have greater particle displacement and carry more energy
• Sound intensity measures the power carried by sound waves per unit area ($W/m^2$)
  • Intensity is directly proportional to the square of the amplitude: $I \propto A^2$
  • Doubling the amplitude quadruples the intensity (4x increase)
• Perceived loudness is a subjective measure of sound strength interpreted by the human ear and brain
  • Loudness depends on both sound intensity and frequency
  • Human hearing is most sensitive to frequencies between 2 kHz and 5 kHz (speech range)
  • Doubling the intensity results in an approximately 10-fold increase in perceived loudness

Decibel scale for sound intensity

• The decibel (dB) scale compares sound intensities over a wide range using logarithms
  • Defined as: $\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)$ where $I_0 = 10^{-12} W/m^2$ (reference intensity)
  • A 10 dB increase represents a 10-fold increase in intensity (10x)
  • A 20 dB increase represents a 100-fold increase in intensity (100x)
• Typical sound levels:
  • Threshold of hearing: 0 dB (barely audible)
  • Whisper: 30 dB (quiet library)
  • Normal conversation: 60 dB (1 meter away)
  • Heavy traffic: 80 dB (busy city street)
  • Jet engine at 30 m: 150 dB (painful and damaging)

Calculations of sound wave intensity

• Sound intensity $I$ at a distance $r$ from a point source with power $P$: $I = \frac{P}{4\pi r^2}$
  • Intensity decreases with the square of the distance from the source (inverse-square law)
  • Example: Doubling the distance from a sound source reduces the intensity by a factor of 4 (2^2)
• Sound intensity level $\beta$ in decibels: $\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)$
  • Calculate intensity if the decibel level is known: $I = I_0 \cdot 10^{\frac{\beta}{10}}$
• Sound pressure $p$ is related to intensity by: $I = \frac{p^2}{2\rho v}$
  • $\rho$ is the density of the medium (air at sea level: 1.225 kg/m^3)
  • $v$ is the speed of sound in the medium (air at 20°C: 343 m/s)
• Acoustic impedance of a medium affects sound transmission and reflection at boundaries

Physiology of sound production and perception

• Human sound production:
  1. Vocal cords in the larynx vibrate, creating pressure waves in exhaled air
  2. Vibration frequency determines pitch (higher frequency = higher pitch)
  3. Vocal tract (throat, mouth, nose) acts as a resonance chamber, amplifying and modifying the sound
• Human sound perception:
  1. Outer ear: Pinna collects and channels sound waves into the ear canal
  2. Middle ear: Eardrum (tympanic membrane) vibrates in response to sound waves
     • Vibrations are transmitted and amplified by three small bones (ossicles): malleus, incus, and stapes
  3. Inner ear: Vibrations from the stapes are transferred to the fluid in the cochlea
     • Basilar membrane in the cochlea vibrates at different positions based on frequency
     • Hair cells on the basilar membrane convert mechanical vibrations into electrical signals
     • Auditory nerve transmits these signals to the brain for processing and interpretation

Wave properties and sound characteristics

• Resonance occurs when an object is driven at its natural frequency, amplifying the sound
• Wave interference can lead to constructive or destructive combination of sound waves
• The frequency spectrum of a sound determines its timbre and quality
• Sound absorption by materials affects the acoustic properties of spaces and can reduce noise levels