Comprehensive Study Guide on Sound Waves, Acoustic Intensity, the Decibel Scale, and the Doppler Effect

Sound Intensity and Acoustic Power Propagation

• Physical Definition of Sound Intensity:

  • Sound intensity represents the acoustic power transmitted per unit area perpendicular to the direction of propagation.
  • Mathematical definition:     $I = \frac{P}{A_{\text{air}}}$
  • Standard International (SI) unit of measurement:     $\text{W/m}^2$

• Spherical Acoustic Wave Propagation:

  • For an isotropic sound source radiating energy uniformly in three dimensions, wavefronts expand as concentric spheres.
  • The surface area over which sound energy distributes at a distance $r$ from the source is given by the surface area of a sphere:     $A = 4\pi r^2$
  • The resulting intensity expression for emitted power $P_{\text{émise}}$ is:     $I = \frac{P_{\text{émise}}}{4\pi r^2}$

• Inverse-Square Law Proportionality:

  • Sound intensity is inversely proportional to the square of the distance from the point source:     $I \propto \frac{1}{r^2}$
  • The symbol $\propto$ denotes direct proportionality, meaning doubling the distance from the sound source decreases the received intensity by a factor of four.

The Decibel Scale and Human Auditory Perception

• Non-Linear Subjective Perception of Loudness:

  • The human auditory system responds non-linearly to acoustic energy.
  • If sound $A$ possesses physical intensity twice that of sound $B$, the human ear does not register sound $A$ as being twice as loud.

• Decibel Level Formulation:

  • Acoustic intensity level (sound level), designated by $\beta$, is calculated logarithmically in decibels ($\text{dB}$):     $\beta = 10 \log\left(\frac{I}{I_0}\right)$
  • Here, $\log$ denotes the base-10 logarithm, and $I_0$ represents the standardized reference intensity.

• Fundamental Auditory Thresholds:

  • Threshold of Audibility ($\text{Seuil d'audibilité}$):
  • Defined as the lowest sound intensity detectable by the human ear:       $I_0 = 10^{-12}\,\text{W/m}^2$
  • Corresponding decibel level:       $\beta = 10 \log\left(\frac{10^{-12}}{10^{-12}}\right) = 10 \log(1) = 0\,\text{dB}$
  • Threshold of Pain ($\text{Seuil de la douleur}$):
  • The intensity level at which sound transitions from acoustic sensation to physical pain:       $I = 1\,\text{W/m}^2$
  • Corresponding decibel level:       $\beta = 10 \log\left(\frac{1}{10^{-12}}\right) = 10 \log\left(10^{12}\right) = 10 \times 12 = 120\,\text{dB}$

Intensity Doubling and Decibel Increments

• Analytical Problem Statement:

  • Determining the precise decibel increase when the acoustic intensity produced by a loudspeaker is doubled.

• Mathematical Setup:

  • Initial intensity and decibel level:     $I_1 \implies \beta_1 = 10 \log\left(\frac{I_1}{I_0}\right)$
  • Doubled intensity and corresponding decibel level:     $I_2 = 2I_1 \implies \beta_2 = 10 \log\left(\frac{2I_1}{I_0}\right)$
  • Net decibel variation:     $\beta_2 - \beta_1 = 10\left[\log\left(\frac{2I_1}{I_0}\right) - \log\left(\frac{I_1}{I_0}\right)\right]$

• Logarithmic Identity Application:

  • Quotient property of logarithms:     $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$

• Step-by-Step Derivation:

  • Substituting into the difference formula:     $\beta_2 - \beta_1 = 10 \log\left(\frac{\frac{2I_1}{I_0}}{\frac{I_1}{I_0}}\right)$
  • Simplifying the fraction:     $\beta_2 - \beta_1 = 10 \log(2)$
  • Numerical evaluation:     $\beta_2 - \beta_1 \approx 3\,\text{dB}$

• Physical Conclusion:

  • Doubling acoustic intensity translates strictly to an increase of $+3\,\text{dB}$.
  • Linear scaling in physical intensity does not equate to linear scaling in perceived decibels:     $2I_{\left(\text{W/m}^2\right)} \neq 2I_{(\text{dB})}$

Theoretical Principles of the Doppler Effect

• Wavelength Modification from Source Motion:

  • When the sound source is in motion, emitted wavefronts compress ahead of the source and stretch behind it.
  • The observed wavelength $\lambda'$ differs from the stationary wavelength $\lambda$ ($\lambda' \neq \lambda$):     $\lambda = \frac{V}{f} \implies f = \frac{V}{\lambda}$
  • Modified wavelength expression:     $\lambda' = \frac{V \pm V_s}{f}$
  • Expressed relative to the stationary wavelength:     $\lambda' = \frac{V \pm V_s}{V} \lambda = \lambda\left(1 \pm \frac{V_s}{V}\right)$

• General Doppler Equation:

  • The perceived frequency $f'$ depends on wave speed in air $V$, observer speed $V_o$, source speed $V_s$, and emitted frequency $f$:     $f' = \left(\frac{V \pm V_o}{V \mp V_s}\right) f$

• Fundamental Single-Component Motion Cases:

  • Case 1: Source at Rest ($S$ au repos), Observer in Motion ($O$ en mouvement):
  • Observer approaches stationary source ($O\text{ s'approche}$):       $f' = \left(\frac{V + V_o}{V + 0}\right) f = \left(\frac{V + V_o}{V}\right) f$$f' > f$
  • Observer recedes from stationary source ($O\text{ s'éloigne}$):       $f' = \left(\frac{V - V_o}{V - 0}\right) f = \left(\frac{V - V_o}{V}\right) f$$f' < f$
  • Case 2: Source in Motion ($S$ en mouvement), Observer at Rest ($O$ au repos):
  • Source approaches stationary observer ($\text{la source s'approche}$):       $f' = \left(\frac{V - 0}{V - V_s}\right) f = \left(\frac{V}{V - V_s}\right) f$$f' > f$
  • Source recedes from stationary observer ($\text{la source s'éloigne}$):       $f' = \left(\frac{V + 0}{V + V_s}\right) f = \left(\frac{V}{V + V_s}\right) f$$f' < f$

• Universal Sign Conventions:

  • Numerator Sign ($V_o$):
  • $+$ when the observer approaches the source ($O\text{ s'approche de }S$).
  • $-$ when the observer moves away from the source ($O\text{ s'éloigne de }S$).
  • Denominator Sign ($V_s$):
  • $-$ when the source approaches the observer ($S\text{ s'approche de }O$).
  • $+$ when the source moves away from the observer ($S\text{ s'éloigne de }O$).

Doppler Effect Case Study: Emergency Fire Truck Scenario

• Scenario Parameters:

  • Wave and Modern Physics Chapter 3: Sound (Ondes et physique moderne Chapitre 3: le son), authored by Mathis Blanchet.
  • Fire truck (source) speed:     $V_s = 135\,\text{km/h} = \frac{135\,000\,\text{m}}{3600\,\text{s}} = 37.5\,\text{m/s}$
  • Automobile (observer) speed:     $V_o = 90\,\text{km/h} = \frac{90\,000\,\text{m}}{3600\,\text{s}} = 25\,\text{m/s}$
  • Siren emission frequency:     $f = 440\,\text{Hz}$
  • Sound velocity in medium (air):     $V = 343\,\text{m/s}$
  • Velocity relationships:     $37.5 - 25 = 12.5\,\text{m/s}$$37.5 + 25 = 62.5\,\text{m/s}$

• Condition (a): Vehicles Driving in Opposite Directions (Contresens), Truck in Front:

  • Relative direction: Observer and source move directly toward each other.
  • Observer approaches source: Numerator sign is positive ($+$).
  • Source approaches observer: Denominator sign is negative (-$).\n - Resulting frequency relation:\n    f' > f\n - Analytical substitution:\n    f' = \left(\frac{V + V_o}{V - V_s}\right) f = \left(\frac{343 + 25}{343 - 37.5}\right) 440\n - Computation:\n    f' = \left(\frac{368}{305.5}\right) 440 = 530\,\text{Hz}\n\n- **Condition (b): Vehicles Driving in Opposite Directions (Contresens), Truck Behind**:\n - Relative direction: Observer and source move directly away from each other after passing.\n - Observer moves away from source: Numerator sign is negative (-$).
  • Source moves away from observer: Denominator sign is positive ($+$).
  • Resulting frequency relation:     $f' < f$
  • Analytical substitution:     $f' = \left(\frac{V - V_o}{V + V_s}\right) f = \left(\frac{343 - 25}{343 + 37.5}\right) 440$
  • Computation:     $f' = \left(\frac{318}{380.5}\right) 440 = 367.7\,\text{Hz}$

• Condition (c): Vehicles Driving in the Same Direction, Truck in Front:

  • Relative configuration: Both vehicles travel in identical direction; the truck is ahead and pulling away since $V_s > V_o$.
  • Observer approaches source: Numerator sign is positive ($+$).
  • Source moves away from observer: Denominator sign is positive ($+$).
  • Resulting frequency relation:     $f' < f$
  • Analytical substitution:     $f' = \left(\frac{V + V_o}{V + V_s}\right) f = \left(\frac{343 + 25}{343 + 37.5}\right) 440$
  • Computation:     $f' = \left(\frac{368}{380.5}\right) 440 = 425.55\,\text{Hz}$

• Condition (d): Vehicles Driving in the Same Direction, Truck Behind:

  • Relative configuration: Both vehicles travel in identical direction; the faster fire truck approaches the car from behind ($V_s > V_o$).
  • Observer moves away from source: Numerator sign is negative (-$).\n - Source approaches observer: Denominator sign is negative (-$).
  • Resulting frequency relation:     $f' > f$
  • Analytical substitution:     $f' = \left(\frac{V - V_o}{V - V_s}\right) f = \left(\frac{343 - 25}{343 - 37.5}\right) 440$
  • Computation:     $f' = \left(\frac{318}{305.5}\right) 440 = 458\,\text{Hz}$

Doppler Effect Case Study: Police Vehicle Pursuit

• Scenario Parameters:

  • Police vehicle velocity (source):     $V_s = 50\,\text{m/s}$
  • Civilian driver velocity (observer):     $V_o = 25\,\text{m/s}$
  • Common direction: Both vehicles travel along the exact same path and direction.
  • Siren source frequency:     $f = 1200\,\text{Hz}$
  • Speed of sound in ambient air:     $V_{\text{air}} = 340\,\text{m/s}$

• Sub-case (a): Police Vehicle Following Driver ($\text{Un policier suit un conducteur}$):

  • Physical setup: Police car travels behind the driver, closing the distance at $50\,\text{m/s}$ while the driver leads at $25\,\text{m/s}$.
  • Observer moves away from source: Numerator sign is negative (-$).\n - Source moves toward observer: Denominator sign is negative (-$).
  • Analytical formula:     $f' = \left(\frac{V - V_o}{V - V_s}\right) f$
  • Calculation:     $f' = \left(\frac{340 - 25}{340 - 50}\right) 1200 = \left(\frac{315}{290}\right) 1200 = 1303.45\,\text{Hz}$

• Sub-case (b): Vehicles in Motion with Source Leading ($\text{Conducteur en mouvement et policier devant}$):

  • Physical setup: The police vehicle is ahead and pulling away, while the driver follows.
  • Observer approaches source: Numerator sign is positive ($+$).
  • Source moves away from observer: Denominator sign is positive ($+$).
  • Analytical formula:     $f' = \left(\frac{V + V_o}{V + V_s}\right) f$
  • Calculation:     $f' = \left(\frac{340 + 25}{340 + 50}\right) 1200 = \left(\frac{365}{390}\right) 1200 = 1123.08\,\text{Hz}$