Formula Sheet: Sound

Table of Contents
1. Sound Wave Fundamentals
2. Sound Intensity and Decibels
3. Doppler Effect
4. Beats
5. Standing Waves and Resonance
6. Wave Interference and Superposition
7. Sound Wave Properties and Characteristics
8. Energy and Power in Sound Waves
9. Additional Important Concepts
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Sound Wave Fundamentals

Wave Properties

• Wave Speed: \[v = f \lambda\] where:
  • \(v\) = wave speed (m/s)
  • \(f\) = frequency (Hz)
  • \(\lambda\) = wavelength (m)
• Period and Frequency Relationship: \[T = \frac{1}{f}\] where:
  • \(T\) = period (s)
  • \(f\) = frequency (Hz)
• Speed of Sound in Air: \[v = 331 + 0.6T_C\] where:
  • \(v\) = speed of sound (m/s)
  • \(T_C\) = temperature in Celsius (°C)
  • At room temperature (20°C): \(v \approx 343\) m/s
  Note: Speed of sound increases with temperature
• Speed of Sound in a Medium: \[v = \sqrt{\frac{B}{\rho}}\] where:
  • \(v\) = wave speed (m/s)
  • \(B\) = bulk modulus (Pa)
  • \(\rho\) = density of medium (kg/m³)
  Note: Greater stiffness (bulk modulus) increases speed; greater density decreases speed

Wave Types and Characteristics

• Longitudinal Waves: Sound waves are longitudinal; particle displacement is parallel to direction of wave propagation
• Compression and Rarefaction: Regions of high pressure (compression) and low pressure (rarefaction) alternate in sound waves
• Transverse vs. Longitudinal: Sound cannot travel as transverse waves through fluids (liquids and gases)

Sound Intensity and Decibels

Intensity

• Intensity Definition: \[I = \frac{P}{A}\] where:
  • \(I\) = intensity (W/m²)
  • \(P\) = power (W)
  • \(A\) = area (m²)
• Intensity for Point Source: \[I = \frac{P}{4\pi r^2}\] where:
  • \(I\) = intensity at distance \(r\) (W/m²)
  • \(P\) = total power of source (W)
  • \(r\) = distance from source (m)
  Note: Intensity follows inverse square law with distance
• Intensity and Amplitude: \[I \propto A^2\] where \(A\) is the amplitude of the wave
• Threshold of Hearing: \[I_0 = 1 \times 10^{-12} \text{ W/m}^2\] Reference intensity at 1000 Hz

Decibel Scale

• Sound Level (Decibels): \[\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)\] where:
  • \(\beta\) = sound level (dB)
  • \(I\) = intensity of sound (W/m²)
  • \(I_0\) = reference intensity = \(1 \times 10^{-12}\) W/m²
  Note: Decibel scale is logarithmic
• Intensity Ratio from Decibel Difference: \[\frac{I_2}{I_1} = 10^{(\beta_2 - \beta_1)/10}\] where:
  • \(\beta_2 - \beta_1\) = difference in sound levels (dB)
• Key Decibel Relationships:
  • +10 dB → intensity increases by factor of 10
  • +20 dB → intensity increases by factor of 100
  • +3 dB → intensity approximately doubles
  • -3 dB → intensity approximately halves

Doppler Effect

General Doppler Equation

• Observed Frequency: \[f_o = f_s \left(\frac{v \pm v_o}{v \mp v_s}\right)\] where:
  • \(f_o\) = observed frequency (Hz)
  • \(f_s\) = source frequency (Hz)
  • \(v\) = speed of sound in medium (m/s)
  • \(v_o\) = speed of observer (m/s)
  • \(v_s\) = speed of source (m/s)
  Sign Conventions:
  • Numerator (observer): Use (+) when observer moves toward source; (-) when moving away
  • Denominator (source): Use (-) when source moves toward observer; (+) when moving away
• Alternative Formulation:
  • Approaching (getting closer): \(f_o > f_s\) (higher pitch)
  • Receding (getting farther): \(f_o < f_s\)="" (lower="">

Special Cases

• Stationary Observer, Moving Source: \[f_o = f_s \left(\frac{v}{v \mp v_s}\right)\] Use (-) when source approaches; (+) when source recedes
• Moving Observer, Stationary Source: \[f_o = f_s \left(\frac{v \pm v_o}{v}\right)\] Use (+) when observer approaches; (-) when observer recedes
• Both Moving in Same Direction: Apply general formula with careful attention to sign conventions
• Doppler Shift: \[\Delta f = f_o - f_s\] Change in frequency due to relative motion

Beats

Beat Frequency

• Beat Frequency: \[f_{\text{beat}} = |f_1 - f_2|\] where:
  • \(f_{\text{beat}}\) = beat frequency (Hz)
  • \(f_1\) = frequency of first wave (Hz)
  • \(f_2\) = frequency of second wave (Hz)
  Note: Beats occur when two waves of slightly different frequencies interfere
• Beat Period: \[T_{\text{beat}} = \frac{1}{f_{\text{beat}}} = \frac{1}{|f_1 - f_2|}\] Time between successive maxima in amplitude
• Condition for Beats: Two waves must have similar frequencies and amplitudes; beats are most prominent when \(|f_1 - f_2|\) is small

Standing Waves and Resonance

Standing Waves in Strings

• Wave Speed on a String: \[v = \sqrt{\frac{T}{\mu}}\] where:
  • \(v\) = wave speed (m/s)
  • \(T\) = tension in string (N)
  • \(\mu\) = linear mass density (kg/m) = \(m/L\)
• Wavelength for String Fixed at Both Ends: \[\lambda_n = \frac{2L}{n}\] where:
  • \(\lambda_n\) = wavelength of nth harmonic (m)
  • \(L\) = length of string (m)
  • \(n\) = harmonic number (1, 2, 3, ...)
• Resonant Frequencies for String Fixed at Both Ends: \[f_n = \frac{nv}{2L} = \frac{n}{2L}\sqrt{\frac{T}{\mu}}\] where:
  • \(f_n\) = frequency of nth harmonic (Hz)
  • \(n\) = 1, 2, 3, ... (harmonic number)
  • \(n = 1\): fundamental frequency (first harmonic)
  • \(n = 2\): second harmonic (first overtone)
• Fundamental Frequency: \[f_1 = \frac{v}{2L} = \frac{1}{2L}\sqrt{\frac{T}{\mu}}\] Lowest resonant frequency

Standing Waves in Open Pipes

• Open Pipe (Both Ends Open):
  • Antinodes at both ends
  • Same formulas as string fixed at both ends
• Wavelength for Open Pipe: \[\lambda_n = \frac{2L}{n}\] where \(n\) = 1, 2, 3, ...
• Resonant Frequencies for Open Pipe: \[f_n = \frac{nv}{2L}\] where:
  • \(n\) = 1, 2, 3, ... (all harmonics present)
  • \(v\) = speed of sound (m/s)
  • \(L\) = length of pipe (m)
• Fundamental Frequency (Open Pipe): \[f_1 = \frac{v}{2L}\]

Standing Waves in Closed Pipes

• Closed Pipe (One End Closed):
  • Node at closed end
  • Antinode at open end
  • Only odd harmonics present
• Wavelength for Closed Pipe: \[\lambda_n = \frac{4L}{n}\] where \(n\) = 1, 3, 5, 7, ... (odd integers only)
• Resonant Frequencies for Closed Pipe: \[f_n = \frac{nv}{4L}\] where:
  • \(n\) = 1, 3, 5, 7, ... (only odd harmonics)
  • \(v\) = speed of sound (m/s)
  • \(L\) = length of pipe (m)
• Fundamental Frequency (Closed Pipe): \[f_1 = \frac{v}{4L}\] Note: Fundamental frequency is half that of an open pipe of same length
• Harmonic Series for Closed Pipe: \[f_n = (2n-1)f_1\] where \(n\) = 1, 2, 3, ... gives the 1st, 3rd, 5th, ... harmonics

Wave Interference and Superposition

Interference Principles

• Principle of Superposition: When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements
• Constructive Interference:
  • Waves in phase
  • Path difference = \(n\lambda\) where \(n\) = 0, 1, 2, 3, ...
  • Phase difference = \(2\pi n\) radians = \(360°n\)
  • Amplitude increases
• Destructive Interference:
  • Waves out of phase
  • Path difference = \((n + \frac{1}{2})\lambda\) where \(n\) = 0, 1, 2, 3, ...
  • Phase difference = \((2n + 1)\pi\) radians = \(180°(2n + 1)\)
  • Amplitude decreases or cancels
• Path Difference for Interference: \[\Delta x = |d_2 - d_1|\] where \(d_1\) and \(d_2\) are distances from two sources to the point of interference

Sound Wave Properties and Characteristics

Pitch, Loudness, and Timbre

• Pitch: Perception of frequency
  • Higher frequency → higher pitch
  • Lower frequency → lower pitch
• Loudness: Perception of intensity/amplitude
  • Greater amplitude → louder sound
  • Related to energy and intensity
• Timbre (Quality): Determined by harmonic content (combination of fundamental and overtones)

Acoustic Phenomena

• Reflection: Sound waves bounce off surfaces (echoes)
• Refraction: Sound waves bend when passing between media of different densities or temperatures
• Diffraction: Sound waves bend around obstacles and spread through openings
• Absorption: Sound energy converted to other forms (usually heat) when encountering materials

Energy and Power in Sound Waves

Energy Relationships

• Energy Density: \[u = \frac{1}{2}\rho v \omega^2 A^2\] where:
  • \(u\) = energy density (J/m³)
  • \(\rho\) = density of medium (kg/m³)
  • \(v\) = wave speed (m/s)
  • \(\omega\) = angular frequency (rad/s)
  • \(A\) = amplitude (m)
• Power and Intensity Relationship: \[P = I \cdot A\] where:
  • \(P\) = power (W)
  • \(I\) = intensity (W/m²)
  • \(A\) = area (m²)
• Intensity Proportionality: \[I \propto \frac{P}{r^2} \propto A^2 \propto f^2\] For point sources and given conditions

Additional Important Concepts

Harmonic Relationships

• Harmonic Series: Integer multiples of the fundamental frequency \[f_n = nf_1\] where \(n\) = 1, 2, 3, ... for systems supporting all harmonics
• Overtones:
  • First overtone = second harmonic (\(f_2 = 2f_1\))
  • Second overtone = third harmonic (\(f_3 = 3f_1\))
  • For closed pipes: overtones are only odd multiples of \(f_1\)

Nodes and Antinodes

• Node: Point of zero displacement in standing wave (destructive interference)
• Antinode: Point of maximum displacement in standing wave (constructive interference)
• Distance Between Adjacent Nodes or Antinodes: \[d = \frac{\lambda}{2}\]
• Distance Between Node and Adjacent Antinode: \[d = \frac{\lambda}{4}\]
• Number of Nodes in Standing Wave: For string or open pipe: \(n + 1\) nodes for nth harmonic
  For closed pipe: varies with harmonic

Phase and Wave Relationships

• Phase Difference and Path Difference: \[\Delta \phi = \frac{2\pi}{\lambda} \Delta x\] where:
  • \(\Delta \phi\) = phase difference (radians)
  • \(\Delta x\) = path difference (m)
  • \(\lambda\) = wavelength (m)
• Angular Frequency: \[\omega = 2\pi f = \frac{2\pi}{T}\] where \(\omega\) is in rad/s
• Wave Number: \[k = \frac{2\pi}{\lambda}\] where \(k\) is in rad/m