Sound Waves

Definition and Nature

Mechanical Longitudinal Waves

Sound waves: mechanical disturbances propagating through elastic media. Particle oscillations parallel to wave direction. Require medium: solid, liquid, gas.

Medium Dependence

Propagation impossible in vacuum. Speed and attenuation dependent on medium density and elasticity. Compression and rarefaction regions alternate.

Energy Transmission

Transport energy without net particle displacement. Energy loss due to friction and absorption causes attenuation.

Frequency Range

Audible range: 20 Hz to 20 kHz. Infrasound: below 20 Hz. Ultrasound: above 20 kHz.

Wave Characteristics

Frequency (f)

Number of oscillations per second. Unit: Hertz (Hz). Determines pitch in audible sound.

Wavelength (λ)

Distance between consecutive compressions or rarefactions. Related inversely to frequency.

Amplitude (A)

Maximum particle displacement from equilibrium. Correlates with loudness/intensity.

Period (T)

Time for one complete oscillation. T = 1/f.

Wave Number (k)

Spatial frequency of wave: k = 2π/λ.

Propagation of Sound Waves

Longitudinal Nature

Particles oscillate parallel to propagation direction. Alternating compressions and rarefactions form wave fronts.

Reflection and Refraction

Sound waves reflect at boundaries between media. Refraction occurs when wave speed changes due to medium variation.

Diffraction

Sound waves bend around obstacles and openings. Diffraction more prominent at longer wavelengths.

Absorption and Attenuation

Energy loss due to medium viscosity and thermal conduction. High frequencies attenuate faster.

Speed of Sound

Dependence on Medium

Speed increases with medium elasticity and decreases with density. Fastest in solids, slower in liquids, slowest in gases.

Temperature Effect

In gases, speed increases approximately 0.6 m/s per °C rise. Due to increased molecular kinetic energy.

Typical Values

Air at 20°C: ~343 m/s. Water: ~1482 m/s. Steel: ~5960 m/s.

Formula for Speed in Gas

v = √(γRT/M), where γ = adiabatic index, R = gas constant, T = absolute temperature, M = molar mass.

Mathematical Description

Wave Equation

One-dimensional sound wave satisfies: ∂²p/∂x² = (1/v²) ∂²p/∂t², p = pressure variation, v = speed of sound.

Harmonic Waveform

p(x,t) = p₀ sin(kx - ωt + φ), where p₀ = amplitude, k = wave number, ω = angular frequency, φ = phase.

Relationship Between Variables

v = f × λk = 2π / λω = 2π fT = 1 / f

Intensity and Power

Intensity I = power per unit area. I ∝ A² v ρ, where ρ = medium density.

Acoustics and Sound Intensity

Sound Intensity Level

Measured in decibels (dB). L = 10 log₁₀(I/I₀), I₀ = 10⁻¹² W/m² (threshold of hearing).

Loudness Perception

Logarithmic scale correlates with human ear sensitivity. Equal increments in dB correspond to perceived doubling of loudness.

Sound Pressure Level

Related to pressure fluctuations: SPL = 20 log₁₀(p/p₀), p₀ = 20 μPa.

Acoustic Impedance

Ratio of acoustic pressure to particle velocity: Z = ρv. Determines reflection and transmission at boundaries.

Doppler Effect

Principle

Change in observed frequency due to relative motion between source and observer. Frequency increases if approaching, decreases if receding.

Mathematical Formula

f' = f (v ± v_o) / (v ∓ v_s)where:f' = observed frequencyf = source frequencyv = speed of sound in mediumv_o = velocity of observer (positive if moving towards source)v_s = velocity of source (positive if moving towards observer)

Applications

Radar speed detectors, medical ultrasound imaging, astronomy (redshift/blueshift), navigation aids.

Limitations

Only valid for velocities significantly less than speed of sound. Complex for media in motion or multiple sources.

Resonance in Sound Waves

Definition

Condition when frequency of external force matches natural frequency of system. Results in maximum amplitude oscillations.

Resonators

Examples: air columns in pipes, strings in musical instruments, cavities. Resonant frequencies determined by geometry and boundary conditions.

Mathematical Conditions

Standing waves form when path length equals integer multiples of half wavelengths: L = n(λ/2), n=1,2,...

Practical Uses

Musical tuning, architectural acoustics, noise control, ultrasonic cleaning.

Wave Interference and Standing Waves

Constructive and Destructive Interference

Superposition leads to amplitude enhancement or cancellation depending on phase difference.

Standing Waves

Formed by interference of two waves traveling in opposite directions. Nodes: zero displacement, antinodes: maximum displacement.

Mathematical Description

y(x,t) = 2A sin(kx) cos(ωt)Nodes at: x = n (λ/2)Antinodes at: x = (2n + 1) (λ/4)

Applications

Musical instrument acoustics, noise cancellation, ultrasonic resonators.

Human Hearing and Perception

Audible Frequency Range

20 Hz to 20 kHz. Sensitivity varies with age and health.

Loudness and Intensity

Loudness subjective, related to sound intensity and frequency. Equal loudness contours illustrate ear response.

Pitch Perception

Determined primarily by frequency. Complex sounds contain multiple frequencies perceived as timbre.

Thresholds and Damage

Threshold of hearing ~0 dB SPL. Prolonged exposure above 85 dB causes hearing damage.

Applications of Sound Waves

Communication

Speech and music transmission via air as medium. Microphones and speakers convert sound to electrical signals and back.

Medical Ultrasound

High-frequency sound waves for imaging soft tissues. Non-invasive, real-time diagnostics.

Sonar and Navigation

Echo-location using sound pulses in water or air. Used in submarines, fish-finders, and bats.

Industrial Uses

Ultrasonic cleaning, flaw detection in materials, welding, and flow measurements.

Entertainment and Acoustics

Design of auditoriums, musical instruments, noise control, and audio technology development.

Experimental Methods

Measuring Speed of Sound

Time-of-flight measurements using pulses over known distances. Resonance tube experiments.

Frequency and Wavelength Determination

Using oscilloscopes, signal generators. Visualizing waveforms and interference patterns.

Intensity and Decibel Level Measurement

Sound level meters with calibrated microphones. Calibration against standard sources.

Visualization Techniques

Acoustic interferometry, Schlieren photography, laser Doppler vibrometry.

References

• Kinsler, L.E., Frey, A.R., Coppens, A.B., Sanders, J.V., Fundamentals of Acoustics, Wiley, 4th ed., 1999, pp. 1-560.
• Pierce, A.D., Acoustics: An Introduction to Its Physical Principles and Applications, Acoustical Society of America, 1989, pp. 1-425.
• Morse, P.M., Ingard, K.U., Theoretical Acoustics, Princeton University Press, 1968, pp. 1-927.
• Rossing, T.D., The Science of Sound, Addison-Wesley, 3rd ed., 2002, pp. 1-480.
• Blackstock, D.T., Fundamentals of Physical Acoustics, Wiley-Interscience, 2000, pp. 1-600.