Wave and sound

Study Summary: Waves

1. Wave Motion Fundamentals

• Definition: A periodic disturbance in a medium that carries energy from one point to another.
• Types of Waves:
  • Longitudinal Waves: Particles vibrate parallel to the direction of wave propagation (e.g., sound waves, compressional waves).
  • Transverse Waves: Particles vibrate perpendicular to the direction of wave propagation (e.g., electromagnetic waves, waves on a string).

2. Wave Nomenclature

2.1. With respect to Distance (Spatial)

• Amplitude (A): Maximum displacement from the equilibrium position.
• Crest: Point of maximum positive displacement.
• Trough: Point of maximum negative displacement.
• Wavelength (λ): The distance over which the wave repeats (e.g., between two consecutive crests or troughs).
• Wave Number (k): The number of wavelengths per 2π units of distance.
  • Formula: k = 2π / λ (measured in m⁻¹)

2.2. With respect to Time (Temporal)

• Period (T): The time over which the wave repeats (e.g., time between two consecutive crests passing a fixed point).
• Angular Frequency (ω): Related to the period.
  • Formula: ω = 2π / T
• Frequency (f): The number of waves that occur in 1 second.
  • Formula: f = 1 / T (measured in Hertz, Hz)
  • Relation: ω = 2πf

3. Wave Equation and Velocity

• General Wave Equation (propagating in positive x-direction):
  • y = A sin(kx - ωt)
  • With initial phase (φ): y = A sin(kx - ωt + φ)
• Phase Velocity (v): The velocity with which a constant phase is propagated.
  • Formula: v = ω / k
  • Relation with frequency and wavelength: v = fλ
  • For wave propagating in negative x-direction: y = A sin(kx + ωt) or velocity is negative.
• Alternative forms of the wave equation:
  • y = A sin(2π(x/λ - ft))
  • y = A sin(2π(x/λ - t/T))
  • y = A sin(2π/λ (x - vt))
  • y = A sin(k(x - (ω/k)t)) = A sin(k(x - vt))

4. Speed of Waves on a String

• For a string under tension (F) with mass per unit length (μ), the speed of transverse waves is:
  • Formula: v = sqrt(F / μ)
  • Observations: Speed is large with large tension and small mass per unit length.

5. Particle Velocity and Acceleration in Wave Motion

• Particle Velocity (u): The velocity of individual particles of the medium, distinct from wave velocity.
  • u = ∂y/∂t = -Aω cos(kx - ωt + φ)
• Particle Acceleration (a):
  • a = ∂²y/∂t² = -Aω² sin(kx - ωt + φ) = -ω²y
  • This shows that each particle performs Simple Harmonic Motion (SHM) about its equilibrium position.

6. Interference of Waves

• Superposition Principle: When two or more waves overlap, the resultant displacement at any point is the vector sum of the displacements of the individual waves.
• Combining two waves with same amplitude, frequency, and wave number but a phase difference (φ):
  • y₁ = A sin(kx - ωt + φ)
  • y₂ = A sin(kx - ωt)
  • Resultant wave: y = y₁ + y₂ = 2A cos(φ/2) sin(kx - ωt + φ/2)
• Types of Interference:
  • Constructive Interference: Occurs when waves combine to produce a larger amplitude.
    • Condition for phase difference: φ = 2nπ (where n is an integer)
    • Condition for path difference: Δx = nλ
  • Destructive Interference: Occurs when waves combine to produce a smaller or zero amplitude.
    • Condition for phase difference: φ = (2n + 1)π (where n is an integer)
    • Condition for path difference: Δx = (n + 1/2)λ

7. Standing Waves

• Formation: Result from the superposition of two identical waves traveling in opposite directions (e.g., incident and reflected waves on a string with rigid supports).
• Equation: y = 2A sin(kx) cos(ωt)
• Characteristics:
  • Nodes: Points where the displacement is always zero.
    • Condition: kx = nπ, so x = nλ/2 (n = 0, 1, 2, ...)
    • Distance between consecutive nodes is λ/2.
  • Antinodes: Points where the displacement is maximum.
    • Condition: kx = (2n + 1)π/2, so x = (2n + 1)λ/4 (n = 0, 1, 2, ...)
    • Distance between consecutive antinodes is λ/2.
  • No net energy propagation.
  • Particles perform SHM, with amplitude varying with position.

8. Resonance

• Definition: When an external forcing frequency matches the natural frequency of a system, leading to a large amplitude of vibration.
• Natural Frequencies for a String Fixed at Both Ends:
  • The length of the string (L) must be an integer multiple of half-wavelengths: L = nλ/2
  • Substituting λ = v/f and v = sqrt(F/μ):
  • Formula: f_n = (n / 2L) * sqrt(F / μ) (where n = 1, 2, 3, ... are the harmonics)

9. Sound Waves

• Nature: Longitudinal mechanical waves, requiring a medium for propagation.
• Mechanism: Involves compressions (regions of higher density/pressure) and rarefactions (regions of lower density/pressure).
• Audible Range: 20 Hz to 20,000 Hz. Frequencies below 20 Hz are infrasonic, above 20,000 Hz are ultrasonic.
• Speed of Sound:
  • In a fluid: v = sqrt(B / ρ₀) (where B = Bulk Modulus, ρ₀ = undisturbed density)
  • In an ideal gas: v = sqrt(γRT / M) (where γ = ratio of specific heats, R = gas constant, T = absolute temperature, M = molecular mass)
  • Approximately for air (0°C): v ≈ 331.11 m/s
  • Generally, v ∝ sqrt(T).
• Pressure Variation in Sound Wave:
  • p = -B(∂y/∂x) = P cos(kx - ωt) (where P is pressure amplitude)

10. Beats

• Formation: Occurs when two sound waves of slightly different frequencies (f₁ and f₂) superimpose.
• Resultant Wave: Has an average frequency f_avg = (f₁ + f₂) / 2 and an amplitude that varies at the beat frequency.
• Beat Frequency: The frequency at which the amplitude of the resultant sound fluctuates (heard as pulsations in loudness).
  • Formula: f_beat = |f₂ - f₁|

11. Doppler Effect

• Definition: The apparent change in frequency of a wave due to the relative motion between the source of the wave and the observer.
• General Formula: f' = f (v ± v₀) / (v ∓ v_s)
  • f': Observed frequency
  • f: Source frequency
  • v: Speed of sound in the medium
  • v₀: Speed of the observer
  • v_s: Speed of the source
  • Sign Convention:
    • Use +v₀ if observer moves towards source, -v₀ if observer moves away.
    • Use -v_s if source moves towards observer, +v_s if source moves away.

12. Pitch, Loudness, Quality/Timbre, Intensity, Decibel

• Pitch: Determined by the frequency of the sound wave. Higher frequency means higher pitch.
• Loudness: A subjective perception related to the intensity of the sound, which is proportional to the square of the wave's amplitude.
• Intensity (I): The power carried by sound waves per unit area in a direction perpendicular to the area.
  • Intensity falls off as the inverse square of the distance from the source: I ∝ 1/r².
• Decibel (dB): A logarithmic scale used to measure sound intensity level (L).
  • Formula: L = 10 log₁₀(I / I₀) dB
  • Reference Intensity (I₀): The threshold of human hearing, approximately 10⁻¹² W/m².
  • A 10 dB increase corresponds to a tenfold increase in intensity.
• Bel (B) and Neper (np): Other logarithmic units for intensity. 1 Bel = 10 dB.
• Quality/Timbre: Determined by the number and relative intensities of overtones (harmonics) present in the sound. This allows differentiation between sounds from different instruments even if they play the same fundamental note.