Waves and Sound

7.1 General Wave Characteristics

Waves can be transverse or longitudinal, described by wavelength, frequency, speed, and amplitude. Superposition, standing waves, and resonance govern their interactions and natural frequencies.

• Wave Types:
  • Transverse waves: oscillation ⟂ to propagation (e.g., wave in stadium)
  • Longitudinal waves: oscillation ∥ to propagation, compressions and rarefactions (e.g., slinky)

• Wave properties:
  • Wavelength: $\lambda$ distance between crests
  • Frequency: $f$ Hz (cycles per second)
  • Propagation speed: $v = \lambda f$
  • Period: $T = 1/f$
  • Angular frequency: $\omega = 2\pi f = 2\pi/T$ (rad/s)
  • Amplitude (A): max displacement from equilibrium

  

• Phase:
  • In phase: crests/troughs align → difference 0
  • Out of phase: crests do not align

• Superposition:
  • Constructive: amplitudes add
  • Destructive: amplitudes subtract (noise cancelling)
• Traveling and standing waves:
  • Traveling: moves, reflected/inverted at boundary
  • Standing: fixed nodes, fluctuating antinodes
• Resonance:
  • Natural frequencies → object vibrates
  • Timbre: quality of sound (single or multiple frequencies)
  • Forced oscillation: amplitude maximized when applied f ≈ natural f
  • Damping reduces amplitude

7.2 Sound

Sound is a longitudinal wave in a medium. Pitch, Doppler effect, intensity, standing waves, and applications like ultrasound are discussed.

• Propagation:
  • Cannot travel through vacuum
  • $v = \sqrt{B/\rho}$ (B = bulk modulus, ρ = density)
  • Air: 343 m/s
• Production:
  • Mechanical disturbance of particles; solids vibrate for musical sounds
  • Pitch depends on air column length
  • Human voice: vocal cords; tension controls pitch
• Frequency and pitch:
  • Low f → low pitch, high f → high pitch
  • <20 Hz: infrasonic, >20 kHz: ultrasonic
• Doppler effect:
  • Perceived f differs if source/detector moves
  • $f' = f \frac{v \pm v_D}{v \mp v_S}$
  • Top sign: toward, bottom: away
  • Applies to sound and light (blue/red shift)

• Shock waves:
  • Object ≥ speed of sound → overlapping wave fronts → sonic boom
• Intensity and loudness:
  • Intensity: $I = P/A$ (power per area)
  • I ∝ A², I ∝ 1/r²
  • Softest sound: $10^{-12}\ W/m^2$
  • Sound level: $\beta = 10 \log(I/I_0)$(dB)
  • Attenuation: damping reduces amplitude
  • Beat frequency: $f_\text{beat} = |f_1 - f_2|$
• Standing waves:
  • 2 waves f traveling opposite → amplitude oscillates
  • Nodes: no movement; antinodes: max movement
• Strings:
  • Secured ends → nodes
  • Wavelength: $\lambda = 2L/n$, f = nv/2L, n = harmonic
• Open pipes:
  • Antinodes at open ends, L = λ/2 for first harmonic
  • Harmonics: L = λ/2, λ, 3λ/2 …
• Closed pipes:
  • Node at closed, antinode at open
  • Only odd harmonics: $\lambda = 4L/n$, f = nv/4L
  • First overtone = 3rd harmonic
• Ultrasound:
  • High f sound waves beyond hearing
  • Imaging via reflection; doppler mode measures flow
  • Applications: prenatal imaging, kidney stone fragmentation, tumor ablation, dental cleaning

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