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PHY 1020 – Chapter 7 Notes: Waves, UFOs, Earthquakes & Music

Fundamentals of Waves

  • A wave = disturbance/oscillation that transports energy through space & matter.
    • Characterized by:
    • Amplitude y → displacement (for transverse) or density/pressure variation (for longitudinal).
    • Wavelength \lambda → spatial period/"size" of the wave (peak-to-peak or compression-to-compression distance).
    • Velocity v → how fast one wavelength passes a fixed point.
    • Measuring the time for a single wavelength to pass allows determination of v.
  • Energy propagation without net mass transport; individual particles mainly oscillate about equilibrium.
  • Real-world importance: governs light, sound, earthquakes, ocean motion, musical instruments, radio transmissions, etc.

Classification of Waves

  • Longitudinal Waves
    • Oscillation direction ⟂ energy flow? No, parallel.
    • Regions of compression & rarefaction (e.g., sound, P-waves in earthquakes).
  • Transverse Waves
    • Oscillation direction ⟂ energy flow (e.g., waves on strings, water surface ripples’ vertical motion, S-waves in earthquakes).
  • Water waves contain both longitudinal & transverse components, especially in deep water.
  • Superposition Principle
    • Two or more waves traveling in the same medium add algebraically.
    • Resultant = another wave of same type (if direction is same) with amplitude determined by vector sum.
    • Lays foundation for interference, beats, standing waves, diffraction.

Mathematical Description of a Harmonic Wave

  • Period (time for one full cycle): T = \frac{\lambda}{v}.
  • Frequency (cycles per second): f = \frac{v}{\lambda}.
    • Relation f = \frac{1}{T}.
  • Interference
    • Constructive → wave crests align, amplitude increases.
    • Destructive → crest meets trough, amplitude decreases or cancels.
    • Responsible for patterns in water waves, sound beats, light fringes, noise-cancelling headphones, etc.
  • Huygens Principle & Diffraction
    • Every point on a wavefront = source of secondary spherical wavelets.
    • For an aperture of size D, spread S after traveling distance R with wavelength L (here $L$ represents \lambda):
      S = \frac{L R}{D}.
    • Small opening (≈ wavelength) → large spread; large opening → beam remains collimated.

Sound: Properties & Perception

  • Pitch vs. Frequency
    • Higher f → higher perceived pitch.
    • Musical scales built on specific frequency ratios (octave = doubling f).
  • Physical Mechanism
    • Sound = longitudinal compression waves in a medium; molecules oscillate, jiggling the eardrum.
    • Speed depends mainly on stiffness (bulk modulus), not on density alone → faster in solids than air.
  • Beats
    • When two close frequencies f1 & f2 overlap, amplitude modulates at beat frequency:
      fB = |f2 - f_1|.
    • Musicians use beats to tune instruments (zero beats = perfect unison).

Sound Intensity & the Decibel Scale

  • Intensity I (W/m²) quantifies power per area.
  • Reference threshold of hearing: I_0 = 1.0 \times 10^{-12}\,\text{W/m}^2.
  • Sound level in decibels (logarithmic, base-10): I{\text{dB}} = 10 \log \left(\frac{I}{I0}\right).
    • Example (vacuum cleaner scenario): I = 2.5 \times 10^{-4}\,\text{W/m}^2 → I_{\text{dB}} \approx 84\,\text{dB}.
  • Selected everyday values (approx.):
    • 140 dB → pain threshold, jet engine close.
    • 100 dB → rock concert (15 ft), lawn mower (3 ft).
    • 60 dB → ordinary speech (3 ft).
    • 30 dB → quiet library.
    • 10 dB → barely audible whisper.
  • Ethical/health angle: prolonged exposure >85 dB can cause hearing loss; informs workplace regulations, urban noise codes, product design.

Doppler Effect

  • Definition: observed change in frequency due to relative motion between source & observer.
    • Approaching source → f{\text{observed}} > f{\text{emitted}} (pitch rises).
    • Receding source → f{\text{observed}} < f{\text{emitted}}.
    • At instant of passing → frequencies match.
  • Applications: radar speed guns, medical ultrasound, astrophysical red/blue shifts, weather Doppler radar.

SOFAR (Sound Fixing And Ranging) Channel & “UFO” Acoustics

  • Mid-ocean depth where sound speed is minimum due to interplay of temperature (drops) & pressure (rises).
  • Acts as a natural waveguide; low-frequency sound can propagate thousands of km with little loss (refraction keeps sound near channel axis).
  • Military & marine-biology relevance: submarine communication, tracking whales, investigating unexplained low-frequency “UFO” noises.

Earthquakes & Seismic Waves

  • Generated by sudden release of tectonic stress.
  • Wave Types:
    • P (Primary or Pressure) Waves
    • Longitudinal; fastest; arrive first.
    • S (Secondary or Shear) Waves
    • Transverse; slower; arrive second; do not travel through liquids (inner core study).
    • L (Long) Waves
    • Surface waves; mixture of longitudinal & transverse; arrive last; cause most damage.
  • Magnitude relation (Richter-type): 10^M = \frac{A}{A0(d)} \quad \Rightarrow \quad M = \log \left(\frac{A}{A0(d)}\right)
    • A = measured amplitude, A_0(d) = reference amplitude at distance d.
  • Societal implications: building codes, early-warning systems, tsunami generation.

Water Waves: Depth Dependence

  • Classification by ratio of wavelength L to water depth D.
    • Deep-Water Wave
    • D > \tfrac{L}{2}; wave base doesn’t touch seafloor; dispersive (speed depends on L).
    • Intermediate-Water Wave
    • \tfrac{L}{20} < D < \tfrac{L}{2}; partial seafloor interaction.
    • Shallow-Water Wave
    • D < \tfrac{L}{20}; whole column feels bottom; non-dispersive; speed approximated by
      v \approx \pi \sqrt{D} or more generally v = \sqrt{gD} (g = gravitational acceleration).
  • Same depth can host any class depending on L; same L can be any class depending on depth.
  • Practical: coastal engineering, surfing conditions, tsunami modeling.

Tsunamis

  • Generated by seafloor displacement (earthquakes, landslides, volcanoes).
  • Extremely long wavelength (>>100 km) → behave as shallow-water waves even in deep ocean.
  • Speed approximated by
    v \approx \sqrt{g D} → for D \approx 3000\,\text{m} ⇒ v \approx 171\,\text{m/s} \approx 386\,\text{mph}.
  • 1000 mile distant event → warning time ≈2.6 h (important for Hawaii, Pacific rims; less critical for continental US east coast).
  • Coastal run-up: wave slows & grows in height, flooding land.
  • Necessity for early-warning networks (buoys, seismic+pressure sensors) to mitigate disasters (e.g., 2004 Indian Ocean, 2011 Japan).

Cross-Topic Connections & Implications

  • Shared mathematics: same \sin/\cos solutions describe guitar strings, ocean tides, EM waves.
  • Interference principles underlie noise-cancelling tech, radio antennas, seismic tomography.
  • Logarithmic decibel scale parallels Richter scale for quakes → human sense organs & hazard metrics both compress large dynamic ranges.
  • Ethical/practical aspects:
    • Engineering for sound comfort vs. hearing safety.
    • Coastal zoning & evacuation planning against tsunamis.
    • Building codes respecting S-wave shear stresses.
    • International monitoring (SOFAR) intersects with marine conservation & defense policies.
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