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.