In-Depth Notes on Waves and Sound

Waves

  • Definition: A wave is a disturbance or oscillation that transfers energy through space and matter.

    • Amplitude ($y$): Displacement or density of the wave.

    • Wavelength ($\lambda$): Size of the wave.

    • Wave Velocity: Determined by time taken for one wavelength to pass a point.

Mathematical Descriptions of Waves

  • Period ($T$): The time for one wavelength to pass a point.

    • Formula: T = \frac{\lambda}{v}

  • Frequency ($f$): Number of wavelengths per second.

    • Formula: f = \frac{v}{\lambda}

Types of Waves

  • Longitudinal Waves: Particle displacement is parallel to wave direction (e.g., sound waves).

  • Transverse Waves: Particle displacement is perpendicular to wave direction (e.g., water waves).

  • Superposition: When two waves meet, they combine to form a new wave (interference).

    • Constructive Interference: Results in an increase in amplitude.

    • Destructive Interference: Results in a decrease in amplitude.

The Huygens Principle

  • Describes how waves propagate.

    • Formula: S = \frac{LR}{D}

    • Where $S$ = spread, $L$ = wavelength, $R$ = distance from the opening, $D$ = size of the opening.

Sound Waves

  • Frequency and Pitch: Higher frequency leads to a higher pitch.

  • Compression Waves: Movement of molecules creating pressure variations (sound).

    • Sound travels faster in stiffer materials rather than denser ones.

Beats

  • Occur due to interference between two sound waves.

Sound Intensity and Decibels

  • Threshold of Hearing: Intensity level below which sound cannot be perceived.

    • Example of intensity: I[dB] = 10 \log \frac{2.5 \times 10^{-4}}{1.0 \times 10^{-12}} = 84 dB

  • Common Sounds:

    • Rock band (110 dB), Whisper (30 dB), Threshold of pain (140 dB).

Doppler Effect

  • A change in frequency and wavelength of a wave in relation to an observer moving relative to its source.

    • Observed with moving vehicles (siren or horn) as the sound source approaches and recedes.

Sound Channel and UFOs

  • SOFAR Channel: A horizontal layer in the ocean where the speed of sound is at a minimum, allowing sound waves to travel long distances without losing energy.

Earthquakes

  • Consist of both compression waves (P-waves) which arrive first and shear waves (S-waves) which arrive second.

    • Equations describing the magnitude of earthquakes:

    • M = \log \frac{A}{A_0} (\delta)

    • Where $M$ = magnitude, $A$ = amplitude, $A_0$ = reference amplitude, and $\delta$ = distance from the epicenter.

Water Waves

  • Behavior of waves is influenced by water depth.

    • Shallow Water Waves: Wave interacts with the sea floor.

    • Deep Water Waves: Wave does not interact with the sea floor.

    • Critical Wave Depth Relationships:

      • Shallow-water waves behave differently than deep-water waves due to depth.

Tsunamis

  • Waves caused by underwater disturbances.

    • Warning time for a tsunami 1000 miles away is about 2.6 hours.

    • Velocity formula: v \approx \pi 3000m \approx 171m/s \approx 386mph.