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.