Wave Types and Interference

Wave Types: Fundamental Wave Characteristics

• Longitudinal waves: oscillations in the same direction as the propagation (e.g., sound waves).
• Transverse waves: oscillations across the direction of propagation (e.g., vibrations on a guitar string).
• Amplitude (Am): the maximum deflection from the x-axis.
• Wavelength (λ m): the distance between two wave crests or troughs.
• Frequency (f): the number of oscillations per second (Hz).
  • F = number of oscillations (n) / time (s)
• Period (T s): time for a complete oscillation (T=1).
• Wave velocity (v): v=λ·f
  • velocity(m/s) = wavelength(m) * frequency (hz)

Interference

• Superposition of waves resulting in a new wave pattern.
• Constructive interference: wave crests meet and reinforce each other.
• Destructive interference: wave crest meets a trough and weakens each other.

Standing Waves

• Waves of the same wavelength and amplitude moving towards each other interfere.
  • Certain points always have destructive interference, while others have constructive interference.
  • A wave appears still, oscillating up and down.

Refraction

• Change in a wave's direction when passing from one medium to another.
• Law of Refraction: sin(i) / sin(b) = v1 / v2
  • i = angle of incidence
  • b = angle of refraction
  • v1 and v2 are wave speeds in each medium.
• Refractive Index: the refraction ratio when light passes from air (vacuum) to the material in question.
• Refractive Index (n) = Speed of light in air (c) / Speed of light in the material (v)
• Optical Grating: a plate with many closely spaced slits that split light into a spectrum.
  • n·λ = d·sin(φ)
    • n is the order number
    • λ is the wavelength
    • d is the grating constant
    • φ is the angle of refraction.

Sound Waves and the Speed of Sound

• Sound waves are mechanical waves that travel through air or other media.
• 343 m/s in air at 20°C.

Sound Waves in tubes

• Frequencies where standing waves are formed are called resonance frequencies.
  • f: Frequency; the number of oscillations per second, measured in Hertz (Hz).
  • n: An integer representing the number of the nth partial or overtone.
  • vlyd: the speed of sound in the given medium, typically air.
    • In air at 20°C, it is about 343 m/s.
  • L: The length of the tube

Open tube

• Has antinodes on both ends and nodes in between.
  • L = n* λη / 2
  • fn = n* vlyd / 2*L

Half-open tube

• Has a node at the closed end and an antinode at the open end.
  • fn = (2n-1)*vlyd / 4L

Doppler Effect

• Change in the frequency of a wave when the source or observer moves.
  • f: the frequency of the wave emitted by the source, measured in Hertz (Hz).
  • f1: the frequency of the wave received by the observer; this frequency can differ from f due to the Doppler effect.
  • vlyd: the speed of sound in the given medium, typically air.
    • In air at 20°C, it is about 343 m/s.
  • u: the velocity of the source (or observer) relative to the medium (e.g., air), which can be positive or negative depending on whether the source is moving towards or away from the observer.
  • Formula for motion toward the observer: f1=vlyd / (vlyd-u) *f
  • Formula for motion away from the observer: f1=vlyd / (vlyd+u) *f