Waves and Oscillations Flashcards

Unit 4: Waves and Oscillations

Oscillations and Simple Harmonic Motion

Oscillations refer to periodic (or repetitive) motion. Key concepts in oscillations include:

• Period (T): The time it takes for one complete cycle of motion.
• Frequency (f): The number of cycles in a given time, measured in Hertz (Hz), where $1 ext{ Hz} = 1 ext{ cycle/sec}$.
• Hooke’s Law describes the restoring force in a spring:
  F_H = -k riangle x
  where $k$ is the spring constant and $ riangle x$ is the displacement from equilibrium.
• Amplitude (A): The maximum displacement from the equilibrium position.
• The relationship between frequency and period is:
  f = rac{1}{T}
• The period of a spring-mass system can also be expressed as:
  T = 2rac{ ext{m}}{k}
  where m is the mass attached to the spring.

The position of a mass in simple harmonic motion can be described with the equation:
riangle x = A ext{cos} rac{2 ext{π}t}{T}

Energy in Simple Harmonic Motion

Energy considerations in simple harmonic motion involve both potential energy and kinetic energy.

• The spring potential energy is expressed as:
  PE_S = rac{1}{2} kx^2
  where x is the displacement from the equilibrium position.
• The total mechanical energy (ME) is conserved in the absence of friction:
  ME = KE + PE_S
• The relationship can be expressed as:
  $$ MEi = MEf
  ightsquigarrow rac{1}{2}kA^2 = rac{1}{2}mv^2 + rac{1}{2}kx^2 $$

Waves: Definitions and Properties

Waves are disturbances that propagate through a medium, transferring energy without the transfer of matter. Key terms include:

• Wave Velocity (vw): Characterized by the relationship:
  vw = rac{λ}{T} = fλ
  where λ is the wavelength and f is the frequency.
• Types of Waves:
  • Transverse Waves: Oscillation is perpendicular to the direction of wave travel.
  • Longitudinal Waves: Oscillation is parallel to the direction of wave travel.

Wave Behavior and Superposition

• Superposition Principle: When two waves meet, they interfere with each other. This can result in:
  • Constructive Interference: When waves reinforce each other, leading to greater amplitude.
  • Destructive Interference: When waves cancel each other out, leading to reduced amplitude.

Standing Waves

• Standing waves occur when two identical waves travel in opposite directions and interfere with one another. Key features include:
  • Nodes: Points that do not move (destructive interference).
  • Antinodes: Points that have maximum amplitude (constructive interference).

The Doppler Effect

• The Doppler Effect describes the change in frequency (or pitch) of a wave in relation to an observer moving relative to the source of the wave. When the source moves towards the observer, the frequency appears higher, and when moving away, it appears lower, expressed by:
  f{observed} = f{source} rac{vw ext{±} v{observer}}{vw ext{±} v{source}}

Example Problem Problem Solving Strategy

When tackling problems:

1. Sketch: Define the reference frame and mark variables.
2. Math: Identify appropriate equations, perform algebra before calculations, check for units and reasonableness of answers.