🍏Principles of Physics I Unit 14 – Oscillations and Waves

Key Concepts and Definitions

• Oscillation involves repetitive back and forth motion about an equilibrium position
• Period $T$ represents the time for one complete oscillation cycle
• Frequency $f$ measures the number of oscillations per unit time, related to period by $f = \frac{1}{T}$
• Amplitude $A$ defines the maximum displacement from equilibrium position
• Angular frequency $\omega$ relates to frequency by $\omega = 2\pi f$
• Wave involves propagation of energy through a medium without net transport of matter
• Wavelength $\lambda$ measures the distance between corresponding points on adjacent waves
• Wave speed $v$ relates to frequency and wavelength by $v = f\lambda$

Types of Oscillations

• Harmonic oscillation exhibits sinusoidal motion with equal periods and amplitudes
  • Examples include pendulums and mass-spring systems
• Anharmonic oscillation deviates from sinusoidal behavior with varying periods or amplitudes
• Linear oscillation involves motion along a straight line (one dimension)
  • Examples include a mass sliding on a frictionless surface attached to a spring
• Angular oscillation involves rotational motion about an axis (two dimensions)
  • Examples include a torsional pendulum or a swinging pendulum
• Coupled oscillation occurs when two or more oscillating systems interact and influence each other's motion
• Self-excited oscillation extracts energy from a steady source to sustain motion
  • Examples include a clock pendulum or a violin string

Simple Harmonic Motion

• Simple harmonic motion (SHM) is a special case of periodic motion where restoring force is directly proportional to displacement
• Follows the equation $F = -kx$, where $k$ is the spring constant and $x$ is displacement from equilibrium
• Period of SHM depends only on mass $m$ and spring constant $k$, given by $T = 2\pi \sqrt{\frac{m}{k}}$
• Displacement $x$ varies sinusoidally with time $t$ according to $x(t) = A \cos(\omega t + \phi)$
  • $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is initial phase angle
• Velocity $v$ in SHM is given by $v(t) = -A\omega \sin(\omega t + \phi)$
• Acceleration $a$ in SHM is given by $a(t) = -A\omega^2 \cos(\omega t + \phi)$
  • Acceleration is always directed towards equilibrium position

Energy in Oscillating Systems

• Total energy $E$ in SHM consists of kinetic energy $K$ and potential energy $U$
  • $E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$
• Kinetic energy varies as $K = \frac{1}{2}mv^2 = \frac{1}{2}kA^2 \sin^2(\omega t + \phi)$
• Potential energy varies as $U = \frac{1}{2}kx^2 = \frac{1}{2}kA^2 \cos^2(\omega t + \phi)$
• Energy oscillates between kinetic and potential forms, with total energy remaining constant in absence of dissipation
• Power $P$ measures rate of energy transfer, given by $P = Fv = -kxv$
  • Average power over a complete cycle is zero in SHM

Damped and Forced Oscillations

• Damped oscillation involves gradual decrease in amplitude due to dissipative forces (friction or resistance)
  • Follows the equation $F = -kx - bv$, where $b$ is damping coefficient
• Critically damped system returns to equilibrium in shortest time without oscillating
• Overdamped system slowly returns to equilibrium without oscillating
• Underdamped system oscillates with gradually decreasing amplitude before reaching equilibrium
• Forced oscillation occurs when an external periodic force drives the oscillating system
  • Driving frequency can be different from natural frequency of the system
• Resonance occurs when driving frequency matches natural frequency, leading to large amplitude oscillations
  • Can be beneficial (musical instruments) or destructive (bridge collapse)

Wave Properties and Behavior

• Waves transport energy through a medium without net transport of matter
• Transverse waves oscillate perpendicular to direction of wave propagation
  • Examples include light waves and vibrations on a string
• Longitudinal waves oscillate parallel to direction of wave propagation
  • Examples include sound waves and pressure waves in fluids
• Reflection occurs when a wave encounters a boundary and reverses direction
  • Angle of incidence equals angle of reflection
• Refraction occurs when a wave changes speed and direction while passing through a boundary between different media
  • Governed by Snell's law: $\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}$
• Diffraction involves bending of waves around obstacles or through openings
  • More pronounced when wavelength is comparable to obstacle size
• Interference occurs when two or more waves overlap and combine
  • Constructive interference results in increased amplitude
  • Destructive interference results in decreased amplitude

Mathematical Descriptions of Waves

• Wave equation relates wave speed $v$, wavelength $\lambda$, and frequency $f$: $v = f\lambda$
• Angular wave number $k$ measures number of wave cycles per unit distance, given by $k = \frac{2\pi}{\lambda}$
• Wave function $y(x,t)$ describes wave shape and propagation: $y(x,t) = A \sin(kx - \omega t + \phi)$
  • $A$ is amplitude, $k$ is angular wave number, $\omega$ is angular frequency, and $\phi$ is initial phase angle
• Phase velocity $v_p$ measures speed of a point of constant phase, equal to wave speed $v$
• Group velocity $v_g$ measures speed of a wave packet or envelope, can differ from phase velocity in dispersive media
• Intensity $I$ measures power per unit area carried by a wave: $I = \frac{1}{2}\rho v\omega^2 A^2$
  • $\rho$ is density of the medium, $v$ is wave speed, $\omega$ is angular frequency, and $A$ is amplitude

Applications and Real-World Examples

• Musical instruments rely on oscillations of strings, air columns, or membranes to produce sound waves
  • Guitar strings exhibit transverse standing waves with nodes at fixed ends
  • Organ pipes and flutes produce longitudinal standing waves in air columns
• Seismic waves from earthquakes include both transverse (S-waves) and longitudinal (P-waves) components
  • P-waves travel faster and arrive first, while S-waves cause more damage due to larger amplitudes
• Electromagnetic waves, including light and radio waves, are transverse waves that can propagate through vacuum
  • Different frequencies correspond to different parts of the electromagnetic spectrum (radio, microwave, infrared, visible, ultraviolet, X-ray, gamma)
• Doppler effect occurs when a wave source or observer is moving relative to the medium
  • Observed frequency increases when source and observer move towards each other
  • Observed frequency decreases when source and observer move away from each other
• Interferometry uses wave interference to make precise measurements of distances or changes in distance
  • Applications include gravitational wave detection (LIGO) and measuring surface flatness in manufacturing