🔋College Physics I – Introduction Unit 16 – Oscillatory Motion and Waves

Key Concepts and Definitions

• Oscillation involves repetitive back-and-forth motion about an equilibrium position
• Period $T$ represents the time required for one complete oscillation cycle
• Frequency $f$ measures the number of oscillations per unit time, related to period by $f = \frac{1}{T}$
• Amplitude $A$ quantifies the maximum displacement from the equilibrium position
• Angular frequency $\omega$ describes the rate of change of the oscillation phase, given by $\omega = 2\pi f$
• Wavelength $\lambda$ is the distance between two consecutive points on a wave with the same phase
• Wave speed $v$ relates wavelength and frequency through the equation $v = f\lambda$

Types of Oscillations

• Harmonic oscillations exhibit sinusoidal motion with a constant amplitude and frequency
  • Examples include an ideal pendulum and a mass-spring system
• Anharmonic oscillations deviate from the sinusoidal pattern and may have varying amplitudes or frequencies
• Damped oscillations gradually decrease in amplitude over time due to energy dissipation
  • Caused by friction, air resistance, or other dissipative forces
• Forced oscillations occur when an external periodic force drives the oscillatory system
  • The driving force can alter the amplitude and frequency of the oscillation
• Coupled oscillations involve the interaction and energy transfer between two or more oscillating systems
  • Demonstrated by coupled pendulums or resonating tuning forks

Simple Harmonic Motion

• Simple harmonic motion $SHM$ is a special case of oscillation with a restoring force proportional to the displacement
• The restoring force $F$ in SHM is given by Hooke's law: $F = -kx$, where $k$ is the spring constant and $x$ is the displacement
• The period of SHM is independent of the amplitude and depends only on the system's properties
  • For a mass-spring system, the period is $T = 2\pi \sqrt{\frac{m}{k}}$, where $m$ is the mass
• The position $x$ as a function of time $t$ in SHM is described by $x(t) = A \cos(\omega t + \phi)$, where $\phi$ is the initial phase
• Velocity $v$ and acceleration $a$ in SHM are given by the first and second derivatives of position, respectively

Energy in Oscillatory Systems

• In SHM, the total energy $E$ remains constant and is the sum of kinetic energy $KE$ and potential energy $PE$
  • $E = KE + PE = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$
• Kinetic energy is maximum at the equilibrium position, where the velocity is highest and the displacement is zero
• Potential energy is maximum at the extremes of the oscillation, where the displacement is greatest and the velocity is zero
• Energy continuously transforms between kinetic and potential forms during the oscillation cycle
• The principle of conservation of energy applies to oscillatory systems in the absence of dissipative forces

Damped and Forced Oscillations

• Damped oscillations experience a gradual decrease in amplitude due to energy dissipation
  • The damping force is proportional to the velocity and opposes the motion
• The damping coefficient $b$ quantifies the strength of the damping force: $F_d = -bv$
• Critically damped systems return to equilibrium in the shortest time without oscillating
• Overdamped systems slowly approach equilibrium without oscillating
• Underdamped systems exhibit decaying oscillations before reaching equilibrium
• Forced oscillations occur when an external periodic force drives the system
  • The driving force can be described by $F_d = F_0 \cos(\omega_d t)$, where $F_0$ is the force amplitude and $\omega_d$ is the driving frequency
• Resonance occurs when the driving frequency matches the system's natural frequency, resulting in large amplitude oscillations

Wave Properties and Behavior

• Waves transport energy through a medium without permanently displacing the medium itself
• Transverse waves oscillate perpendicular to the direction of wave propagation $e.g.$ light waves$
• Longitudinal waves oscillate parallel to the direction of wave propagation $e.g.$ sound waves$
• Reflection occurs when a wave encounters a boundary and bounces back, with the angle of incidence equal to the angle of reflection
• Refraction happens when a wave changes direction as it passes from one medium to another due to a change in wave speed
• Diffraction is the bending of waves around obstacles or through openings, more prominent when the wavelength is comparable to the obstacle size
• Interference occurs when two or more waves overlap, resulting in constructive $amplitude increase$ or destructive $amplitude decrease$ interference
  • Constructive interference happens when the waves are in phase, while destructive interference occurs when the waves are out of phase

Mathematical Descriptions of Waves

• The wave equation is a partial differential equation that describes the propagation of waves: $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$
  • $y(x,t)$ represents the wave displacement as a function of position $x$ and time $t$
• For a sinusoidal wave, the displacement can be expressed as $y(x,t) = A \sin(kx - \omega t + \phi)$
  • $k$ is the wave number, defined as $k = \frac{2\pi}{\lambda}$
• The phase velocity $v_p$ is the speed at which a point of constant phase moves, given by $v_p = \frac{\omega}{k}$
• The group velocity $v_g$ is the speed at which the envelope of a wave packet propagates, determined by $v_g = \frac{d\omega}{dk}$
• Fourier analysis allows the decomposition of complex waveforms into a sum of simple sinusoidal components
  • The Fourier transform converts a time-domain signal into its frequency-domain representation

Applications and Real-World Examples

• Musical instruments rely on oscillations to produce sound waves
  • String instruments $guitar, violin$ utilize the vibration of strings, while wind instruments $flute, trumpet$ rely on the oscillation of air columns
• Seismic waves generated by earthquakes are used in geophysical exploration to study the Earth's interior structure
• Electromagnetic waves, including light and radio waves, exhibit oscillatory behavior and are used in various applications
  • Radio and television broadcasting, wireless communication, and radar systems
• Oscillations in electrical circuits, such as LC $inductor-capacitor$ and RLC $resistor-inductor-capacitor$ circuits, are fundamental to electronic devices
• Mechanical oscillations are employed in various engineering applications
  • Vibration isolation systems, shock absorbers in vehicles, and seismic protection for buildings
• Medical imaging techniques, such as ultrasound and magnetic resonance imaging $MRI$, rely on the properties of acoustic and electromagnetic waves, respectively
• Lasers exploit the coherent oscillation of electromagnetic waves to produce highly focused and monochromatic light beams
  • Used in optical communication, material processing, and medical procedures