Waves and Sound I: Introductory Physics Study Notes

Periodic Motion and Restorative Forces

Review of Periodic Motion

Periodic motion occurs when an object moves back and forth in a regular, repeating pattern. A restorative force is required to maintain this motion, always acting to return the system to its equilibrium position.

• Restorative Force: The force that brings the system back to equilibrium (e.g., spring force, gravity for a pendulum).

• Vectors in Periodic Motion: Position (r), velocity (v), and acceleration (a) vectors all rotate together with angular frequency ω.

• Equations for x-components:

  • Position: $x = A \cos(\omega t + \delta)$

  • Velocity: $v = -\omega A \sin(\omega t + \delta)$

  • Acceleration: $a = -\omega^2 A \cos(\omega t + \delta)$

• Phase Constant (δ): Sets initial conditions for position and velocity.

Applications of Periodic Motion

• Mass on a spring

• Mass on a pendulum

• Physical pendula (e.g., leg for a kicker, swing of a bat)

• Oscillations in molecules

• Radio/TV/Communication signals

• Economic cycles (supply and demand)

Introduction to Waves

What is a Wave?

A wave is a disturbance that travels through a material (medium), transferring energy from one location to another without the bulk movement of the medium itself.

• Energy Transfer: Waves carry energy, not matter.

• Medium Behavior: The medium does not move with the wave (no net current).

• Examples:

  • Sound waves: Molecules oscillate but do not travel with the wave.

  • Light waves: Electric and magnetic fields oscillate.

Example: Lightning and Thunder

• Lightning produces both light and sound waves.

• Light travels faster than sound, so you see lightning before hearing thunder.

• Both waves transfer energy through the air.

Types of Waves

Transverse Waves

In a transverse wave, the disturbance is perpendicular to the direction of wave travel.

• Example: Wave on a string, light waves.

• Mathematical description: $y = A \sin(kx - \omega t)$

• If x is constant, the disturbed point moves up and down in periodic motion.

Longitudinal Waves

In a longitudinal wave, the disturbance occurs along the same axis as the wave's direction of travel.

• Example: Sound waves, ultrasound imaging.

• Medical Application: Ultrasound uses sound waves to image tissues.

• Music Application: The physics of sound underlies musical acoustics.

Water Waves and Slinkys

• Water waves are both longitudinal and transverse.

• A slinky can demonstrate both types of waves depending on how it is moved.

Wave Properties

Basic Properties

Waves are characterized by several key properties:

• Wavelength (λ): The distance over which the wave pattern repeats.

• Amplitude (A): The maximum displacement from equilibrium.

• Crests and Troughs: Highs and lows of the wave, respectively.

Traveling Waves

Traveling waves move along an axis, carrying energy from one place to another.

• Wave Speed Equations:

  • $v = \lambda f$

  • $v = (\lambda / 2\pi) \cdot (2\pi f)$

  • $v = \omega / k$

• Angular Frequency: $\omega = 2\pi f$

• Wave Number: $k = 2\pi / \lambda$

Surfing the Wave

• To "surf" a wave, stay at the same height as the wave moves.

• For a transverse wave: $y = A \sin(kx - \omega t)$

• As time increases, x must increase to keep $(kx - \omega t)$ constant (i.e., to move with the wave).

Periodic Motion in Waves

Periodic Motion and Phase

• At a fixed location (x constant), the wave exhibits periodic motion.

• General form: $y = A \sin(kx - \omega t) = A \sin(-\omega t + \delta)$

• Each position has its own phase angle ($\delta = kx_0$).

Wave Behavior: Reflection and Transmission

Reflection at Boundaries

When a wave encounters a boundary between two media, part of the wave is reflected and part is transmitted.

• If a lighter string meets a heavier string (fixed end), the reflection is inverted.

• If a heavier string meets a lighter string (free end), the reflection is not inverted.

Transmission and Medical Imaging

• Ultrasound waves in air directed at human tissue behave like a wave on a lighter string encountering a heavier string.

• The transmitted wave is never inverted.

• Reflection is enhanced when the wavelength is smaller than the feature being studied.

• Reflection is a particle-like property; diffraction and interference are wave-like properties.

Summary Table: Types of Waves

Type of Wave	Direction of Disturbance	Examples	Applications
Transverse	Perpendicular to wave travel	Waves on a string, light waves	Communication, optics
Longitudinal	Parallel to wave travel	Sound waves, ultrasound	Medical imaging, acoustics
Water Waves	Both perpendicular and parallel	Ocean waves	Environmental science

Additional info:

• Linear superposition and standing waves are upcoming topics, relevant for music and quantum mechanics.

• Wave properties such as interference and diffraction are essential for understanding advanced wave phenomena.