Waves: Key Concepts and Equations (Last-Minute)

Wave Basics

  • A wave is a traveling disturbance that carries energy as it propagates.

Wave Classifications

  • Transverse: displacement perpendicular to the direction of travel; examples include radio waves, light, microwaves. Key features: amplitude, wavelength, crest, trough.
  • Longitudinal: displacement parallel to the direction of travel; examples include sound waves, seismic P-waves. Features: compression, rarefaction.
  • Water waves: particle motion is circular, involving both transverse and longitudinal components; not purely one type.

Periodic Waves

  • Periodic waves repeat cycles produced by the source.
  • Key quantities: wavelength (\lambda), amplitude (A), period (T), frequency (f), velocity (v).
  • Relationships:
    • v = f \lambda
    • \lambda = \dfrac{v}{f}
    • f = \dfrac{1}{T}
    • T = \dfrac{1}{f}

Wave Speed on a String

  • Wave speed on a string: v = \sqrt{\dfrac{T}{\mu}} where T is tension and \mu is mass per unit length (\mu = m/L).
  • Alternative form: v = \sqrt{\dfrac{F}{\mu}} when using force F as tension.
  • Increasing tension or decreasing linear density increases wave speed.

Mathematical Description of a Wave

  • Displacement for a traveling wave: y(x,t) = A \sin(kx - \omega t) (or cosine).
  • Phase: \phi = kx - \omega t (in radians).
  • Definitions:
    • k = \dfrac{2\pi}{\lambda}
    • \omega = 2\pi f
    • v = \dfrac{\omega}{k} = f\lambda

Wave Speed vs Particle Speed

  • The speed of a wave is not the same as the speed of individual particles in the medium; particles oscillate with smaller speeds while the wave propagates.

Reflection, Refraction, Diffraction

  • Reflection: waves bounce back at barriers.
  • Wavefronts: surfaces of constant phase; direction of motion is perpendicular to wavefronts. Plane waves have nearly straight fronts.
  • Refraction: bending of waves when entering a different medium due to a change in speed.
  • Diffraction: bending around barriers/obstacles; more noticeable when obstacle size is comparable to the wavelength.

Interference and Superposition

  • Interference: overlapping waves combine to form a new pattern; can be constructive or destructive.
  • Superposition principle: the resultant displacement is the sum of the individual displacements: y{total} = y1 + y_2 + \dots

Wave Intensity

  • Intensity: energy per unit area per unit time; proportional to the square of the amplitude: I \propto A^2
  • For a 3D spherical wave, energy spreads over a sphere, so I(r) \propto \dfrac{1}{r^2}
  • If intensities at two distances are I1 at r1 and I2 at r2, then
    • \dfrac{I2}{I1} = \left( \dfrac{r1}{r2} \right)^2

Problem-Solving with Intensity

  • Given I1 at r1, find I2 at r2 using the inverse-square law.

AM vs FM Radio Waves

  • AM (Amplitude Modulation): varies carrier amplitude to transmit information.
  • FM (Frequency Modulation): varies carrier frequency to transmit information.
  • FM generally offers better sound quality and is less susceptible to static; AM can travel farther due to ionospheric reflection.

Quick Reference Formulas

  • Wave relations:
    • v = f \lambda
    • \lambda = \dfrac{v}{f}
    • If period is known: f = \dfrac{1}{T} and T = \dfrac{1}{f}
  • String waves:
    • v = \sqrt{\dfrac{T}{\mu}}
  • Wave description:
    • y(x,t) = A \sin(kx - \omega t)
    • k = \dfrac{2\pi}{\lambda},\quad \omega = 2\pi f
  • Intensity and amplitude:
    • I \propto A^2
  • Solid angle and spherical waves:
    • I(r) \propto \dfrac{1}{r^2}
  • Point-source intensity example:
    • I2 = I1 \left( \dfrac{r1}{r2} \right)^2
  • Superposition:
    • y{total} = y1 + y_2

Note on water-wave particle motion

  • Water waves involve circular particle motion; thus particle displacement is not purely up-down nor purely back-and-forth in one direction.