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: $y<em>{total} = y</em>1 + y_2 + \dots$

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
  • $\dfrac{I<em>2}{I</em>1} = \left( \dfrac{r<em>1}{r</em>2} \right)^2$

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:
  • $I<em>2 = I</em>1 \left( \dfrac{r<em>1}{r</em>2} \right)^2$
• Superposition:
  • $y<em>{total} = y</em>1 + y_2$

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.