Waves

Waves

Wave Basics

A progressive wave transfers energy in the same direction as wave travel without transferring matter.

Definitions and Parts of a wave

• Displacement x (m) - distance of a point on a wave from the mid-point
• Amplitude A (m) - maximum distance from the mid-point
• Wavelength λ (m) - length of one whole oscillation
• Period T (s) - time taken for one full oscillation
• Frequency f (Hz) - number of full oscillations per unit time Frequency = 1 / Time period
• Phase - how far a point is along an oscillation
• Phase difference (radians or degrees) - how far along an oscillation one point is compared to another, either on the same wave or two different waves. 2π = 360° = one wavelength
• Path difference - the no. of wavelengths between two points on a wave or on two different waves. Measured in terms of λ.

Wave Speed

${\begin{alignedat}{0}{\text{speed of wave}}={\text{wavelength}}\times {\text{frequency}}\\v=\lambda f\end{alignedat}}$

Refraction and Refractive Index

Refractive Index:

$n={c \over v}$

Snell's Law:

$n_{1}sin(\theta _{1})=n_{2}sin(\theta _{2})$

Critical Angle:

$sin(\theta _{c})={n_{2} \over n_{1}}$

Diffraction and Two-Source Interference

$\lambda \approx {ax \over D}$

Stationary/Standing Waves

A standing wave, also known as a stationary wave is the superposition of two progressive waves with the same wavelength, moving in opposite directions.

Animation of a standing wave (red) created by the superposition of a left traveling (blue) and right traveling (green) wave

Features of standing waves:

• No net energy transfer

How standing waves form:

• The incident wave is reflected off of the closed end (of the string or tube)
• The incident and reflected waves superpose
• Where the two waves are in-phase (phase diff. of 0, 2π, 4π etc.), total constructive interference occurs, creating anti-nodes.
• Where the two waves are out of phase (phase diff. of π, 3π, 5π etc.), total destructive interference occurs, creating nodes (fixed points on the wave which don't oscillate).

Harmonics

A diagram showing the multiple harmonics.

n	Length of string	Wavelength	Frequency
1	${1 \over 2}\lambda$	2L	$f_{0}$
2	$\lambda$	$L$	$2f_{0}$
3	${3 \over 2}\lambda$	${2 \over 3}L$	$3f_{0}$