Wave properties and the wave equation

Key ideas

A wave transfers energy from place to place without transferring matter — the medium vibrates but does not travel with the wave.

Every wave is described by the same set of quantities:

Symbol	Quantity	Meaning	Unit
$\lambda$	wavelength	length of one full cycle	m
$f$	frequency	cycles passing a point each second	Hz
$T$	period	time for one full cycle	s
$v$	speed	how fast the wave travels	m s⁻¹
$A$	amplitude	maximum displacement from rest	m

Transverse vs longitudinal waves

• Transverse — the medium vibrates at right angles to the direction the wave travels.
  • Example: light, water surface waves, a wave on a rope.
• Longitudinal — the medium vibrates back and forth along the direction of travel, making compressions and rarefactions.
  • Example: sound in air.

The wave equation

The speed of a wave is its wavelength times its frequency:

$v = f\lambda$

• $v$ — wave speed (m s⁻¹)
• $f$ — frequency (Hz)
• $\lambda$ — wavelength (m)

Frequency and period are reciprocals:

$f = \frac{1}{T}$

And for a wave (or pulse) crossing a known distance in a known time:

$v = \frac{d}{t}$

:::tip When a wave crosses into a new material its frequency never changes — the source sets it. What changes is the speed, and therefore the wavelength ($v = f\lambda$ with $f$ fixed). If your working keeps $f$ the same across a boundary, you are on the right track. :::

Practice question

A water wave travels $12$ m in $4.0$ s and has a wavelength of $2.0$ m. Find its frequency.

Worked solution: speed first, $v = d/t = 12/4.0 = 3.0$ m s⁻¹. Then $f = v/\lambda = 3.0/2.0 = 1.5$ Hz.