Progressive waves
The stadium wave
- In a stadium "wave", people stand and sit — but nobody runs around the stadium.
- The wave travels; the people stay put.
- A wave carries energy from place to place without moving matter overall.
What a wave is
- The particles of the medium oscillate about fixed rest positions.
- Only the disturbance — and its energy — moves along. This is a progressive wave.
A progressive wave carries energy without moving matter along overall.
Yes — the particles only oscillate about fixed points; the disturbance and its energy are what travel.
The key words
- amplitude $A$ — biggest displacement from rest; wavelength $\lambda$ — distance between repeats.
- period $T$ — time for one cycle; frequency $f = \dfrac{1}{T}$ (in Hz).
A wave has a period of $0.020\ \text{s}$. What is its frequency?
$f = \dfrac{1}{T} = \dfrac{1}{0.020} = 50\ \text{Hz}$.
Phase difference
- Points one wavelength apart move together — they are in phase.
- Points half a wavelength apart are exactly out of phase.
Two points exactly one wavelength apart on a wave are:
One whole wavelength is one full cycle ($2\pi$), so the two points move together — in phase.
The wave equation
- In one period the wave advances one wavelength, so $v = \dfrac{\lambda}{T} = f\lambda$.
- $v = f\lambda$ works for every progressive wave.
A wave has frequency $50\ \text{Hz}$ and wavelength $4.0\ \text{m}$. What is its speed?
$v = f\lambda = 50 \times 4.0 = 200\ \dfrac{\text{m}}{\text{s}}$.
Intensity
- Intensity is the power per unit area: $I = \dfrac{P}{A}$ (in $\dfrac{\text{W}}{\text{m}^2}$).
- It grows with the square of the amplitude: $I \propto A^{2}$.
If the amplitude of a wave doubles, its intensity becomes:
$I \propto A^{2}$, so doubling $A$ multiplies the intensity by $2^{2} = 4$.
Spreading from a point
- A point source spreads energy over a sphere: $I = \dfrac{P}{4\pi r^{2}}$, so $I \propto \dfrac{1}{r^{2}}$.
- Double the distance → a quarter of the intensity.
A point source gives an intensity of $100\ \dfrac{\text{W}}{\text{m}^2}$ at distance $r$. What is the intensity at $2r$?
$I \propto \dfrac{1}{r^{2}}$, so at twice the distance the intensity is $\dfrac{100}{2^{2}} = 25\ \dfrac{\text{W}}{\text{m}^2}$.
You've got it
- a wave carries energy, not matter; $f = \dfrac{1}{T}$
- the wave equation: $v = f\lambda$
- intensity $I = \dfrac{P}{A}$, with $I \propto A^{2}$ and $I \propto \dfrac{1}{r^{2}}$ from a point source