A progressive (or travelling) wave transports energy and momentum through a medium without permanent displacement of the medium itself. The wave front moves forward, and the disturbance at any point in the medium varies with time.
Suggested diagram: A sinusoidal progressive wave travelling to the right, showing successive crests and troughs.
Key Quantities
Amplitude ($A$): maximum displacement from equilibrium.
Wavelength ($\lambda$): distance between successive points in phase.
Frequency ($f$) and Angular frequency ($\omega = 2\pi f$).
Wave speed ($v$): $v = f\lambda$.
Power ($P$): rate at which energy is transmitted by the wave.
Intensity ($I$): power per unit area perpendicular to the direction of propagation.
Intensity Definition
The intensity of a wave is defined as
$$
I = \frac{P}{A_{\perp}}
$$
where $P$ is the average power carried by the wave and $A_{\perp}$ is the area of a surface normal to the direction of propagation.
Relationship Between Intensity and Amplitude
For a harmonic progressive wave in a linear medium, the average power is proportional to the square of the amplitude. Consequently, the intensity is also proportional to $A^{2}$:
$$
I \propto A^{2}
$$
Derivation (outline):
The instantaneous energy density $u$ of a wave is the sum of kinetic and potential contributions:
$$u = \frac{1}{2}\rho v^{2} + \frac{1}{2}kA^{2}\sin^{2}(kx-\omega t)$$
where $\rho$ is the medium density and $k$ the wave number.
Both terms contain $A^{2}\sin^{2}(kx-\omega t)$. The time‑averaged energy density $\langle u\rangle$ therefore contains a factor $A^{2}$.
Power is the product of energy density and wave speed: $P = \langle u\rangle v A_{\perp}$.
Dividing by $A_{\perp}$ gives intensity $I = \langle u\rangle v$, which retains the $A^{2}$ dependence.
Practical Use of $I = P/A_{\perp}$ and $I \propto A^{2}$
These relationships allow you to:
Calculate the intensity of a sound or light beam when the power and cross‑sectional area are known.
Predict how the intensity changes when the amplitude of a wave is altered (e.g., turning up a speaker).
Compare intensities of different waves by examining their amplitudes.
Example Problem
Problem: A speaker emits a sinusoidal sound wave with an average power of $2.0\ \text{W}$ uniformly over a circular area of radius $0.10\ \text{m}$. Find the intensity at a distance where the wavefront is still approximately planar.
Use the intensity definition:
$$I = \frac{P}{A_{\perp}} = \frac{2.0\ \text{W}}{3.14\times10^{-2}\ \text{m}^{2}} \approx 63.7\ \text{W m}^{-2}$$
Common Misconceptions
Intensity is not the same as amplitude. Intensity depends on the square of the amplitude, so halving the amplitude reduces intensity by a factor of four.
Intensity is not a vector. It is a scalar quantity representing power per unit area.
Area must be perpendicular to the direction of propagation. Using an oblique area will give an apparent intensity that is too low because the effective area is $A_{\perp}=A\cos\theta$.
Summary Table
Quantity
Symbol
Definition / Relation
Intensity
$I$
$I = \dfrac{P}{A_{\perp}}$
Power
$P$
Average rate of energy transport by the wave.
Amplitude
$A$
Maximum displacement of the medium.
Intensity–Amplitude Relation
$I \propto A^{2}$
For a harmonic progressive wave in a linear medium.
Wave Speed
$v$
$v = f\lambda = \dfrac{\omega}{k}$
Key Take‑aways
Intensity quantifies how much power passes through a unit area perpendicular to the wave’s travel direction.
Because $I \propto A^{2}$, small changes in amplitude produce large changes in intensity.
When solving problems, always verify that the area used is the projected (perpendicular) area.