recall and use intensity = power/area and intensity ∝ (amplitude )2 for a progressive wave

Progressive Waves

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.

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):

1. 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.

2. Both terms contain \$A^{2}\sin^{2}(kx-\omega t)\$. The time‑averaged energy density \$\langle u\rangle\$ therefore contains a factor \$A^{2}\$.
3. Power is the product of energy density and wave speed: \$P = \langle u\rangle v A_{\perp}\$.
4. 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.

Solution:

1. Calculate the area:

   \$A_{\perp} = \pi r^{2} = \pi (0.10\ \text{m})^{2} = 3.14\times10^{-2}\ \text{m}^{2}\$

2. 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

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.