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

What is a Progressive Wave?

A progressive wave is a wave that travels through a medium, carrying energy from one place to another without transporting matter. Think of a ripple that spreads across a pond when you drop a stone – the water moves, but the stone stays where it was.

Intensity of a Progressive Wave

The intensity (I) tells us how much power is carried by the wave per unit area. It is given by:

$$I = \frac{P}{A}$$
where P is the power (Watts) and A is the area (m²) over which the wave spreads.

For many mechanical waves (sound, water, light in a medium), intensity is also proportional to the square of the amplitude (A):

$$I \propto A^2$$
In practice we write: $$I = k\,A^2$$ where k is a constant that depends on the medium.

Why Amplitude Matters

• Doubling the amplitude makes the intensity increase by a factor of 4 (since $2^2 = 4$).
• Halving the amplitude reduces the intensity to one‑quarter.
• In sound, louder sounds have higher amplitudes.

Example Problem

A speaker emits a sound with a power of 10 W that spreads uniformly over a sphere of radius 2 m. What is the intensity at the surface of the sphere?

1. Find the area of the sphere: $A = 4\pi r^2 = 4\pi (2\,\text{m})^2 = 16\pi\,\text{m}^2$.
2. Use $I = P/A$: $I = 10\,\text{W} / (16\pi\,\text{m}^2) \approx 0.20\,\text{W m}^{-2}$.

🎯 Answer: $I \approx 0.20\,\text{W m}^{-2}$.

Exam Tips 📚

• Always write the formula in full: $I = \dfrac{P}{A}$.
• Check units – power in Watts, area in m², intensity in W m⁻².
• When asked about how intensity changes with amplitude, remember the square law.
• Use a diagram to show the wave and the area over which it spreads.
• For sound waves, remember that the medium is air and the area is usually a sphere or a circle.

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