Wave motion: (amplitude) 2 .

The idea

This statement is the second part of the intensity relationship: intensity is proportional to the square of the amplitude. Amplitude is the maximum displacement of a particle from its equilibrium position, measured in metres. Squaring it means that small changes in amplitude produce much larger changes in intensity. For example, if amplitude increases by 20%, intensity increases by a factor of 1.2² = 1.44, a 44% rise. The relationship applies to progressive waves and follows from the energy carried by the oscillation. When comparing two waves, write the ratio of intensities as the square of the ratio of amplitudes, so I₁/I₂ = (A₁/A₂)². This lets you predict how brightness or loudness changes when amplitude changes.

What earns the marks

• Amplitude is the maximum displacement from the equilibrium position and is measured in metres.
• Intensity is proportional to the square of the amplitude, written I ∝ A².
• The ratio of two intensities equals the square of the ratio of their amplitudes, I₁/I₂ = (A₁/A₂)².
• A 20% increase in amplitude gives a factor 1.2² = 1.44 increase in intensity.
• Doubling the amplitude multiplies the intensity by four, and halving the amplitude divides it by four.

Common mistakes and how to put them right

• Squaring the percentage change instead of the multiplying factor.

  Put it right: A 20% increase uses 1.2², not 20².

• Confusing amplitude with wavelength or frequency when substituting into the proportionality.

• Forgetting to take the square root when working backwards from an intensity ratio to an amplitude ratio.

Exam tips

• Convert percentage changes into decimal multiplying factors before squaring.
• Show the ratio equation I₁/I₂ = (A₁/A₂)² so the examiner can follow your reasoning.
• Check whether the question asks for a ratio or an actual value, and give units where a value is required.