For a wave such as a sound wave or light wave, the intensity of the wave is proportional to which mathematical function of its amplitude?

Introduction / Context:
This question is about the relationship between the amplitude of a wave and its intensity. Intensity tells us how much energy a wave transports per unit area per unit time. For many types of waves, including sound waves and electromagnetic waves such as light, there is a simple mathematical relationship between amplitude and intensity. Understanding this relationship is important for explaining why small changes in amplitude can cause large changes in perceived loudness or brightness.

Given Data / Assumptions:

• We have a wave characterised by an amplitude A.
• We are interested in how intensity I depends on A.
• Options include direct proportionality to amplitude, square of amplitude, square root of amplitude and cube of amplitude.
• We assume linear wave behaviour in a medium where standard wave theory applies.

Concept / Approach:
According to wave theory, the energy carried by a wave is proportional to the square of its amplitude. Since intensity is energy transmitted per unit area per unit time, it also becomes proportional to the square of the amplitude. For sound, doubling the amplitude increases intensity by a factor of four. For light, electric and magnetic field amplitudes lead to intensity also being proportional to the field amplitude squared. Therefore, intensity is proportional to the square of amplitude, not to the amplitude itself or other powers listed.

Step-by-Step Solution:
Step 1: Let the amplitude of the wave be A. This represents the maximum displacement of particles (for sound) or maximum field strength (for light). Step 2: Write the qualitative relationship from wave theory: energy carried by the wave ∝ A^2. Step 3: Recognise that intensity I is defined as energy per unit area per unit time, so I is also proportional to the energy transported by the wave. Step 4: Therefore, I ∝ A^2 for a given medium and frequency. Step 5: Translate this into words: intensity is proportional to the square of amplitude. Step 6: Compare with the options and identify square of amplitude as the correct choice. Step 7: Conclude that a small increase in amplitude leads to a larger increase in intensity because of the square relationship.

Verification / Alternative check:
For sound waves, decibel scales relate to the logarithm of intensity ratios. A small increase in amplitude results in a much larger increase in intensity and perceived loudness, consistent with a square dependence. For light, interference patterns depend on the square of the sum of amplitudes. When two coherent light waves interfere constructively, the resultant intensity is four times that of one wave because the amplitude doubles and intensity scales with amplitude squared. These experimental facts support the I ∝ A^2 relationship.

Why Other Options Are Wrong:
Direct proportionality to amplitude would underestimate the energy change when amplitude changes and does not match experimental observations. Proportionality to the square root of amplitude would make intensity change too slowly with amplitude, contradicting energy considerations. Proportionality to the cube of amplitude is not supported by basic wave theory and would make intensity grow too fast with amplitude.

Common Pitfalls:
Some learners confuse amplitude and intensity as if they were directly proportional. It is important to remember that intensity relates to energy, and energy for many oscillatory systems scales as amplitude squared. Keeping in mind the key relation I ∝ A^2 helps you reason correctly about changes in loudness, brightness and energy transmission whenever amplitude changes.

Final Answer:
The intensity of a wave is proportional to the Square of amplitude of the wave.