When the amplitude of a wave is doubled, the energy of the wave increases by a factor of four (2²) if we assume the wave is described by a simple harmonic motion.
The energy of a wave is directly proportional to the square of its amplitude. Mathematically, the energy (E) of a wave is given by the equation:
E ∝ A²
Where A represents the amplitude of the wave. When the amplitude is doubled (2A), the energy becomes:
E' = (2A)² = 4A²
Therefore, doubling the amplitude of a wave results in its energy increasing by a factor of four.