Yes, doubling the amplitude of a wave does cause its energy to increase. The energy of a wave is directly proportional to the square of its amplitude. When the amplitude of a wave doubles, its energy increases by a factor of four.
Mathematically, the energy (E) of a wave is given by the equation:
E ∝ A^2
Where A represents the amplitude of the wave. The symbol "∝" denotes proportionality. This equation shows that the energy is proportional to the square of the amplitude. Therefore, if the amplitude doubles (i.e., A → 2A), the energy will increase by a factor of (2A)^2 = 4.
It's important to note that doubling the amplitude of a wave while keeping its frequency constant increases the wave's energy, but it does not affect its frequency.