The energy carried by a wave is directly proportional to the square of its amplitude. Mathematically, this relationship can be expressed as E∝A2E\propto A^2E∝A2, where EEE is the energy and AAA is the amplitude of the wave

. This means that if the amplitude of a wave doubles, the energy it carries increases by a factor of four (since 22=42^2=422=4). Similarly, tripling the amplitude results in a nine-fold increase in energy, and quadrupling the amplitude leads to a sixteen-fold increase in energy

. The reason behind this relationship is that the energy in a wave is related to the displacement of particles in the medium through which the wave travels. The amplitude represents the maximum displacement from the rest position, so greater amplitude means more energy is imparted to the medium. This principle applies to various types of waves, including sound waves, water waves, and electromagnetic waves, where the energy stored in the wave's fields is proportional to the square of the amplitude

. In summary:

• Energy EEE of a wave increases with the square of the amplitude AAA.
• Doubling amplitude → energy increases by 4 times.
• Tripling amplitude → energy increases by 9 times.
• Quadrupling amplitude → energy increases by 16 times.

This quadratic relationship is fundamental in understanding how waves transport energy