The energy carried by a wave is directly proportional to the square of its amplitude. Therefore, if you cut the amplitude of a wave in half, the energy it carries will decrease by a factor of four (2 squared).
Mathematically, if E1 represents the initial energy carried by the wave with the original amplitude (A1), and E2 represents the energy carried by the wave with the halved amplitude (A2), the relationship can be expressed as:
E2 = (A2^2) / (A1^2) * E1
Since A2 = (1/2) * A1, we can substitute it into the equation:
E2 = ((1/2 * A1)^2) / (A1^2) * E1 = (1/4) * E1
This indicates that reducing the amplitude of a wave by half will reduce its energy to one-fourth of its initial value.