The energy of a wave is directly proportional to the square of its amplitude. Therefore, if the amplitude of a wave becomes half of its original value, the energy of the wave decreases to one-fourth (1/4) of its original value.
Mathematically, if E represents the energy of the wave and A represents the amplitude, the relationship between energy and amplitude can be expressed as:
E ∝ A^2
If the original amplitude is A, and the new amplitude is A/2, then the corresponding energies would be:
Original energy (E1) ∝ A^2 New energy (E2) ∝ (A/2)^2 = (A^2)/4
Therefore, the ratio of the new energy to the original energy is:
E2/E1 = (A^2)/4 / A^2 = 1/4
This shows that the energy becomes one-fourth (1/4) of its original value when the amplitude is reduced to half.