The total energy during an oscillation (for example, a pendulum on a string) is constant as long as no energy is lost to the environment:

The energy in the system changes from potential to kinetic and back every half cycle, but the total energy in the system is constant at all times (the dotted line is the sum of the P.E. and K.E.)

Note: That energy changes from KE to PE and back again twice in every cycle.

Total energy = Kinetic energy + potential energy

This applies to a mass oscillating on a spring, so we can easily calculate the total energy using the equations for kinetic energy of a mass and the potential energy stored in a spring.

KE = 1/2 mv² and PE = 1/2 ks²

So,

Total energy = 1/2 mv² + 1/2 ks²

Where:

m = mass on the spring (kg)

v = velocity of the mass (ms^-1)

k = spring constant

s = displacement of mass (m)

The amplitude of a wave gives an indication of the amount of energy the oscillator has. This makes sense if you think of the spring and mass. The greater the amplitude the larger the amount of energy stored in the spring when it is extended. However,

PE = 1/2 ks² so energy must be proportional to the amplitude².

Therefore, if the Amplitude of a wave is halved, then the Energy is quartered. As the the Energy is proportional to the Amplitude squared.