Relation between Energy and Amplitude

Assuming that the frequency of the wave is unchanged, an increase in Amplitude will result in an increase in Energy and a decrease in Amplitude will result in a decrease in Energy.

In fact, we find that the Energy of a wave is directly proportional to the square of the Amplitude it contains: $E \propto A^2$

Thus, a $2$-fold increase in Amplitude will result in a $4$-fold increase in Energy($2^2$) or a $100$-fold increase in Energy would mean the Amplitude increased $10$-fold.

Let us think of a simple example, the most commonly thought of simple harmonic wave-the spring.

In a spring, the Amplitude($A_w$) of the wave corresponds to the farthest displacement from equilibrium the spring can go($A_s$) and the Energy of the wave($E_w$) corresponds to the amount of mechanical energy the spring has(E_s).

At the point of maximum amplitude($A_s$) during the springs oscillation, all of the spring's energy is Potential Energy so $E_s=PE_s=\frac{kx^2}{2}$. However the $x$(displacement from equilibrium) is equal to the Amplitude of the spring so we get $E_s=\frac{kA_s^2}{2}$.

Since $E_w$ corresponds to $E_s$ and $A_w$ corresponds to $A_s$. Thus $E_s=\frac{kA_s^2}{2}$ can become $E_w=\frac{kA_w^2}{2}\rightarrow E_w \propto A_w$