Question

As the amplitude increases,
the wave energy

Answer 1

It increases as the square of the amplitude. For a sinusoidal wave (e.g. y = A sin(kx − ωt)) the energy (or intensity/power transmitted) is proportional to A^2, so doubling A gives four times the energy. (For EM waves, intensity ∝ E-field amplitude^2; for mechanical waves on a string P_avg = 1/2 μ ω^2 A^2 v, etc.)