Wave energy and power

Every piece of a medium carrying a mechanical wave is itself oscillating, and its energy — like any oscillator's — depends on the square of how far it swings, its amplitude, exactly as a mass on a spring stores energy proportional to the square of how far it is stretched. A wave is simply many such oscillators linked together, each passing energy to its neighbour, so the same squared relationship carries over to the whole wave: energy (and therefore power and intensity) scales with amplitude squared, not amplitude itself.

That squared dependence has real consequences that are easy to underestimate. Doubling a wave's amplitude does not double the energy it carries — it quadruples it. Tripling amplitude increases energy ninefold. This is why a seemingly modest increase in how "tall" a wave looks corresponds to a dramatic increase in the destructive power of an ocean wave, the loudness of a sound, or the intensity of light.

Amplitude and frequency are independent properties of a wave — nothing forces them to move together. A guitar string plucked harder produces a larger-amplitude, louder wave at essentially the same frequency (the same pitch), because amplitude is set by how much energy was put in at the start, while frequency is set by the string's length, tension and mass per length. Confusing "louder" with "higher-pitched" is confusing energy with frequency, two genuinely separate wave properties that happen to both affect what a listener perceives, but through entirely different physical mechanisms.