Energy in Waves
Energy density moving with wave. Builds on Traveling Waves.
You’re generating waves on a rope by shaking one end. You decide to shake with TWICE the amplitude (twice as far up and down), keeping the frequency the same.
By what factor does the energy carried by the wave increase?
The energy quadruples! Wave energy is proportional to the SQUARE of the amplitude: E ∝ A². Double the amplitude means 2² = 4× the energy. This is a universal property of waves — whether on strings, in sound, or in light. It’s why a tsunami double the height is four times more destructive.
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Adjust the wave amplitude and watch the energy meter. Double the amplitude and see the energy bar jump to 4×. The color intensity along the wave shows the local energy density — brightest at the peaks.
P = ½μA²ω²v
Energy goes as A² AND ω². A high-frequency, large-amplitude wave carries enormous power. This is why ultrasound can shatter kidney stones — high frequency + focused amplitude = intense energy.
A tsunami only 1 meter high in the open ocean carries modest energy. But when it shoals to 10 meters near shore, the energy density increases 100-fold (A²). That’s devastating.
A radio station doubles its signal amplitude to reach farther. The transmitted power quadruples, requiring 4× the electrical input. Amplitude is expensive!
Doubling the perceived ‘loudness’ requires about 10× the sound intensity. Since I ∝ A², you need √10 ≈ 3.16× the pressure amplitude. Decibels exist because of this nonlinear relationship.
The Richter scale is logarithmic because seismic wave energy scales as amplitude squared. Each magnitude step represents ~31.6× more energy — magnitude 8 releases about 1000× more energy than magnitude 6.
“Double the wave, quadruple the energy. This A² law governs everything from gentle ripples to catastrophic tsunamis.”
Wave Energy Lab
Explore how wave energy depends on amplitude and frequency. Visualize energy density along the wave and measure transmitted power. Compare how changing A vs. changing f affects the total energy.
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Set amplitude to 1 and frequency to 1. Note the power reading. Now double the amplitude to 2 — power jumps to 4×. Reset and instead double the frequency — power also jumps to 4×! Both A and ω contribute equally through the squared terms. Try tripling amplitude (9× power) to really feel the nonlinearity.
Wave energy is proportional to BOTH amplitude squared and frequency squared. This A²ω² scaling is universal across all types of waves and explains why high-frequency, large-amplitude waves are so energetic.