How can we show mathematically that a wave transfer energy?
Maxwell's equations and the Friis equation for antenna power transfer? https://en.wikipedia.org/wiki/Friis_transmission_equation
I just want a simple explanation. For a wave on a string, how can we show energy transfer.
Look up "Feynman's sticky beads". Feynman used this (embarassingly simple) argument to show that gravitational waves carry energy. Modifying it for a mechanical wave is simple.
I thought you wanted the math. If you want a simple explanation, never ask for the math behind something.
On the other hand, explanations without math are either more complicated than with math or wrong.
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