Concepts : Destructive waves vs. equilibrium

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SUMMARY

The discussion centers on the distinction between destructive waves and true equilibrium in the context of wave propagation along a stretched string. Two waves, one with amplitude +A and the other with amplitude -A, overlap completely, resulting in zero displacement. Despite this cancellation, energy remains present in the system, as the kinetic energy (KE) reaches its maximum while potential energy (PE) is zero at that instant. This highlights that zero displacement does not equate to a lack of energy in the waves.

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Homework Statement


An upward and downward pulse, otherwise identical in shape are propagating in opposite directions along a stretched string. At the instant they overlap completely, the displacement of the string is exactly zero everywhere. How does this situation differ from true equilibrium?
HINT: Where is the wave energy?


Homework Equations





The Attempt at a Solution


I'm assuming that for wave 1, amplitude = +A and for wave 2, amplitude = -A.
y1(x, t) = A1 cos((k1)x - (w1)t + phi1)
y2(x, t) = -A1 cos ((k2)x + (w2)t + phi2)
No phase change so phi's drop, +/- symbols to indicate opposite directions. Now, while they cancel, I'm assuming that even if there is 0 displacement, there is still energy in both waves. However I'm having trouble proving it.
Thanks guys and I hope I posted this properly.
 
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The spring has both kinetic and potential energy. At the instant the displacement is zero along the whole spring, PE=0 and KE is maximum.

ehild
 

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