End of Classical String - Boundary Condition Explained

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The boundary condition for a classical vibrating string states that the spatial derivative of the string at the free end must equal zero. This is derived from the principle that disturbances reflecting off the end of the string undergo 100% reflection without inversion. Consequently, when summing the incident and reflected waves, the displacement doubles while the derivative cancels out. The discussion highlights the importance of understanding these boundary conditions in classical physics.

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The free end of a classical vibrating string imposes the boundary condition that the spatial deriviative of the string at the end must be zero. I can hand wavingly argue this with free body diagrams and manipulate the differential force approximations but i can't come up with a terse intuitive explanation of this boundary equation. any help?
 
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Another way of getting the result is that there is a 100% reflection of any disturbance propagating toward the end, and this reflection is uninverted. When you sum the incident and reflected waves, y(x) doubles, but y'(x) cancels.

This question belongs in the Classical Physics forum.
 

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