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1D String in Desmos

I learned the wave equation in my PDE class this semester and have been having some fun with it on Desmos.

Suppose a one dimensional string is tied at its two ends and is acted on by gravity and its own internal tensions. The string extends along the x axis and has vertical displacement along the y axis, and it varies according to t , time.

Assuming small string oscillations, we let \frac{dy}{dx} of the string be << 1, which turns the wave equation into a linear PDE with uniform Tension across the string, T(x, t) = T(t). Then, with one extra step, assuming constant density p(x, t) = p_0 and constant tension T(t) = T_0, the 1D wave equation is simplified into

T_0 \frac{\partial^2 y(x, t)}{\partial x^2} = p_0 (\frac{\partial^2 y(x, t)}{\partial t^2} + g)

By defining Boundary conditions for the wave equation as y(.

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Dan bi

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