impedence Z = \frac{force on string in y direction}{amplitude of velocity in y direction}

Z = \frac{F_{y}}{v_{y}}

F_{y} = T_{y} = T sin(θ) = T (\frac{T_{y}}{\sqrt{T_{x}^{2} + T_{y}^{2}}} )

\sqrt{T_{x}^{2} + T_{y}^{2}} ≃ T_{x}

F_{y} ≃ T (\frac{T_{y}}{T_{x}} ) = T (slope at P) = T \frac{∂y}{∂x}

F_{y} = −T \frac{∂y}{∂x}

y(x, t) = A_{i} sin(k_{I}x − ωt)

v_{y} = \frac{∂y}{∂t} = −ωA_{i} cos(k_{I}x − ωt)

F_{y} = −T [k_{I}A_{i} cos (k_{I}x − ωt)]

Z_{I} = \frac{F_{y}}{v_{y}} = \frac{−T k_{I}A_{i} cos(k_{I}x − ωt)}{−ωA_{i} cos(k_{I}x − ωt)}

Z_{I} = \frac{T k_{I}}{ω}

Z_{I} = \frac{T}{v_{I}}

v_{I} = \sqrt{\frac{T}{μ_{I}}}

Z_{I} = \sqrt{T μ_{I}}

A_{r} = (\frac{Z_{II} − Z_{I}}{Z_{I} + Z_{II}} ) A_{i}

A_{t} = (\frac{2 Z_{I}}{Z_{I} + Z_{II}} ) A_{i}

Z_{II} = Z_{I}

Z_{I} = \sqrt{T_{I}μ_{I}} = \sqrt{T_{II}μ_{II}} = Z_{II}

F = −bv  ⇒  b = \frac{F}{v}

b = Z_{I}

Z = \frac{V}{I}