v_{I} = \sqrt{\frac{T}{μ_{I}}} = \frac{ω}{k_{I}}

k_{I} = ω \sqrt{\frac{μ_{I}}{T}}

k_{II} = ω \sqrt{\frac{μ_{II}}{T}}

y_{i}(x, t) = A_{i} sin(k_{I}x − ωt)

y(x, t) = A_{i} sin (ω[t − \frac{1}{v_{I}} x])

y_{r}(x, t) = A_{r} sin(k_{I}x + ωt)

y_{t}(x, t) = A_{t} sin(k_{II}x − ωt)

y_{i} + y_{r} = y_{t}

v_{yi} + v_{yr} = v_{yt}

A_{i} sin(0 − ωt) + A_{r} sin(0 + ωt) = A_{t} sin(0 − ωt)

−A_{i} + A_{r} = −A_{t}

A_{i} = A_{r} + A_{t}

−ωA_{i} cos(0 − ωt) + ωA_{r} cos(0 + ωt) = −ωA_{t} cos(0 − ωt)

−A_{i} + A_{r} = −A_{t}

\frac{∂y_{i}}{∂x} + \frac{∂y_{r}}{∂x} = \frac{∂y_{t}}{∂x}

k_{I}A_{i} cos(0 − ωt) + k_{I}A_{r} cos(0 + ωt) = k_{II}A_{t} cos(0 − ωt)

k_{I}A_{i} + k_{I}A_{r} = k_{II}A_{t}

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

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

y_{r} = A_{i} sin(k_{I}x + ωt)