y(x, t) = f(x, t)

\sqrt{L^{2} + (y_{p+1} − y_{p})^{2}} ∼ L

sin (\frac{y_{p+1} − y_{p}}{L} ) ∼ \frac{y_{p+1} − y_{p}}{L}

sin(θ_{p+1}) ∼ \frac{y_{p+1} − y_{p}}{L}

F_{y} = T (\frac{y_{p+1} − y_{p}}{L} ) − T (\frac{y_{p} − y_{p−1}}{L} ) = \frac{T}{L} (y_{p+1} − 2y_{p} + y_{p−1})

\frac{d^{2}y_{p}}{dt^{2}} = \frac{T}{mL}(y_{p+1} − 2y_{p} + y_{p−1})

y_{p}(t) = A_{p} cos(ωt)

−ω^{2}A_{p} cos(ωt) = \frac{T}{mL}[A_{p+1} cos(ωt) − 2A_{p} cos(ωt) + A_{p−1} cos(ωt)]

−ω^{2}A_{p} = \frac{T}{mL}[A_{p+1} − 2A_{p} + A_{p−1}]

(\frac{2T}{mL} − ω^{2}) A_{p} = \frac{T}{mL}[A_{p+1} + A_{p−1}]

\frac{A_{p+1} + A_{p−1}}{A_{p}} = [2 − \frac{mL}{T}ω^{2}]

A_{p} = C sin(pα)

A_{1} = C sin(α)

A_{2} = C sin(2α)

A_{3} = C sin(3α)

\frac{A_{p+1} + A_{p−1}}{A_{p}} = \frac{C sin [(p − 1)α] + C sin [(p + 1)α]}{C sin (pα)}

sin β + sin γ = 2 sin[\frac{1}{2}(β + γ)] cos[\frac{1}{2}(β − γ)]

C sin [(p − 1)α] + C sin [(p + 1)α] = 2C sin(pα) cos(α)

\frac{A_{p+1} + A_{p−1}}{A_{p}} = \frac{2C sin(pα) cos(α)}{C sin (pα)} = 2 cos(α)

A_{0} = 0 ⇒ C sin(0α) = 0

A_{N+1} = 0 ⇒ C sin((N + 1)α) = 0

(N + 1)α = π, 2π, 3π, ..., nπ

α = \frac{nπ}{(N + 1)}

A_{p} = C sin (\frac{pnπ}{N + 1})

y_{p}(t) = C sin (\frac{pnπ}{N + 1}) cos(ωt)

2 cos[\frac{nπ}{N + 1}] = (2 − \frac{mL}{T}ω^{2})

(\frac{mL}{T}) ω^{2} = 2 − 2 cos[\frac{nπ}{N + 1}]

ω^{2} = (\frac{2T}{mL})[1 − cos (\frac{nπ}{N + 1})]

2 sin^{2}(β) = 1 − cos(2β)

β = \frac{nπ}{2(N + 1)}

ω_{n}^{2} = (\frac{2T}{mL}) 2 sin^{2} (\frac{nπ}{2(N + 1)})

ω_{n} = \sqrt{\frac{4T}{mL}} sin (\frac{nπ}{2(N + 1)}) v = \sqrt{\frac{T}{μ}} = \sqrt{\frac{T}{M/L}}

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