F_{y} = −T sin(θ) + T sin(θ + Δθ)

F_{y} = −T (θ) + T (θ + Δθ) = T Δθ

μ = \frac{M}{L}

F_{y} = (μ Δx) \frac{∂^{2}y}{∂t^{2}} = T Δθ

θ = slope of line = \frac{∂y}{∂x}

Δθ = \frac{∂θ}{∂x} Δx

\frac{∂θ}{∂x} = \frac{∂^{2}y}{∂x^{2}}

(μ Δx) \frac{∂^{2}y}{∂t^{2}} = T \frac{∂^{2}y}{∂x^{2}} Δx

\frac{∂^{2}y}{∂x^{2}} = \frac{μ}{T} \frac{∂^{2}y}{∂t^{2}}

speed of disturbance along string v = \sqrt{\frac{T}{μ}}

\frac{∂^{2}y}{∂x^{2}} = \frac{1}{v^{2}} \frac{∂^{2}y}{∂t^{2}}

y(x, t) = f(x) cos(ωt)

\frac{∂^{2}y}{∂x^{2}} = \frac{∂^{2}f}{∂x^{2}} cos(ωt)

\frac{∂^{2}y}{∂t^{2}} = −ω^{2} f(x) cos(ωt)

\frac{∂^{2}f}{∂x^{2}} cos(ωt) = (\frac{1}{v^{2}} ) − ω^{2} f(x) cos(ωt)

\frac{∂^{2}f}{∂x^{2}} = − (\frac{ω^{2}}{v^{2}} ) f(x)

f(x) = A cos(\frac{ω}{v} x) + B sin(\frac{ω}{v} x)

sin(\frac{ω}{v} L) = 0 ⇒ \frac{ω}{v} L = nπ

\frac{ω}{v} = \frac{nπ}{L}

f(x) = B sin (\frac{nπx}{L})

ω_{n} = \frac{nπv}{L} = \frac{nπ}{L} \sqrt{\frac{T}{μ}}

ω_{1} = \frac{π}{L} \sqrt{\frac{T}{μ}}

ω_{n} = n ω_{1}

y(x, t) = B sin (\frac{nπx}{L}) cos(ω_{n}t)

slope = \frac{∂y}{∂x} = θ  ⇒  \frac{∂θ}{∂x} = \frac{∂}{∂x} (\frac{∂y}{∂x}) = \frac{∂^{2}y}{∂x^{2}}