x = location on the rod
z = displacement from original location
dm = ρ V = ρ A dx
stress = \frac{force}{area} = \frac{F_{2} − F_{1}}{A}

strain = \frac{change in length}{original length} = \frac{Δz}{Δx}

stress = Y strain

(\frac{F_{2} − F_{1}}{A} ) = Y (\frac{Δz}{Δx})

F_{2} − F_{1} = (ρ A Δx) \frac{∂^{2}z}{∂t^{2}}

\frac{F_{2} − F_{1}}{A} = (ρ Δx) \frac{∂^{2}z}{∂t^{2}}

(\frac{F_{2} − F_{1}}{A} ) = Y (\frac{Δz}{Δx})

Δz = \frac{∂z}{∂x} Δx

\frac{Δz}{Δx} ≃ \frac{∂}{∂x}(Δz) = \frac{∂}{∂x} (\frac{∂z}{∂x} Δx) = \frac{∂^{2}z}{∂x^{2}} Δx

(\frac{F_{2} − F_{1}}{A} ) = Y (\frac{Δz}{Δx}) ⇒ (ρ Δx) \frac{∂^{2}z}{∂t^{2}} = Y \frac{∂^{2}z}{∂x^{2}} Δx

\frac{∂^{2}z}{∂x^{2}} = (\frac{ρ}{Y} ) \frac{∂^{2}z}{∂t^{2}}

\frac{1}{v^{2}} = \frac{ρ}{Y} ⇒ v = \sqrt{\frac{Y}{ρ}}

λ_{n} = \frac{4L}{2n − 1}

f_{n} = \frac{v}{λ_{n}} = \sqrt{\frac{Y}{ρ}} (\frac{2n − 1}{4L})