dKE = \frac{1}{2} dm v^{2} = \frac{1}{2} (μ dx) (v_{y}^{2})

y(x, t)

v_{y} = \frac{∂y}{∂t}

dKE = \frac{1}{2} (μ dx) (\frac{∂y}{∂t} )^{2}

original length = dx

stretched length = ds = \sqrt{dx^{2} + dy^{2}}

ds − dx = \sqrt{dx^{2} + dy^{2}} − dx

ds − dx = \sqrt{dx^{2} [1 + (\frac{dy}{dx})^{2}]} − dx

= dx \sqrt{1 + (\frac{dy}{dx})^{2}} − dx

= dx [1 + (\frac{dy}{dx})^{2}]^{1/2} − dx

≃ dx [1 + \frac{1}{2} (\frac{dy}{dx})^{2}] − dx

≃ \frac{1}{2} (\frac{dy}{dx})^{2} dx

(\frac{dy}{dx})^{2} dx

\frac{1}{2} (\frac{dy}{dx})^{2} dx

\frac{1}{2} (\frac{dy}{dx}) dx

(\frac{dy}{dx}) dx

ds − dx ≃ \frac{1}{2} (\frac{∂y}{∂x})^{2} dx

Work = ∫ F⃗ · d⃗x

dW = F⃗ · d⃗x

dW = T (ds − dx) = T \frac{1}{2} (\frac{∂y}{∂x})^{2} dx

dPE = \frac{1}{2} T (\frac{∂y}{∂x})^{2} dx

dE = dKE + dPE = \frac{1}{2} (μ dx) (\frac{∂y}{∂t} )^{2} + \frac{1}{2} T (\frac{∂y}{∂x})^{2} dx

\frac{dE}{dx} = \frac{1}{2} μ (\frac{∂y}{∂t} )^{2} + \frac{1}{2} T (\frac{∂y}{∂x})^{2}

Δx = v Δt

(energy of section) = \frac{dE}{dx} Δx = (\frac{1}{2} μω^{2}A^{2}) (v Δt)

Power = \frac{energy of section}{Δt} = \frac{(\frac{1}{2} μω^{2}A^{2}) (v Δt)}{Δt}

y(x, t) = A sin(kx − ωt)

\frac{∂y}{∂t} = −ωA cos(kx − ωt)

\frac{∂y}{∂x} = kA cos(kx − ωt)

\frac{dE}{dx} = \frac{1}{2} μω^{2}A^{2} cos^{2}(kx − ωt) + \frac{1}{2} T k^{2}A^{2} cos^{2}(kx − ωt)

Power = \frac{1}{2} μvω^{2}A^{2}

Power = \frac{1}{2} (μv) (ω^{2}A^{2})

KE = \frac{1}{2} mv^{2} PE = \frac{1}{2} k(Δx)^{2} = ∫_{0}^{Δx} kx⃗ · d⃗x = ∫_{0}^{Δx} F⃗ · d⃗x

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