y(t) = A cos(ωt + ϕ)

y(x) = A cos(kx + ϕ)

v = \frac{ω}{k}

y(x, t) = A cos(kx − ωt + ϕ)

v_{y}(x, t) = \frac{∂y(x, t)}{∂t}

v_{y}(x, t) = ωA sin(kx − ωt + ϕ)

a_{y}(x, t) = \frac{∂v_{y}(x, t)}{∂t}

a_{y}(x, t) = −ω^{2}A cos(kx − ωt + ϕ)

KE = \frac{1}{2}Mv_{y}^{2}

KE = \frac{1}{2}M (−ωA sin(kx − ωt + ϕ))^{2}

KE_{avg} = \frac{1}{2}M (ω^{2}A^{2} \frac{1}{2})

total E =  KE_{avg}  +  PE_{avg}

= \frac{1}{2}M(ω^{2}A^{2} \frac{1}{2}) + \frac{1}{2}M(ω^{2}A^{2} \frac{1}{2})

total E = \frac{1}{2}Mω^{2}A^{2}

μ ≡ \frac{M}{d}

M = μvt

total E = \frac{1}{2} (μvt) ω^{2}A^{2}

power P = \frac{1}{2} (μv) ω^{2}A^{2}