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

L_{tot} = NL

n = 1 λ_{1} = \frac{2L_{tot}}{1}

n = 2 λ_{2} = \frac{2L_{tot}}{2}

n = 3 λ_{3} = \frac{2L_{tot}}{3}

n = n λ_{n} = \frac{2L_{tot}}{n}

ω_{n} = \sqrt{\frac{4T}{mL}} sin (\frac{nπ}{2(N + 1)})

N >> 1 n << N

\frac{nπ}{2(N + 1)} << 1

sin (\frac{nπ}{2(N + 1)}) ≃ \frac{nπ}{2(N + 1)} = (\frac{π}{2(N + 1)}) n

ω_{n} ≃ \sqrt{\frac{4T}{mL}} (\frac{π}{2(N + 1)}) n

v = λf = λ\frac{ω}{2π}

v_{n} = (\frac{2L_{tot}}{n} ) \frac{1}{2π} \sqrt{\frac{4T}{mL}} (\frac{π}{2(N + 1)}) n

n ∼ \frac{1}{2} N

sin (\frac{nπ}{2(N + 1)}) ≠ \frac{nπ}{2(N + 1)}

ω_{p} ≃ 56 × 10^{3}\sqrt{n_{e}} rad/s

v_{pl} = c \sqrt{1 − \frac{ω_{p}^{2}}{ω^{2}}}

v_{pl} = c \sqrt{1 − \frac{9.4 × 10^{7}}{ω^{2}}}

t_{0} = \frac{D}{c} = 6500 years ≃ 2 × 10^{11}sec

t_{2} − t_{1} = Δt = \frac{D}{c} (  −  )

Δt = \frac{D}{c} ( )

D = \frac{cΔt}{( )}