y_{1}(x, t) = A sin(kx − ωt)

y_{2}(x, t) = A sin(kx + ωt)

y_{1} + y_{2} = A[sin(kx − ωt) + sin(kx + ωt)]

sin(α) cos(β) = \frac{1}{2}[sin(α − β) + sin(α + β)]

y_{1} + y_{2} = 2A [sin(kx) cos(ωt)]

y_{1}(x, t) = A sin(k_{1}x − ω_{1}t)

y_{2}(x, t) = A sin(k_{2}x − ω_{2}t)

k_{1} ≃ k_{2} ⇒ k_{2} = k_{1} + Δk

ω_{1} ≃ ω_{2} ⇒ ω_{2} = ω_{1} + Δω

v_{1} = \frac{ω_{1}}{k_{1}}  =  v_{2} = \frac{ω_{2}}{k_{2}}

y_{1} + y_{2} = A sin(k_{1}x − ω_{1}t) + A sin(k_{2}x − ω_{2}t)

sin(α) + sin(β) = 2 sin(\frac{α + β}{2} ) cos(\frac{α − β}{2} )

α = k_{1}x − ω_{1}t , β = k_{2}x − ω_{2}t

y_{1} + y_{2} = 2A sin (\frac{k_{1}x − ω_{1}t + k_{2}x − ω_{2}t}{2}) cos (\frac{k_{1}x − ω_{1}t − k_{2}x + ω_{2}t}{2})

= 2A sin ((\frac{k_{1} + k_{2}}{2} )x − (\frac{ω_{1} + ω_{2}}{2} )t) cos ((\frac{k_{1} − k_{2}}{2} )x − (\frac{ω_{1} − ω_{2}}{2} )t)

= 2A sin (k_{avg} x − ω_{avg} t) cos (\frac{Δk}{2} x − \frac{Δω}{2} t)

y_{1} + y_{2} = 2A sin (k_{avg} x − ω_{avg} t) cos (\frac{Δk}{2} x − \frac{Δω}{2} t)

v_{p} = \frac{ω_{avg}}{k_{avg}}

v_{g} = \frac{Δω/2}{Δk/2} = \frac{Δω}{Δk}

v_{g} = \frac{dω}{dk}

ω_{1} = ω (no change)

λ_{1} = \frac{λ}{n}

k_{1} = \frac{2π}{λ_{1}} = \frac{2πn}{λ} = nk

v_{p} = \frac{ω_{1}}{k_{1}} = \frac{ω}{nk} = \frac{1}{n} \frac{ω}{k}

v_{p} = \frac{c}{n}

v_{p} = \frac{ω}{k}  ⇒  ω = k v_{p}

ω = \frac{kc}{n}

\frac{dω}{dk} = \frac{d}{dk} (\frac{kc}{n})

v_{g} = \frac{dω}{dk} = \frac{d}{dk} (\frac{kc}{n})

\frac{dω}{dk} = \frac{c}{n} + kc \frac{d}{dk} (\frac{1}{n}) = \frac{c}{n} + kc (−\frac{1}{n^{2}} \frac{dn}{dk})

\frac{dω}{dk} = \frac{c}{n} − \frac{kc}{n^{2}} \frac{dn}{dk}

v_{g} = \frac{dω}{dk} = \frac{c}{n} − \frac{c}{n} (\frac{k}{n} \frac{dn}{dk}) = \frac{c}{n} (1 − \frac{k}{n} \frac{dn}{dk})

v_{g} = v_{p} (1 − \frac{k}{n} \frac{dn}{dk})

\frac{dn}{dλ} > 0  or  \frac{dn}{dk} < 0