\begin{aligned}y_{1}(x, t) & = A_{1} cos(k_{1}x − ω_{1}t + ϕ_{1}) \\ y_{2}(x, t) & = A_{2} cos(k_{2}x − ω_{2}t + ϕ_{2}) \end{aligned}

y_{1}(x, t) + y_{2}(x, y) = ... messy ...

\begin{aligned}y_{1}(x, t) & = A cos(kx − ωt + ϕ_{1}) \\ y_{2}(x, t) & = A cos(kx − ωt + ϕ_{2}) \end{aligned}

Φ ≡ ϕ_{2} − ϕ_{1}

\begin{aligned}y_{1}(x, t) & = A cos(kx − ωt) \\ y_{2}(x, t) & = A cos(kx − ωt + Φ) \end{aligned}

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

y_{1} + y_{2} = 2 cos (\frac{Φ}{2}) A cos (kx − ωt + \frac{Φ}{2})

\begin{aligned}y_{1}(x, t) & = A_{1} cos(k_{1}x − ω_{1}t) \\ y_{2}(x, t) & = A_{2} cos(k_{2}x − ω_{2}t) \end{aligned}

ω_{1} ≃ ω_{2}

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

\begin{aligned}y_{1}(x, t) & = A cos(k_{1}x − ω_{1}t + ϕ_{1}) \\ y_{2}(x, t) & = A cos(k_{2}x − ω_{2}t + ϕ_{2}) \end{aligned}

\begin{aligned}y_{1}(x, t) & = A cos(ω_{1}t) \\ y_{2}(x, t) & = A cos(ω_{2}t) \end{aligned}

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

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

ω = 2π f

\begin{aligned}y_{1}(x, t) & = A cos(2πf_{1} t) \\ y_{2}(x, t) & = A cos(2πf_{2} t) \end{aligned}

y_{1} + y_{2} = 2A cos (2π(1 Hz) t) cos (2π(499 Hz) t)

f_{beat} = 2 (\frac{f_{1} − f_{2}}{2} ) = (f_{1} − f_{2})