beats of two travelling waves

classical physics – problem 3: interference of two sound waves, group velocity and beat envelope.

A = 1.00
ξ₁ = A cos(ω₁ t − k₁ z), ξ₂ = A cos(ω₂ t − k₂ z)
ω₁ = 800,  ω₂ = 630
k₁ = 2.0,  k₂ = 1.5

space profile – ξ(z, t) and envelope at fixed time

time trace – ξ(z₀, t) at a probe point