Group velocity

A wave is the superposition of two harmonic waves of similar wave numbers k₁ ≅ k₂ and frequencies ω₁ ≅ ω₂:
u(t,x) = A₁ cos(k₁ x - ω₁ t) + A₂ cos(k₂ x - ω₂ t) .

Below one can choose the amplitudes A_i, the wave numbers k_i and the frequencies ω_i.
One may also select the animation step Δt, as well as the space interval x₁ ≤ x ≤ x₂ and the number of points to compute and display.
Put the mouse pointer over an element to see the corresponding tooltip.

Activities

Compute the phase velocities v_i = ω_i/k_i and the group velocity v_g = |Δω/Δk| in the cases discussed below.
Check that when v₁ = v₂ we have v_g = v₁ = v₂. Use the simulation to check that the wave maxima and the amplitude maximum all move with same speed.
What would change if you select an higher (lower) value for k₂-k₁? Check your answer.
If now v₁ < v₂ what would you expect? Use the simulation to check your answer.
What happens if v₁ > v₂?
And when A₁ ≠ A₂? Why?

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.