Wave packet

An approximation to a wave packet with wave numbers in the interval k₁ ≤ k ≤ k₂,
u(t,x) = A/(k₂-k₁) ∫ cos(k x - ω t) dk,
is constructed by summing up N harmonic waves:
u(t,x) = A/(N-1) ∑ cos(k_ix - ω_it), (i = 0, 1, ..., N-1),
where k_i = k₁ + i (k₂-k₁)/(N-1) and ω_i = ω(k_i).

Below one can choose in the first row of values the minimum and maximum wave numbers k₁ and k₂ and the number N of waves, as well as the constants a and b in dispersion relation ω(k) = a k^b and the amplitude A.
In the second row one may select the time interval t₁ ≤ t ≤ t₂ and 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 velocity v = ω/k and the group velocity v_g = dω/dk in terms of the constants a and b.
Check that when b = 1 we have v = v_g = const., i.e., the medium is not dispersive and all harmonic components propagate at the same velocity. The packet width will be constant and each points one will have beats with constant frequency (which one?). Use the simulation to check your conclusions.
What would change if you select an higher (lower) value for k₂-k₁? Check your answer.
What about decreasing (increasing) the number N of waves?
What would happen in the limit N → ∞?
If now b < 1, what would you expect? Use the simulation to check your answer.
What happens if b > 1?

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.