Standing waves in a pipe

Let us consider a narrow pipe along the OX axis. Each end may be open or closed. The simulation will display the first 5 normal modes, which are

u(t,x) = A sin(n π x) cos(ω t + δ) when both ends are closed.
u(t,x) = A sin((n-1/2) π x) cos(ω t + δ) when the left end is closed and the right end open.
u(t,x) = A cos((n-1/2) π x) cos(ω t + δ) when the left end is open and the right end closed.
u(t,x) = A cos(n π x) cos(ω t + δ) when both ends are open.

Units are arbitrary.
Below you may choose the mode n = 1, ...,5, as well as the animation step Δt.
The upper animation shows the displacement field u(t,x) and the pressure p(t,x) as functions of x at each time t.
In the lower animation you may see the evolution of the position x + u(t,x) of several points and a contour plot of p(t,x) (lighter/darker blue means higher/lower pressure).
Optionally one can see the nodes where the displacement wave vanishes at all times.
Scale has been arbitrarily enhanced to make things visible; but keep in mind that we are considering very small displacements and pressure changes in a narrow pipe.
Put the mouse point over an element to get the corresponding tooltip.

Activities

Compute the position of the nodes for mode number n in the four considered cases.
Use the simulation to check your calculation.
Where are the pressure nodes in the different cases?
Which is the relationship between the displacement and pressure waves? How does it appears in the animation?

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.