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
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u(t,x)
= A sin(n π x) cos(ω t + δ) when
both ends are closed.
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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.
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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.
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u(t,x)
= A cos(n π x) cos(ω t + δ) when
both ends are open.
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Units are arbitrary.
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Below you may choose the mode n = 1, ...,5,
as well as the animation step Δt.
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The upper animation shows the displacement field u(t,x)
and the pressure p(t,x) as functions of x
at each time t.
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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).
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Optionally one can see the nodes where the
displacement wave vanishes at all times.
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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.
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Put the mouse point over an element to get the
corresponding tooltip.