http://weelookang.blogspot.sg/2015/07/ejss-standing-wave-in-pipe-model.html
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.