= Pressure and motion in a closed acoustic tube
{title2=$p'=P\cos(kx)\cos(\omega t),\quad u=P\sin(kx)\sin(\omega t)/(\rho_0c)$}
Consider a uniform tube $0\leq x\leq L$ with rigid ends, constant undisturbed <mass density> $\rho_0$ and <speed of sound> $c$. Neglect losses and transverse structure. A free <normal mode> has
$$
k=\frac{n\pi}{L},\qquad \omega=ck,\qquad n=1,2,\ldots,
$$
and a consistent choice of <sound pressure>, <acoustic particle velocity> and <acoustic displacement> is
$$
\begin{aligned}
p'(x,t)&=P\cos(kx)\cos(\omega t),\\
u(x,t)&=\frac{P}{\rho_0c}\sin(kx)\sin(\omega t),\\
\xi(x,t)&=-\frac{P}{\rho_0c\omega}\sin(kx)\cos(\omega t).
\end{aligned}
$$
Here $P>0$ is a peak <sound pressure>. Direct differentiation verifies $\rho_0\partial_tu=-\partial_xp'$ and $\partial_tp'=-\rho_0c^2\partial_xu$, the one-dimensional <linear homentropic acoustic equations>. The rigid ends enforce $u=0$ there. A driven experiment only approximates this free-mode geometry; the formulas do not predict a finite resonant <amplitude> for an undamped driven tube.
The maxima of <sound pressure> amplitude occur at $kx=j\pi$, where <acoustic particle velocity> vanishes. The maxima of <acoustic particle velocity> amplitude instead occur at $kx=(j+1/2)\pi$, where <sound pressure> vanishes. The two families are displaced by $\lambda/4$, but each family repeats after $\lambda/2$. At a position where both are nonzero, the time dependence is also a quarter cycle apart. Using consecutive maxima of <sound pressure> amplitude therefore gives
$$
\boxed{c=2f d},\qquad f=\frac{\omega}{2\pi},\qquad d=\frac{\lambda}{2}.
$$
The relation also applies to consecutive minima of <sound pressure> amplitude, provided the same family is used throughout. Mixing maxima and minima would halve the spacing and give an erroneous <speed of sound>.
The local <acoustic energy density> is $p'^2/(2\rho_0c^2)+\rho_0u^2/2$. Averaging over one <wave period> gives $P^2/(4\rho_0c^2)$, independent of $x$. Likewise $\langle p'u\rangle=0$: this ideal <standing wave> has no mean longitudinal <acoustic energy flux>, although its <pressure> and fluid <velocity> oscillate. Energy transport into a real lossy tube is a different question from the free-mode pattern.
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