Traveling Waves In A Tube With Open Ends

Longitudinal waves propagating in a tube can be modeled approximately by a one dimensional waveguide. The standard wave variable used for analysis is the pressure $p(x,t)$ at any point $x$ along the tube and time $t$. Recall from the traveling waves laboratory assignment that the wave variable $p(x,t)$ can be decomposed into left-going $p_l(x,t)$ and right-going $p_r(x,t)$ traveling wave components.

$$p(x,t) = p_l(x,t) + p_r(x,t)$$ (1)

From the virtual acoustic tube lab, we know that when two cylindrical tubes are joined together, the discontinuity will cause traveling wave components to reflect according to the reflectance $k$. A traveling wave leaving a tube with radius $r_1$ and entering a tube with radius $r_2$ will reflect back into the tube with radius $r_1$ according to

$$k = \frac{r_1^2 - r_2^2}{r_1^2 + r_2^2}.$$ (2)

Contemplate what traveling pressure wave components might do at the open end of a tube. We could approximate the region outside of the tube as being a tube with infinite radius. To find the reflectance, we can write

$$k_{open} = \lim_{r_2 \to \infty} \frac{r_1^2 - r_2^2}{r_1^2 + r_2^2} = \lim_{r_2 \to \infty} \frac{-r_2^2}{r_2^2} = -1$$ (3)

$k_{open}=-1$ means that pressure waves reflect approximately with a simple sign inversion from an open end. This means that the reflecting wave will cancel the impinging wave at the open end, creating a node. For more intuition on standing waves in wind instruments, see some depictions of standing waves on the web; however, to avoid confusion, make sure you understand that pressure waves have nodes where displacement waves have anti-nodes and vice versa.