# Why does energy loss cause end corrections?

Several people have said that end corrections occur because of acoustic radiation or something similar, where energy is used in vibrating the air outside the pipe.

How exactly does energy loss cause the frequency to be lowered? I thought that radiation would only cause the amplitude of the returning wave to be lowered, which does not change the frequency of the standing wave.

Normally the problem is described in terms of acoustic impedances. So, if one end of the pipe is driven (say, inside your loudspeaker) with input impedance $Z_0$, and if the exit of the pipe is terminated with impedance $Z_\mathrm{exit}$ (this is the side that will create an end correction), then the impedances are related by
$$Z_0 = \frac{Z_\mathrm{exit}+i\tan(kL)}{1+iZ_\mathrm{exit}\tan(kL)}\; .$$
This will be recognized as the equation for a transmission line. Here, $Z_0$ and $Z_\mathrm{exit}$ are normalized (dimensionless) mechanical impedances: $$Z_0 = \frac{\mathrm{Z}_{m,0}}{\rho \, c_s S} \quad\text{and}\quad Z_\mathrm{exit} = \frac{\mathrm{Z}_{m,\mathrm{exit}}}{\rho \, c_s S} \; ,$$ where $\rho$ is the density of air, $c_s$ is the sound speed, and $S$ is the cross-sectional area of the pipe. The frequenecy-dependence of $Z_\mathrm{exit}$ depends on the type of exit (fully baffled, partially baffled, open) and the calculation is quite complicated in all cases. The simplest case is the fully baffled (flanged) case, which at low frequency becomes $$Z_\mathrm{exit} \sim i \frac{8}{3\pi} k a \; ,$$ where $a$ is the radius of the pipe. In the limit of long-wavelength ($k \ll 1$), the impedance takes the simple form
$$Z_0 \sim i \left( kL + \frac{8}{3\pi} ka \right) \; .$$ Thus, at the input end, the pipe looks like it has length $$L_\mathrm{effective} = L + \frac{8a}{3\pi}$$ The second term is the end correction. In terms of the effective length, the resonant frequencies are $$f_n = n \, \frac{c_s}{2L_\mathrm{effective}} \; .$$ You can see this by retaining the $\tan$ function and solving for the zeros.