Apparent contradiction in wavelength of standing waves in an open-ended organ pipe

Question

Recently I came across the "end correction" in organ pipes, and got to know that it affected only displacement waves, i.e it shifts the antinode of these displacement waves further outward.

Now if we consider this mathematically,

Here, the effective phase difference between the twice reflected wave and the newly sent by the source are related as follows, if we consider constructive interference (as it is the condition for resonance, i.e standing waves)

$$ \Delta \phi = \frac{2\pi}{\lambda}\cdot 2(L + 2e) = 2N\pi $$

NOTE- this derivation is for displacement waves. this yields

$$ \lambda = \frac{2\pi \cdot 2(L + 2e)}{2N\pi} = \frac{2(L + 2e)}{N}$$ where $$N$$ can take values 1,2,3... (harmonic modes)

however, if we consider the pressure wave, we know that these are out of phase by $\pi/2$ radians. and the pressure wave is unaffected by the "end correction"

so we have

$$ \lambda = \frac{2\pi \cdot 2(L)}{2N\pi} = \frac{2(L)}{N}$$

and this result is highly contradictory (I have probably gone wrong somewhere, can't find out where) as the wavelengths of both these waves should be the same (determined by source and medium).

Answer 1

If you think the displacement antinode is shifted some distance outward, then so is the pressure wave's node, the same distance and vice versa.

This is because the pressure change is result of displacement of the molecules of air. When, the molecules are displacement maximally from a point at a time, rarefaction occurs. Similarly, when the molecules are not displaced at a point at some time, compression occurs. This shows that at place of displacement node, pressure antinode occur and vice versa.

So, when the displacement antinode is forming a distance $e$ away from the end, the pressure antinode must be at that same place outside the tube. This is the result of the above described relation between displacement wave and pressure wave.

I think, after this, you might be able to see this in this post mentioned above by John Rennie.

Answer 2

I had done a term paper on something similar, so here’s my two cents.

In an open cylindrical pipe, both pressure and displacement waves are affected by the so-called end correction, which accounts for the radiation of sound into the surrounding air and the failure of the boundary condition $p = 0$ exactly at the pipe’s physical end. Instead, the correct boundary condition at the open mouth is $p(L) = Z_{\mathrm{rad}}\, u(L)$, where the radiation impedance is given in the low-frequency limit by

$$ Z_{\mathrm{rad}}(kr) = \rho_0 c \left[ \tfrac{1}{2}(kr)^2 + i\, 0.6133\, kr \right], $$

for a circular piston of radius $r$. The imaginary part of this impedance corresponds to added inertance — it behaves like a small column of air outside the pipe that still participates in the oscillation. This inertial load is equivalent to extending the pipe by a length $e \approx 0.61\,r$ at each open end. Inside the pipe, the pressure and velocity fields are described by counter-propagating waves:

$$ p(x) = A e^{ikx} + B e^{-ikx}, \qquad u(x) = \frac{A e^{ikx} - B e^{-ikx}}{\rho_0 c}, $$

and applying the open-end boundary condition leads to

$$ \tan(kL) \approx -k e, $$

or equivalently, the quantization condition $k(L + e) = m\pi$, so the allowed wavelengths become

$$ \lambda_m = \frac{2(L+e)}{m}. $$

Since both $p(x)$ and the displacement field $\xi(x) \propto \sin(kx)$ depend on the same wavenumber $k$, they share the same wavelength but differ in phase. In particular, the pressure node and displacement antinode both occur at $x = L + e$ — not at the geometric end. Hence, the total effective length is $L + 2e$, and the correct standing-wave condition becomes

$$ \lambda = \frac{2(L + 2e)}{N}, \quad \text{with } e \approx 0.61\,r \text{ per open end}. $$

Bottom line: radiation of sound into free space shifts the pressure node outward and effectively increases the pipe’s length. This explains why end correction must be included to satisfy boundary conditions at the open end.

Reference: For the derivation of the impedance expression and its relation to the end correction, see Levine & Schwinger, Phys. Rev. 73, 383 (1948).