Open Organ Pipe

In a pipe open at both ends, both ends must be displacement antinodes. Since an antinode-to-antinode spacing (like a node-to-node spacing) is $\lambda/2$, the pipe's length must fit a whole number of half-wavelengths: $L=\frac{n\lambda_n}{2}, \qquad n=1,2,3,\dots \;\Rightarrow\; \lambda_n=\frac{2L}{n}, \qquad f_n=\frac{nv}{2L}=n\,f_1, \qquad f_1=\frac{v}{2L}$ where $v$ is the speed of sound in the air filling the pipe. This is structurally identical in form to the stretched-string result of the previous section (both ends the same type of boundary point give the identical $L=n\lambda/2$ condition), so an open pipe, just like a string fixed at both ends, supports the complete harmonic series: fundamental $f_1$, second harmonic $f_2=2f_1$ (first overtone), …