A pipe sings the note that fits a standing wave inside it — but cap one end and the SAME pipe sounds a whole octave LOWER, and loses its even harmonics (which is why a clarinet, a closed pipe, sounds hollow next to a flute). The end conditions pick the notes. Slide from an open pipe to a closed one and watch the wave and the pitch drop.
Standing waves in a pipe fit the boundary conditions: an OPEN end forces a pressure node (displacement antinode), a CLOSED end a pressure antinode (displacement node). An open–open pipe of length L resonates at fₙ = n·c/2L (all harmonics); a closed–open pipe at f = (2n−1)·c/4L — ONLY odd harmonics, and a fundamental at c/4L, exactly HALF the open pipe’s c/2L: an octave lower from the same tube. A fail-loud self-check throws unless the closed fundamental is half the open one. ◆ real acoustics, node-verified.
Ideal pipe with a small end-correction ignored (real pipes sound slightly flat of c/2L); the octave drop on closing one end and the odd-only harmonics are the exact, defining results (flute vs clarinet).