Modes of Standing Waves: II

Boundary conditions describe how air molecules behave at the tube's ends. At the open end, molecules are free to vibrate, creating an antinode. At the closed end, molecules cannot vibrate, creating a node. These fixed boundary conditions determine which standing wave modes can exist in the tube.

In the fundamental mode of a tube open at one end and closed at the other, the tube's length equals one-fourth the wavelength. This relationship arises from the antinode at the open end and node at the closed end. Subsequent overtones follow a pattern where length equals odd multiples of one-fourth wavelength.

The wavelengths of standing wave modes follow a pattern determined by odd integers. This pattern reflects the boundary conditions: an antinode at the open end and a node at the closed end. The corresponding frequencies, or harmonics, are calculated using the wave speed and this wavelength relationship.

Resonant frequencies are calculated from the wave speed and wavelength determined by boundary conditions. Since sound speed depends on temperature, resonant frequencies change with temperature. This is why musicians warm wind instruments to room temperature before playing and why organs in unheated cathedrals experience noticeable frequency shifts.

In a tube open at both ends, air molecules are free to vibrate at both ends, creating antinodes at both boundaries. This differs from the open-closed tube, which has an antinode at one end and a node at the other. The resulting harmonic pattern matches that of standing waves on a string with nodes at both ends.

The wavelength in a tube open at both ends equals an integer multiple of half the tube's length. This relationship emerges from having antinodes at both ends. The harmonics follow the same mathematical pattern as standing waves on a string, allowing predictable calculation of resonant frequencies.

Overtones progress by odd multiples of the fundamental wavelength. The first overtone has length equal to three-fourths wavelength, the second overtone equals five-fourths wavelength, and so on. This odd-integer pattern reflects the fixed antinode-node boundary condition configuration unique to open-closed tubes.