An open-open organ pipe is 84.2 cm long. An open-closed pipe has a fundamental frequency equal to the third harmonic of the open-open pipe. How long is the open-closed pipe?
Harmonics are divisions of standing fundamental frequencies that are created by dividing the fundamental into smaller waves, which produce higher frequencies. Harmonics are numbered starting with the fundamental, which is one cycle the entire length of the vibrating column, which is in this case, an organ pipe. So the fundamental vibrates a length of 84.2 centimeters. The next harmonic would split that length into two equal cycles, the representative column length would be 84.2 cm/2, or an equivalent fundamental pipe length of 41.1 cm. The third harmonic would take the original pipe length and divide it into three separate cycles, so take the original pipe length and divide it by 3, which would be 84.2 cm/3, which would give an equivalent fundamental pipe length of 28.07 cm. Do not confuse harmonics with overtones; the fundamental represents the first harmonic, the second harmonic is the second harmonic. The fundamental in the overtone series is first, while the 1st overtone actuall represents the second harmonic.