Harmonics

The frequencies at which standing waves can exist in a given rope with each end fixed are the natural frequencies or resonant frequencies of that rope. A rope, a spring and even the air in an air column have many natural frequencies, which are often labeled harmonics.

The first harmonic is the simplest mode of vibration and accounts for the fundamental tone. In a rope this means that the rope moves in only one segment, like a jump rope. Overtones are the modes of vibrations that a string, in this case, vibrates in more than one segment. The second harmonic produces the first overtone. The third harmonic produces the second overtone, and so on. In a rope with both ends tied there are only certain ways that this can occur, the frequencies of the overtones are whole number multiples of the fundamental frequency. Almost all vibrating objects produce overtones, which combine with the fundamental. One reason that tuning forks are so important to the study of sound is that their overtones vanish quickly, leaving only the fundamental.

The appearance of a wave, its waveform, is determined by the number and relative intensity levels of the harmonics in its vibration. The quality of the sound, important to music and other things, is a function of its overtones.

Air columns, such as those in musical instruments, have many harmonics. In a pipe with one side open and one side closed, the wavelength of the fundamental is four times the length of the pipe. Using the wave equation, we can see that the corresponding fundamental frequency is the velocity of sound in air divided by four times the length of the pipe. The first three harmonics of such a pipe are illustrated below. As you can see, a pipe with one side open and one side closed has only the odd harmonics.

first harmonic

third harmonic

fifth harmonic

In a closed pipe, the wavelength of the fundamental is twice the length of the pipe. Thus, the fundamental frequency is the velocity of sound in air divided by twice the length of the pipe. In a closed pipe all harmonics are possible. The following illustrations depict the first three harmonics in a closed pipe.

. first harmonic

second harmonic

third harmonic

An open pipe is very similar to a closed pipe. As nodes are forced in the closed ends of a closed pipe, antinodes are forced in the open ends of an open pipe. The wavelength of the fundamental is twice the length of the pipe, so the fundamental frequency is the speed of sound in air divided by twice the length of the pipe. In an open pipe, you can produce all possible harmonics. The first three harmonics of an open pipe follow.

first harmonic

second harmonic

third harmonic

 Type of pipe Wavelength Fundamental frequency One side open, one closed 4L v/4L Closed 2L v/2L Open 2L v/2L

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