Strings on the bass are fixed at both ends, so they have
nodes at both ends. Recall from the previous page that nodes
are areas where the waves cancel creating areas of minimum
amplitude. The frequency of a wave is then determined by the
speed of the wave divided by its wavelength, distance
between points of maximum amplitude.
All waves in a string travel with the same speed, so waves
with different wavelengths produce different frequencies.
The lowest-frequency standing wave occurs at a wavelength of
2L where L is the length of the string. This frequency is
known as the fundamental frequency of a string. The
fundamental frequency is also known as the first mode of
vibration. Mode of vibration means style or way of
vibrating and there are many different modes of vibration
within a string.
The different modes and their sounds are called harmonics
of the string. The whole number frequencies, for example f,
2f, 3f, 4f, 5f, etc, are called the harmonic series. The
following diagram shows the first four waves in the harmonic
series with their wavelengths.
The top wave is the fundamental harmonic,
then the second harmonic, the third harmonic, and the fourth
harmonic. The wavelengths, from top to bottom, are 2L, L,
2L/3, L/2. Thus, their frequencies are f, 2f, 3f, and 4f
based on the fact that the fundamental frequency is f.
The general formula for the frequency of the
nth wave in the harmonic series is nf.
To produce these different frequencies, or pitches, on the bass, lightly touch the string at the point 1/n of it's length where n=1, 2, 3, 4, etc, for example touch the string at 1/4 of it's length. Then, pluck the string and release which creates a node where you touch and gives the mode, or harmonic, n. So if you touch at 1/4 of the string length, you will get the fourth harmonic, 4f.
Photos from: University
New South Wales