# Music Theory: String Vibrations

## A.   Wave Motion and Terminology

The typical wave motion of a string might be described as shown in Figure 3- 1. The “amplitude” is the height or depth of the wave and the “node” is where the amplitude is zero, that is to say, crossing the horizontal line as shown below.   Notice that the ends of the wave are also nodes.

The “wavelength,” is typically represented by the symbol, l, (l is the 11th letter of the Greek alphabet pronounced lambda) and in this application is the distance between alternate nodes, adjacent maxima, or adjacent minima, of the amplitude, as outlined in Figure 3- 1. (All of us old folks frequently use “maxima” for the plural of maximum.   The old rule for spelling plural words ending in “um” is to replace the um with an “a”. Thus maximum becomes maxima, minimum becomes minima, datum becomes data, but languages change with time and these usages may be losing preference.)

If you have more than the usual math background you might recognize that the waveform is that of a cosine. Figure 3- 1

Test your understanding of these concepts by working the problems shown below. Please don’t look at the answers at the end of the chapter until you have given a serious effort to solving the problems.

Problem 3-1.   Sketch a waveform that has the same amplitude but a shorter wavelength as that in Figure 3- 1, which shows two wavelengths.

## B.    The Frequency-Wavelength Relationship

The frequency, f, is related to the pitch of the note, that is, the higher the frequency the higher the pitch. In this discussion, I will use the terms frequency and pitch interchangeably although for different types of waves there may be a difference.   When you think of a wave, think of one wavelength as outlined above, that is, showing one part above the line and one part below. Although the wave concepts we discuss generally apply to all waves, we will focus on sound waves and string vibration waves. The math Equation 3- 1 can be simplified for our study because for a particular string at a given temperature, all the variables, tension, T, string length, L, and string mass per unit length, m, can be kept constant. That allows us to absorb all those variables, and the 2 in the denominator, into a “new” constant (uppercase) K, to simplify the equation to that shown Equation 3- 2. Each string on the guitar would have a different value for K because the mass per unit length, m, would be different. However, two identical strings such as two A strings on the mandolin should have identical values for K.

Equation 3- 2

This frequency-wavelength relationship can easily be demonstrated with a graph. If we let K take a value of some arbitrary number that is easy to plot, let’s say 10, then we can calculate some values of l and plot a graph.   These results are shown in Figure 3- 2 below when K = 10.   When l takes on the values shown in the left column the value of f is shown in the right column according to the equation.   For example, if l is 2, f = 10/2 or 5.0 so f = 5.00. The red points on the graph are the values for the frequency, f for each value of l.

Figure 3- 2

This graph is a pretty vivid demonstration of the relationship between f and l, that is, as the wavelength, l, increases the frequency decreases. Or, as the wavelength decreases the frequency increases and it is the latter that we see vividly as we analyze string harmonics of increasing complexity.