Since each loop is equivalent to one-half a wavelength, the length of the string is equal to three-halves of a wavelength. Thus, there are three loops within the length of the string. The third harmonic pattern consists of three anti-nodes. The same reasoning pattern can be applied to the case of the string being vibrated with a frequency that establishes the standing wave pattern for the third harmonic. Since each loop is equivalent to one-half a wavelength, the length of the string is equal to two-halves of a wavelength. Thus, there are two loops within the length of the string. The second harmonic pattern consists of two anti-nodes. Now consider the string being vibrated with a frequency that establishes the standing wave pattern for the second harmonic. That is, the length of the string is equal to one-half the length of a wave. In comparing the standing wave pattern for the first harmonic with its single loop to the diagram of a complete wave, it is evident that there is only one-half of a wave stretching across the length of the string. Thus, one loop is equivalent to one-half of a wavelength. As shown in the animation, one complete wave in a standing wave pattern consists of two loops. The animation below depicts this familiar pattern. A complete wave in a pattern could be described as starting at the rest position, rising upward to a peak displacement, returning back down to a rest position, then descending to a peak downward displacement and finally returning back to the rest position. The vibration of the rope in this manner creates the appearance of a loop within the string. This antinode position along the rope vibrates up and down from a maximum upward displacement from rest to a maximum downward displacement as shown. The pattern for the first harmonic reveals a single antinode in the middle of the rope. To illustrate, consider the first harmonic standing wave pattern for a vibrating rope as shown below.

Furthermore, there is a predictability about this mathematical relationship that allows one to generalize and deduce a statement concerning this relationship. A careful study of the standing wave patterns reveal a clear mathematical relationship between the wavelength of the wave that produces the pattern and the length of the medium in which the pattern is displayed. These frequencies and their associated wave patterns are referred to as harmonics. Each frequency is associated with a different standing wave pattern.

$long division using equation editor 3.0$

Such standing wave patterns are produced within the medium when it is vibrated at certain frequencies. As discussed in Lesson 4, standing wave patterns are wave patterns produced in a medium when two waves of identical frequencies interfere in such a manner to produce points along the medium that always appear to be standing still.

Long division using equation editor 3.0