

Lesson 1: The Nature of a WaveLesson 2: Properties of a WaveEnergy Transport and Amplitude
Lesson 3: Behavior of WavesReflection, Refraction, and Diffraction
Waves Generated by Moving
Sources
Lesson 4: Standing WavesTraveling Waves vs. Standing Waves 
Lesson 4: Standing WavesMathematics of Standing WavesAs 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 which always appear to be standing still. Such standing wave patterns are produced within the medium when it is vibrated at certain frequencies. Each frequency is associated with a different standing wave pattern. These frequencies and their associated wave patterns are referred to as harmonics. A careful study of the standing wave patterns reveal a clear mathematical relationship between the wavelength of the wave which produces the pattern and the length of the medium in which the pattern is displayed. Furthermore, there is a predictability about this mathematical relationship that allows one to generalize and deduce a statement concerning this relationship. To illustrate, consider the first harmonic standing wave pattern for a vibrating rope as shown below. The pattern for the first harmonic reveals a single antinode in the middle of the rope. 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 vibration of the rope in this manner creates the appearance of a loop within the string. 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 animation below depicts this familiar pattern. As shown in the animation, one complete wave in a standing wave pattern consists of two loops. Thus, one loop is equivalent to onehalf of a wavelength. 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 onehalf of a wave stretching across the length of the string. That is, the length of the string is equal to onehalf the length of a wave. Put in the form of an equation:
Now consider the string being vibrated with a frequency that establishes the standing wave pattern for the second harmonic. The second harmonic pattern consists of two antinodes. Thus, there are two loops within the length of the string. Since each loop is equivalent to onehalf a wavelength, the length of the string is equal to twohalves of a wavelength. Put in the form of an equation:
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. The third harmonic pattern consists of three antinodes. Thus, there are three loops within the length of the string. Since each loop is equivalent to onehalf a wavelength, the length of the string is equal to threehalves of a wavelength. Put in the form of an equation:
When inspecting the standing wave patterns and the lengthwavelength relationships for the first three harmonics, a clear pattern emerges. The number of antinodes in the pattern is equal to the harmonic number of that pattern. The first harmonic has one antinode; the second harmonic has two antinodes; and the third harmonic has three antinodes. Thus, it can be generalized that the nth harmonic has n antinodes where n is an integer representing the harmonic number. Furthermore, one notices that there are n halves wavelengths present within the length of the string. Put in the form of an equation:
This information is summarized in the table below.
For standing wave patterns, there is a clear mathematical relationship between the length of a string and the wavelength of the wave which creates the pattern. The mathematical relationship simply emerges from the inspection of the pattern and the understanding that each loop in the pattern is equivalent to onehalf of a wavelength. The general equation which describes this lengthwavelength relationship for any harmonic is:
Test your understanding of this relationship by answering the questions below.

2. The string at the right is 1.5 meters long and is vibrating as the first harmonic. The string vibrates up and down with 33 complete vibrational cycles in 10 seconds. Determine the frequency, period, wavelength and speed for this wave. 

3. The string at the right is 6.0 meters long and is vibrating as the third harmonic. The string vibrates up and down with 45 complete vibrational cycles in 10 seconds. Determine the frequency, period, wavelength and speed for this wave. 

4. The string at the right is 5.0 meters long and is vibrating as the fourth harmonic. The string vibrates up and down with 48 complete vibrational cycles in 20 seconds. Determine the frequency, period, wavelength and speed for this wave. 

5. The string at the right is 8.2 meters long and is vibrating as the fifth harmonic. The string vibrates up and down with 21 complete vibrational cycles in 5 seconds. Determine the frequency, period, wavelength and speed for this wave. 

Lesson 4: Standing Waves
 Traveling Waves vs. Standing Waves
 Formation of Standing Waves
 Nodes and Antinodes
 Harmonics and Patterns
 Mathematics of Standing Waves
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19962007