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Natural Frequencies and Standing Wave Patterns:
A standing wave pattern is characterized by the presence of nodes and antinodes that are always present at the same position along the medium. The fundamental frequency or first harmonic has the smallest possible number of nodes and antinodes. The standing wave patterns for the other harmonics - second, third, fourth, etc. - have an increasing number of nodes and antinodes.
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A guitar string is clamped at both of its ends. When the string is plucked, the ends are unable to vibrate. The ends then become nodes - points of no disturbance. Each consecutive node must be separated by an antinode. So for the lowest possible frequency (fundamental or first harmonic), there must be one antinode between the two ends. The standing wave patterns for the other harmonics have additional nodes and antinodes in comparison to the first harmonic. So if the first harmonic has two nodes (on the ends) and one antinode, then the second harmonic has three nodes (two of which are on the ends of the string) and two antinodes. The third harmonic has four nodes and three antinodes. The fourth harmonic has five nodes and four antinodes. The fifth harmonic has ... - and so on.
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Length-Wavelength Relationship:
A standing wave pattern shows a unique relationship between the wavelength of the waves that create the pattern and a length measured along the medium between two points on the pattern. Every nodal position on the pattern is separated from the next adjacent nodal position by one-half of a wavelength. Similarly, every antinodal position on the pattern is separated from the next adjacent antinodal position by one-half of a wavelength.
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