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Various standing wave patterns for a resonating guitar string are shown below. Which one of the diagrams represents the standing wave pattern for the first harmonic?
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Natural Frequencies and Standing Wave Patterns:
All objects have a natural frequency or a set of natural frequencies at which they vibrate at. Each frequency in the set is referred to as a harmonic frequency and is associated with a unique standing wave pattern. The 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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The diagram shows the pattern made by the medium at two instants in time during a complete wave cycle. This is a very common way of representing a wave pattern. The pattern is drawn when the antinodal positions are at their maximum displacement from the resting position - for instance, up at a peak. And the mirror image is drawn to represent the pattern formed one-half period later when the antinodal positions have reached a maximum displacement in the opposite direction.
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Suppose a guitar string is clamped at both of its ends. The ends are unable to vibrate when the string is plucked. The ends become nodes - points of no disturbance. Every pair of nodes 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 other harmonics in the set of harmonic frequencies are associated with standing wave patterns that 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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