When a wave reflects off a fixed end, the incident wave and the reflected wave travel in opposite directions and overlap. If the conditions are right, the result no longer appears to propagate: it is a standing wave. Some points, the nodes, remain always still; others, the antinodes, oscillate with maximum amplitude. The figure does not run along the medium, but vibrates in place, like the string of a plucked guitar.
Standing waves on a string fixed at both ends: the first three harmonics (). The red dots are the nodes (zero amplitude).
Not every wavelength can form a standing wave: only those that “fit” a whole number of times into the medium, respecting the constraints at the ends. It follows that the system can vibrate only at a discrete set of resonance frequencies, its harmonics. This is the reason a string of fixed length emits well-defined notes.
Key formula — Resonance frequencies
String fixed at both ends (length ), or open–open tube: Tube open at one end and closed at the other (only odd harmonics):
The lowest frequency () is the fundamental and determines the pitch of the note; the others are its harmonics, which enrich its timbre. Note the difference between the two cases: the string (and the open–open tube) contains all the harmonics, while the closed–open tube contains only the odd ones — which is why a clarinet (a tube closed at one end) sounds different from a flute (an open tube) of equal length.
Note
In the fixed string the ends are compulsory nodes (they cannot move); at the open end of a tube the air is free to oscillate, so there is an antinode there. It is this asymmetry between the closed and open ends that halves the fundamental frequency and eliminates the even harmonics in the closed–open tube.
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Topics: Onde Concepts: Onde stazionarie · Periodo e frequenza
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