Transverse standing waves produced on a string, as shown in Fig. 1, are a common demonstration of standing wave patterns that have nodes at both ends. Longitudinal standing waves can be produced on a helical spring that is mounted vertically and attached to a speaker, as shown in Fig. 2, and used to produce both node-node (NN) and node-antinode (NA) standing waves. The resonant frequencies of the two standing wave patterns are related with theory that is accessible to students in algebra-based introductory physics courses, and actual measurements show good agreement with theoretical predictions.
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Standing waves in pipes can be confusing for students due to terminology. Though an open end of a pipe is a pressure node, it is an air-displacement antinode. Pressure is considered here due to similarity to the spring system, as commonly analyzed open-open pipes are considered a node-node system in terms of pressure.
A video depiction of this process and more is available at goo.gl/axsol.
The spring was attached to a stationary force detector and pulled toward a reversed motion detector. The slope of the resulting position-versus-time graph yields the spring constant.
The overriding uncertainty is that the built-in function generator on a Vernier Labquest changes frequency in steps of 0.125 Hz from 0–10 Hz, steps of 0.25 Hz from 10–30 Hz, and steps of 0.5 Hz for the values tested thereafter.