Investigation of Transverse Standing Waves in Beams Free at Both Ends

Standing waves provide a fertile area of learning for physics students at both introductory and advanced levels. We present results of an inexpensive laboratory investigation of the lowest unforced transverse modes of oscillation of PVC beams that are free at both ends. The damped oscillations are slow enough to make nodes and antinodes easy to observe. Through video analysis of standing waves in beams of identical composition and cross section but variable length, we study the relationship between oscillation frequency and length. By using beams with different cross sections, we also investigate the effect of the area moment of inertia of the beam on the oscillation frequency. Our measurements agree quantitatively with the dispersive frequency vs length relationship predicted by the Euler-Bernoulli theory for transverse waves traveling in beams. Beams that are free at both ends provide a medium that is complementary to other commonly used materials (for example, strings or springs fixed at both ends) for investigating transverse standing waves.