(1) In an earthquake, it is noted that a footbridge...

(1) In an earthquake, it is noted that a footbridge oscillated up and down in a one-loop (fundamental standing wave) pattern once every 1.5 $\mathrm{s}$ . What other possible resonant periods of motion are there for this bridge? What frequencies do they correspond to?

Key Concepts

Frequency-Period Relationship

The frequency of oscillation and the period of motion are inversely related: frequency is the reciprocal of the period. This relationship allows conversion between the time it takes for one complete oscillation (period) and the number of oscillations per unit time (frequency), which is essential when analyzing natural vibrations and resonances.

Harmonics

Harmonics are the higher vibrational modes that occur at integer multiples of the fundamental frequency. They represent the additional resonant frequencies of the structure, each corresponding to more complex standing wave patterns with more nodes and antinodes.

Fundamental Frequency

The fundamental frequency is the lowest natural frequency at which a structure oscillates, corresponding to its simplest mode of vibration (often called the one-loop or one-half wavelength mode). It serves as the basis for identifying higher vibrational modes.

Standing Waves

Standing waves are stationary wave patterns established in a medium or structure when two waves of the same frequency and amplitude traveling in opposite directions interfere. For a bridge, the standing wave patterns correspond to the different modes of vibration (or eigenmodes) that the structure can support.

Resonance

Resonance occurs when a physical system is driven at a frequency that matches one of its natural frequencies, leading to a large amplitude response. In structures like bridges, external forces, such as those from an earthquake, can excite these resonant modes resulting in significant oscillations.

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