Chapter 49: Modes – Reflection & Natural Frequencies

Applying the condition that the displacement must be zero at the fixed point ensures the incident wave is fully reflected. When the incident wave is sinusoidal, the superposition of the incident and reflected waves results in a stationary pattern known as a standing wave, characterized by points of perpetual zero motion called nodes. Generalizing this to a string fixed at both ends, the wave is constrained to oscillate only at specific, allowed wavelengths which correspond to unique natural frequencies. These allowed patterns, or modes, dictate that the resulting frequencies are simple whole-number multiples of the lowest possible frequency. The concept then expands to two dimensions, using the example of a rectangular vibrating plate or drumhead, where the boundary conditions necessitate sinusoidal displacement in both planar directions. A crucial distinction is made here: unlike the simple one-dimensional string, the natural frequencies for these complex two-dimensional modes are generally not simple integer multiples of one another. The analysis then shifts to systems with only a finite number of moving parts, specifically analyzing two coupled pendulums connected by a spring. Solving this simpler system reveals two specific modes associated with two distinct natural frequencies: one mode where the pendulums swing entirely in phase, and one higher-frequency mode where they swing exactly out of phase. The chapter concludes by establishing a profound general principle for all linear vibrating systems: the entire, complex motion of such a system can be understood as a superposition (a combination) of its characteristic pure sinusoidal oscillations, or modes. This principle is foundational to wave mechanics and extends its relevance even into the field of quantum mechanics.