Linear to Nonlinear Relations in Wave Science (Acoustics)

In the realm of acoustics or wave science, the transition from linear to nonlinear physics marks a significant evolution in the understanding of tones and their generation. The foundation of this understanding dates back to Pythagoras, who established a linear relationship between the length of a plucked string and the resultant musical tone. This principle posited that the pitch produced by a string can be modulated linearly by altering its length.

A shift occurred in the 1580s with Vincenzo Galilei, the father of the renowned Galileo, challenging the prevailing linear thinking. Galilei’s experiments revealed a more complex, nonlinear relationship in the production of musical tones, particularly when it came to varying the tension of a string. Contrary to the linear assumption that increasing tension produced higher pitches in a directly proportional manner, Galilei discovered that the pitch interval was related to the square of the string’s tension, $T^2$. This finding showed a nonlinear relationship in the generation of acoustic tones, extending beyond strings to wind instruments, where the pitch interval varied as the cube of the vibrating air volume, $V^3$.

The implications of Galilei’s discovery were profound, demonstrating that an interval of a perfect fifth could be achieved through multiple nonlinear pathways: strings differing in length by a ratio of 3:2, in tension by a factor of 9:4, or wind instrument air volumes by a ratio of 27:8. This nonlinear understanding fundamentally altered the approach to musical acoustics, showing a method for a exploration of the intricate relationships that govern the generation of musical tones. This led to the study of nonlinear systems.