Wave Decay and Considerations Through Limits

This record explores the phenomenon of wave decay from a mathematical perspective.
While watching ripples in a bathtub, Reiji began pondering: “What happens to a wave as time approaches infinity?” From this question, he analyzed the behavior of waves using decay functions, limits, and undefined cases.
The attempt seeks to bridge intuitive physical understanding and mathematical reasoning by translating real-world phenomena into equations.

Reiji's Observations

• This note captures thoughts that arose while observing ripples in bathwater. As the waves diminished, I wondered if they could be described mathematically.
• The first equation, \(\lim\limits_{x \to t} e^{-x} \sin(x)\), describes convergence to 0 as \(t \to \infty\). In the real world, we can’t wait for infinity, but values approach zero asymptotically.
• In the next equation, assuming \(\beta = a\) (where \(a\) is a real number), we get \(\lim\limits_{x \to \beta} \sin(x) = q\) where \(-1 \le q \le 1\). Since \(\lim\limits_{x \to \infty} \sin(x)\) is undefined, this represents a wave that continues indefinitely.
• \(\lim\limits_{x \to \infty} \frac{1}{x} = 0\) serves as an example of a wave that fades with distance. Meanwhile, \(\lim\limits_{x \to 0} \frac{1}{x}\) is undefined, modeling divergence.
• All the expressions ultimately converge to one of three states: \(\infty\), undefined, or 0. I found it fascinating that wave behavior could be interpreted within these boundaries.

Hand-drawn notes visualizing wave decay through equations and graphs

Behavior of \(\lim_{x \to t} e^{-x} \sin(x)\)

Comparison between \(\lim_{x \to \infty} \frac{1}{x}\) and \(\lim_{x \to 0} \frac{1}{x}\)

Classification of \(\lim_{x \to \beta} \sin(x)\) into undefined, finite, or converging values