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--- title: Why sine waves decay summary: Drag one slider and watch an envelope tame an infinite wave. tags: [math, waves] difficulty: 1 model: claude-fable-5 prompt_public: true --- # The wave that calms itself Take a sine wave and divide it by $x$. The result, $\frac{\sin(kx)}{x}$, ripples forever — but each ripple is smaller than the last. Right now the steepness is {k}; drag it and watch. :::params ```yaml k: { min: 0.5, max: 8, init: 2, step: 0.1, label: Steepness } ``` ::: :::plot ```yaml height: 280 x: { min: -15, max: 15 } series: - { name: "sin(kx)/x", expr: "sin(k*x)/x" } - { name: "envelope", expr: "1/abs(x)" } ``` ::: :::aside{kind="intuition"} The $1/x$ **envelope** is a budget: the wave can wiggle however fast it likes, but never above the envelope. ::: :::deeper{title="What happens exactly at x = 0?" difficulty=2} The formula divides by zero, yet the curve looks fine. The limit $\lim_{x \to 0} \frac{\sin(kx)}{x} = k$ fills the gap — this is why the peak rises as you raise the slider. ::: :::quiz ```yaml question: As k grows, the central peak… options: ["gets shorter", "gets taller", "stays the same"] answer: 1 explanation: The limit at zero equals k itself, so steeper waves peak higher. ``` :::