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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.
```
:::