Sine waves are everywhere, including our logo

Press a piano key, watch the tide, track daylight through the year: each repeats in a smooth cycle. The mathematics underneath is the same curve that runs through our logo.

One curve, four dials

Any simple wave can be written

$$y = A\sin(\omega t + \varphi) + d$$

and each letter turns one dial: $A$ the height, $\omega$ the speed, $\varphi$ where it starts, $d$ where it is centred.

Sound

The note A above middle C vibrates at $f = 440$ cycles per second. Then

$$\omega = 2\pi \times 440 \approx 2765 \text{ rad/s}, \qquad T = \frac{1}{440} \approx 2.27 \text{ ms}$$

Double the frequency to $880$ and you hear the same note one octave higher.

Daylight

In a mid-latitude city daylight swings roughly between 9 and 15 hours across the year. That is a midline of $d = 12$ hours and an amplitude of $A = 3$ hours, over a period of one year, so $\omega = 2\pi / 365$ per day.

Radians or degrees?

Why this curve?

Because $y = \sin(\omega t)$ solves $y'' = -\omega^2 y$, the equation for anything pulled back in proportion to how far it has moved. Springs, small pendulum swings and electrical circuits all obey it, so they all trace this curve.

Move each dial yourself in the Sine and Cosine explainer.