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Trigonometry · 03 · The anatomy of a wave · 8 min

Reading & building waves

Graph your height as you circle the unit circle and out comes the sine wave — the most important curve in physics. Every wave has just two dials: how high it swings, and how often it repeats.

Build the intuition

Amplitude: the swing

In y = A sin(x), the A scales every swing: how loud the sound, how bright the light, how high the tide. It stretches the wave vertically without touching its timing.

Frequency and period: the rhythm

In y = sin(Bx), the B compresses the rhythm: bigger B, faster repetition. The period — length of one full cycle — is 2π/B. Frequency and period are inverses: a 440 Hz note repeats 440 times each second, so each cycle lasts 1/440 s.

y=Asin⁡(Bx),period=2πBy = A \sin(Bx), \qquad \text{period} = \frac{2\pi}{B}

Adding waves: the secret of sound

Play two notes and the air carries their sum. Remarkably, this works in reverse: any repeating signal — a violin's tone, a vowel, a square wave — can be decomposed into pure sines added together. That idea (Fourier's) powers MP3s, JPEGs, and noise-cancelling headphones.

See it move

InteractiveBuild a wave
1.5
1
y = 1.5 sin(1x): swings 1.5 high (amplitude — loudness, brightness, tide height) and repeats every 6.28 units (period — pitch, frequency, season length).

Two dials, every wave: amplitude sets the swing, frequency sets the rhythm. The faint wave is plain sin(x) for comparison.

A worked example

Model a tide

  1. A harbor's tide swings ±2 m around its mean, peaking every 12 hours.

  2. Amplitude 2; period 12 h needs B = 2π/12:

    h(t)=2sin⁡(π6t)h(t) = 2 \sin\left(\tfrac{\pi}{6} t\right)

  3. Check t = 3 (quarter cycle): sin(π/2) = 1, so h = 2 m — high tide, as built. Harbor masters publish exactly these curves.

Out in the world

Noise-cancelling headphones

Your headphones sample incoming noise, compute its wave, and emit the same wave flipped upside down. Wave + anti-wave = silence. That's subtraction of sines, running live on your ears.

Common confusion, cleared

“A faster wave is a taller wave.”

Speed of repetition (frequency) and size of swing (amplitude) are independent dials. A whisper can be high-pitched; a foghorn can be low and loud.

“sin and cos waves are different species.”

Identical wave, shifted start: cos x = sin(x + π/2). Cosine is sine that left a quarter-cycle earlier.

Check yourself

PracticeQuick check

  1. In y = 3 sin(2x), the amplitude and period are…

  2. Which everyday thing is NOT naturally modeled by sin/cos?

Recap

  • A sin(Bx): A is amplitude (swing), B sets period 2π/B (rhythm).
  • Frequency and period are inverses.
  • Complex signals are sums of simple sines — the basis of audio and image tech.