TwiddleLab / Math / Trig and Waves

Sound Is a Sine

Lesson 6 of 11 · 7 min · Free, no account needed

What you will learn

• A pure tone is a sine wave. Its frequency f (cycles per second, Hz) is the pitch; its amplitude is the loudness.
• The period is T = 1 ÷ f. Doubling f raises the note by one octave.
• Waves add point by point (superposition). Two close frequencies make beats: f₂ - f₁ wobbles a second.

Start the lesson

How the lesson goes

1. Why does a guitar sound different from a flute playing the same note?

   Play both. Then bet: what is different?

   Pitch is how often the wave repeats; both instruments repeat 330 times a second. The guitar's wave carries extra higher sines on top, so its shape and its color differ.

2. Turn the pitch lever.

   The scope shows 10 milliseconds of the wave. Match three dashed waves; press Play to hear yours.

3. Now the loudness lever.

   Make the 440 Hz wave half as loud. Then make your own note and keep it.

4. Sound is a sine in the air.

   A pure tone pushes the air back and forth like a sine wave. Frequency f is how many cycles happen each second, in hertz (Hz).

   One cycle lasts the period T = 1 ÷ f. At 300 Hz that is about 3.3 ms.

   The amplitude is how hard the air is pushed: the loudness.

5. Two strings, almost in tune: 440 Hz and 444 Hz.

   Play them together and watch their sum over one second.

6. Add two waves point by point.

   At each dashed line, drag the blue dot to a + b.

   Two sounds in the same air simply add: at every moment the pressure is a + b. Where one is negative, it pulls the sum down.

7. A computer stores sound as 44,100 samples every second.

   A 441 Hz tone is recorded. How many samples land on one cycle?

   Samples per cycle = samples per second ÷ cycles per second = 44,100 ÷ 441 = 100.

8. Game: tune your string to 440 Hz.

   Your string is at 452 Hz. Watch the beats and move it until they stop.

9. Pitch is frequency, loudness is amplitude, and sounds add.

   Three keys built on your note. Tap at least two to make a chord.

Previous lesson: Waves: Amplitude, Period, Phase · Next lesson: Pythagoras on the Circle · Part of the course: Trig and Waves

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