TwiddleLab / Math / Trig and Waves
Waves: Amplitude, Period, Phase
What you will learn
- In y = A sin(Bx + C), A is the amplitude: how far the crest rises above the middle.
- B squeezes the wave: the period is 2π ÷ B, so bigger B means more cycles.
- C is the phase: it slides the wave sideways. sin(x + π/2) is cos x.
How the lesson goes
Tides rise and fall twice a day. Can you build the tide with three knobs?
This wave gives only one high tide a day. What has to change to get two?
Two high tides in 24 hours means one cycle every 12 hours: the period halves. Height and start time do not change how often it repeats.
Knob A: match the dashed wave.
Move the A lever. Watch the formula follow your hand.
Knob B: squeeze or stretch the wave.
Match two dashed waves. The card shows how long one cycle takes.
Three knobs, three names.
Every wave like this is y = A sin(Bx + C). A is the amplitude, the height of a crest above the middle.
B sets the period, the length of one cycle: period = 2π ÷ B.
C is the phase: it slides the whole wave sideways.
Knob C: slide the wave onto the dashed one.
C moves in steps of π/4.
Game: build the tide.
A harbour measured its water level every 3 hours. Set height, period and shift until every dot sits on your wave.
Match each formula to its graph.
Drag each card under the graph it draws.
2 sin x reaches 2; 0.5 sin x reaches 0.5; sin 2x repeats twice in 2π; sin(x + π/2) is shifted left by π/2, so it starts at its top.
AI models give each word a place code made of waves.
Try both codes for a sentence, then pick the one where no two places look the same.
One wave repeats its values on the way down. A second wave with a shorter period separates those twins, so each position gets a unique set of numbers.
A sets height, B sets the period, C slides it.
Your tide from the game. Your boat needs at least 1 m of water. Pick a safe hour to leave.
Your tide is 2 sin(2π(t - 3) ÷ 12). It is at least 1 m from 4 h to 8 h and from 16 h to 20 h.
Previous lesson: Sine Is a Shadow · Next lesson: Sound Is a Sine · Part of the course: Trig and Waves
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