Hang a weight on a string, set it swinging. Tape a marker to the weight. Slowly pull a strip of paper underneath. The marker traces — a sine wave.
wave speed v = f λ = 0.40 × 140 = 56 px/s
Reading the pendulum trace
A (amplitude) is how far the pendulum swings from the centre — wider pull, bigger A. b (frequency) is how fast it swings — a shorter string swings quicker, bumping b up and squeezing the waves together. The shape never changes; only its stretch does.
You pull the pendulum twice as far back before releasing. What happens to the sine trace?
Any periodic wiggle — a violin note, a heartbeat, mains electricity — is a sum of sines like this one with different A's and b's. That decomposition is the Fourier series.
Why a sine?
A pendulum's horizontal position oscillates between left and right with smooth turning at each extreme. That's exactly what sin(t) does: oscillate between −1 and +1 forever.
Almost everything that vibrates — strings, springs, AC current, sound — moves as a sine wave (or a sum of them). Fourier proved it.