Sinusoidal Model

Four numbers turn the bare sine curve into a model of something real. A is the amplitude, half the distance from trough to peak. c is the midline the curve oscillates about. d slides it horizontally. And k squeezes or stretches it, setting the period at 2π/k. Worked example: daylight in Ottawa runs from about 8.7 h at the winter solstice to about 15.5 h at the summer one, so the amplitude is (15.5 − 8.7)/2 = 3.4 h and the midline is (15.5 + 8.7)/2 = 12.1 h. Over a 365-day year k = 2π/365 = 0.01721 per day, and the curve crosses its midline going up at the spring equinox, about day 80, so d = 80. On day 172 the model gives 12.1 + 3.4 sin(0.01721 × 92) = 12.1 + 3.4 sin(1.583) = 12.1 + 3.4(0.99992) = 15.5 h, the solstice, as it should.

The choice of sine over cosine is only a choice of where to start the clock, since cos θ = sin(θ + π/2); a model fitted as a cosine differs from the same model fitted as a sine by a quarter period in d and nothing else. Joseph Fourier's 1822 Théorie analytique de la chaleur is the reason this one shape is worth so much attention: he showed that any reasonable periodic function is a sum of sinusoids like this one, which is why tides, alternating current, sound and heat flow all reduce to the same four parameters repeated. Two traps. k is not the period — it is 2π divided by it, so a bigger k means a shorter cycle. And the bracket reads (x − d), so a model written sin(k(x + 3)) is shifted three units left, not right.