(4-6) ① Since the wave is moving to the left, the values of x where the peaks are found (for example) are decreasing with time: after time t, these have decreased by vt. Therefore at time t, you need to add vt to a given x value to get the value whose sine gives the correct value of y. Hence the leftward-bound wave has the equation:
y = Asin(2π(x+vt)/λ)
② If wave 1 is moving right and wave 2 is moving left, when they superpose, you get:
y1 + y2 = Asin(2π(x-vt)/λ) + Asin(2π(x+vt)/λ)
Using the addition formula, this can be rewritten as:
y1 + y2 = 2Asin(2π(x-vt)/2λ + 2π(x+vt)/2λ)cos(2π(x-vt)/2λ - 2π(x+vt)/2λ)
= 2Asin(2πx/λ)cos(2πvt/λ)
③ The remarkable thing here is that the (x±vt) term – which is what describes the travelling of the wave – has disappeared. Instead, the x-dependent term, sin(2πx/λ), corresponds just to a static sine curve. It is multiplied, however, by the time-dependent cos(2πvt/λ) term. This means that the amplitude of the non-travelling wave oscillates in time in a sinusoidal way.
So we still have an oscillating function – but the positions of the peaks are now fixed in space and only their amplitudes change. This is a standing wave.
