Final answer:
Standing waves are created through the superposition of two identical sinusoidal waves moving in opposite directions. The waves interfere constructively or destructively to produce nodes and antinodes. The equation of a standing wave can be derived by algebraically adding the displacements of the individual waves.
Step-by-step explanation:
Standing waves are formed by the superposition of two identical sinusoidal waves moving in opposite directions. When two waves, represented by y₁ (x, t) = A sin (kx - wt) for the wave moving in one direction, and y₂ (x, t) = A sin(kx + wt) for the wave moving in the opposite direction, are superimposed, they interfere with each other. This interference can be constructive or destructive, leading to the formation of standing waves, which have nodes at integer multiples of half wavelengths and antinodes at odd multiples of quarter wavelengths.
The resulting standing wave appears to be stationary and its equation can be obtained by algebraically adding the wave functions of the individual waves. At points where the crests or troughs align, constructive interference occurs and the amplitude of the resulting wave is maximized, up to twice the original amplitude, due to the cosine term cos (wt), which oscillates between +1. This gives the antinodes their characteristic oscillating appearance between y = ±2A.