How Does Superposition of Two Waves Create a Standing Wave?

Two waves create a standing wave when they have the same frequency, wavelength and amplitude, travel in opposite directions, and undergo superposition. Their displacements combine to produce fixed nodes and antinodes rather than a wave pattern that travels through the medium.

How the Superposition Works

Consider two identical waves travelling in opposite directions:

$$y_1=A\sin(kx-\omega t)$$ $$y_2=A\sin(kx+\omega t)$$

The resultant displacement is the sum of their individual displacements:

$$y=y_1+y_2=2A\sin(kx)\cos(\omega t)$$

This equation separates position and time. The factor $2A\sin(kx)$ determines the amplitude at each position, while $\cos(\omega t)$ describes the oscillation with time. Therefore, each point has a fixed amplitude and the pattern does not travel.

Adjacent nodes, or adjacent antinodes, are separated by $\frac{\lambda}{2}$. A node and its nearest antinode are separated by $\frac{\lambda}{4}$. Points between the same pair of nodes oscillate , while points in adjacent sections oscillate in antiphase.

A common example is a wave reflecting from the fixed end of a string. The incident and reflected waves travel in opposite directions and superpose. In an ideal standing wave, there is no net transfer of energy along the medium.

Common misconception: the particles are not stationary. Except at nodes, particles oscillate about equilibrium; it is the overall wave pattern that remains fixed.

IB Exam Technique

This is SL and HL content in C.4 Standing waves and resonance. For an Explain question, link opposite wave directions, superposition, destructive interference at nodes and constructive interference at antinodes. On a sketch, label nodes and antinodes and show the $\frac{\lambda}{2}$ spacing between adjacent nodes.