Superposition

• Principle of superposition: the resultant displacement is the sum of the individual displacements.
• A stationary wave forms when two waves of the same frequency travel in opposite directions; adjacent nodes are λ/2 apart.
• Diffraction is the spreading of a wave through a gap or round an edge; it is greatest when the gap is about one wavelength.
• Interference fringes need coherent sources (constant phase difference).
• Double slit: λ = ax/D.
• Diffraction grating: d sin θ = nλ.

Nodes, antinodes and wavelength

Principle of superposition: when two or more waves meet at a point, the resultant displacement is the sum of the displacements of the individual waves.

A stationary wave forms when two progressive waves of the same frequency (and similar amplitude) travel in opposite directions, usually a wave and its reflection. Drawing the two waves at successive instants and adding them shows points that are always at rest (nodes) and points that oscillate with maximum amplitude (antinodes). Unlike a progressive wave, a stationary wave transfers no energy; all points between adjacent nodes oscillate in phase.

Adjacent nodes (or adjacent antinodes) are λ/2 apart; a node and the nearest antinode are λ/4 apart.

• Stretched string (vibrator at one end, pulley and masses at the other): nodes at both fixed ends; fundamental has L = λ/2.
• Microwaves: a transmitter facing a metal plate; a detector moved along the line finds alternate maxima and minima λ/2 apart.
• Air column: tube closed at one end has a node at the closed end and an antinode at the open end; fundamental L = λ/4. Open at both ends: antinodes at both ends, fundamental L = λ/2 (end corrections ignored).

Spreading through gaps

Diffraction is the spreading of a wave as it passes through a gap or around an edge. The wavelength does not change.

In a ripple tank, plane waves passing through a gap much wider than the wavelength stay nearly straight with only slight spreading at the edges. As the gap is narrowed towards the size of the wavelength the emerging waves become almost circular: diffraction is most noticeable when the gap width is similar to the wavelength.

Coherence and two-source fringes