Wave Interference

The superposition principle

When two waves meet, they don't bounce off each other or fight for space. They simply add together. At every point in the medium, the total displacement is the sum of each individual wave's displacement at that location. This is the superposition principle, and it governs the behavior of all linear waves — water waves, sound waves, light waves, even quantum probability amplitudes.

The ripple tank above demonstrates this directly. Each point source emits concentric circular waves. Where those circles overlap, the waves combine. The resulting pattern can be stunningly complex, but it emerges from the simplest possible rule: just add.

Constructive and destructive interference

When a peak of one wave aligns with a peak of another, the combined amplitude is larger than either wave alone. This is constructive interference — the waves reinforce. When a peak aligns with a trough, they cancel: destructive interference. The water surface there barely moves at all.

With two sources oscillating at the same frequency, the interference pattern is stable. Bright bands (antinodal lines) radiate outward where path differences are integer multiples of the wavelength. Dark bands (nodal lines) appear where the path difference is a half-integer multiple. These patterns are not accidental — they are geometric consequences of the relationship between wavelength and source separation.

Young's double-slit experiment

In 1801, Thomas Young directed light through two narrow slits and observed an interference pattern on a distant screen — alternating bright and dark bands. This was the decisive evidence that light behaves as a wave. The double-slit preset in the simulation above recreates this classic arrangement with water waves: two point sources acting as the slits, producing the same geometry of constructive and destructive interference.

The significance of Young's experiment extends far beyond optics. When the same pattern appears with electrons, with atoms, even with large molecules, it reveals that quantum objects exhibit wave-like behavior. The double slit is not just a demonstration of wave physics — it is the central mystery of quantum mechanics.

Diffraction

When a wave encounters a barrier with an opening, it doesn't simply pass through in a straight line. It bends around the edges. This is diffraction, and its extent depends on the ratio of wavelength to slit width. When the slit is comparable to the wavelength, the wave spreads dramatically, emerging from the gap as a new circular wavefront. When the slit is much larger than the wavelength, the wave passes through mostly unaffected, with only slight bending at the edges.

The diffraction preset places a barrier across the tank with a single opening. Watch how the wave reforms on the other side — it's not a shadow with sharp edges, but a spreading circular pattern. This is why you can hear someone speaking around a corner (sound wavelengths are comparable to doorway widths) but can't see them (light wavelengths are far too short for the doorway to matter).

The wave equation

The physics of this simulation are governed by the two-dimensional wave equation: ∇²u = (1/c²) ∂²u/∂t². Here u(x,y,t) is the displacement of the surface at position (x,y) and time t, and c is the wave speed. The Laplacian ∇²u measures the curvature of the surface — how much each point differs from its neighbors. The wave equation says that curvature drives acceleration: a dip in the surface accelerates upward; a peak accelerates downward.

The simulation discretizes this equation on a grid. At each time step, the new displacement at every cell is computed from the current and previous values, plus the Laplacian approximated by the four nearest neighbors: u[t+1] = 2u[t] - u[t-1] + c²dt²(u_left + u_right + u_up + u_down - 4u). A small damping factor is applied each step to prevent the energy from growing without bound — real water has viscosity, and our simulation does too.

Ripple tanks in the real world

Physical ripple tanks — shallow trays of water with vibrating sources and overhead projectors — have been used in physics classrooms since the nineteenth century. They were among the first tools that made wave phenomena directly visible: interference, diffraction, reflection, refraction. When you illuminate a ripple tank from above, the curved water surface acts as a lens, focusing and defocusing light onto a screen below. The bright and dark patterns on that screen are the same patterns you see in this simulation.