2D Wave Interference Simulator

The core of the simulator is the superposition of cylindrical waves from each source. Each wave's amplitude decays as it propagates outward in two dimensions, and its value at any point in time and space is given by a sinusoidal function.

Here, $\psi$ is the total wave field (what you see visualized). $A$ is the source amplitude, $r_i$ is the distance from the $i$-th source to the calculation point, $k = 2\pi/\lambda$ is the wavenumber, $\omega = 2\pi f$ is the angular frequency, and $\phi_i$ is the phase offset of the $i$-th source. The $\sqrt{r_i}$ term is the geometric spreading factor for 2D waves.

The condition for constructive or destructive interference is determined by the phase difference between waves arriving at a point. For two sources with the same phase ($\phi_1 = \phi_2$), a simplified condition emerges based on the path difference $\Delta r = |r_1 - r_2|$.

Here, $n$ is any integer (0, 1, 2...). This is why you see alternating bright and dark bands (fringes). When you adjust the frequency $f$ in the simulator, you're changing $\lambda$, which directly changes the spacing of these fringes according to these equations.

Phased-Array Ultrasonic Testing (UT): In non-destructive testing of pipelines or aircraft wings, an array of ultrasonic transducers emits sound waves into the material. By precisely controlling the phase of each source (like you can in the simulator), engineers can "steer" and focus the resulting interference pattern to scan for cracks without disassembly.

Speaker Array Beam-forming: Concert venues and modern smart speakers use arrays of speakers. By adjusting the phase and delay between them, audio engineers can shape the interference pattern to direct sound energy towards the audience and away from walls, reducing echoes and improving clarity.

Acoustic FEA & the Helmholtz Equation: In Computer-Aided Engineering (CAE), software like ANSYS or COMSOL solves the Helmholtz equation to model steady-state wave interference in complex 3D geometries (e.g., car interiors, concert halls). This simulator visualizes the core principle that these expensive FEA simulations are calculating.

Underwater Acoustic Propagation: Sonar systems use arrays of hydrophones to detect submarines or map the seafloor. Modeling how sound waves interfere and propagate in 2D/3D water layers, which involves damping and multi-source effects, is directly analogous to what you can explore here by adding sources and adjusting damping.

First, remember the premise that "the 'wave sources' in the simulation are ideal points". Real slits or speakers have width, so they are not perfect point sources. For example, if a slit is too wide, the sharp interference fringes you see in this simulator become blurred. When adjusting parameters, also pay attention to the difference between 'phase' and 'initial phase'. The slider changes the start timing of the wave emitted by the source (the initial phase φ), which is different from the delay due to distance (the phase kr). The phase difference Δφ is determined by both.

Also, avoid discussing results with the attenuation setting left as 'none'. Two-dimensional water surface waves or sound waves experience amplitude decay as energy spreads out. Calculating without attenuation (plane wave approximation) shows uniform, strong fringes continuing far away, but in reality, contrast decreases with distance. In practical antenna directivity design, failing to account for this decay can lead to insufficient field strength at distances farther than expected.