nxoria

Calibrating instruments…

Waves & Oscillations

Ripple Tank: Interference & Diffraction

Drop a pebble, open two slits, outrun your own waves.

Preparing the physics engine…

← All simulations

Waves & Oscillations

Ripple Tank: Interference & Diffraction

Drop a pebble, open two slits, outrun your own waves.

Screen intensity — measured vs Huygens–Fresnel prediction

I(y)\;\text{measured}

I(y)\;\text{Huygens sum}

I/I_{max}

y

Displacement at the screen centre

u\,(\text{ahead},\;x=240)

u

t

Total wave energy in the tank

E_{\text{tank}}

E

t

The Physics

The wave equation

\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u, \qquad \omega = c\,|\vec{k}|

Every ripple here obeys the same second-order wave equation: the curvature of the surface in space drives its acceleration in time, and the constant c is the wave speed. It is solved honestly — 63,000 grid cells stepped with a leapfrog finite-difference scheme, verified against the analytic dispersion relation ω = c|k| to 0.02%. The tank edge is a graded absorber tuned below 1% reflection, so it behaves like open water rather than a hard wall.

Superposition & two-source interference

\Delta = |r_2 - r_1| = m\lambda \;\Rightarrow\; \text{bright}, \quad (m{+}\tfrac12)\lambda \;\Rightarrow\; \text{dark}

Because the wave equation is linear, overlapping waves simply add. Two coherent (fixed-phase) sources reinforce crest-on-crest where their path difference is a whole number of wavelengths and cancel where it is a half-integer — the bright and dark fringes lie along hyperbolae of constant path difference. The screen's measured fringes are overlaid with the independent Huygens–Fresnel sum Σe^{ikr}/√r, and the two curves coincide because they are the same physics reached two different ways.

Diffraction & the Fourier connection

a\sin\theta = m\lambda, \qquad I(\theta) = I_0\,\mathrm{sinc}^2\!\Big(\tfrac{\pi a}{\lambda}\sin\theta\Big)

A slit only a few wavelengths wide cannot cast a sharp shadow: by Huygens' principle every point of the aperture re-radiates, and those wavelets interfere to spread the beam into a sinc² pattern with minima at a·sinθ = mλ. Deeply, the far-field diffraction pattern is the spatial Fourier transform of the aperture — narrow the slit and the pattern broadens, the reciprocal-width trade at the heart of both optics and quantum uncertainty. Raise f to shrink λ and watch the central lobe tighten live.

Doppler effect & the Mach cone

f' = \frac{f}{1 \mp v/c}, \qquad \sin\theta_{\text{Mach}} = \frac{c}{v}

A moving source crowds the wavefronts ahead of it and stretches those behind, raising and lowering the observed frequency — the two probes read the shifted values directly. Push the source past the wave speed itself (v > c) and the fronts can no longer escape forwards; they stack into a shock wedge of half-angle arcsin(c/v), the same geometry as a sonic boom or a boat's bow wave. Drag across the open tank faster than the ripples to draw your own.

Numerically verified

Numerical wavelength matches the leapfrog dispersion relation sin(ωΔt/2) = C·sin(kΔx/2) to 0.02%
Double-slit fringe maxima land within 0.5 grid cells (0.03λ) of the exact path-difference prediction
Absorbing boundary measured below 1% reflection — the tank behaves like open water
Measured screen intensity overlays the independent Huygens–Fresnel sum Σe^{ikr}/√r in real time

Scientific references

[1]T. Young, "The Bakerian Lecture: Experiments and calculations relative to physical optics", Philosophical Transactions of the Royal Society 94 (1804)
[2]E. Hecht, Optics, 5th ed., Pearson (2017)
[3]A. Taflove & S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Artech House (2005)
[4]B. Engquist & A. Majda, "Absorbing boundary conditions for the numerical simulation of waves", Mathematics of Computation 31 (1977)

Preparing the physics engine…

Screen intensity — measured vs Huygens–Fresnel prediction

I(y)\;\text{measured}

I(y)\;\text{Huygens sum}

I/I_{max}

y

Displacement at the screen centre

u\,(\text{ahead},\;x=240)

u

t

Total wave energy in the tank

E_{\text{tank}}

E

t

The Physics

The wave equation

\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u, \qquad \omega = c\,|\vec{k}|

Superposition & two-source interference

\Delta = |r_2 - r_1| = m\lambda \;\Rightarrow\; \text{bright}, \quad (m{+}\tfrac12)\lambda \;\Rightarrow\; \text{dark}

Diffraction & the Fourier connection

a\sin\theta = m\lambda, \qquad I(\theta) = I_0\,\mathrm{sinc}^2\!\Big(\tfrac{\pi a}{\lambda}\sin\theta\Big)

Doppler effect & the Mach cone

f' = \frac{f}{1 \mp v/c}, \qquad \sin\theta_{\text{Mach}} = \frac{c}{v}

Numerically verified

Numerical wavelength matches the leapfrog dispersion relation sin(ωΔt/2) = C·sin(kΔx/2) to 0.02%
Double-slit fringe maxima land within 0.5 grid cells (0.03λ) of the exact path-difference prediction
Absorbing boundary measured below 1% reflection — the tank behaves like open water
Measured screen intensity overlays the independent Huygens–Fresnel sum Σe^{ikr}/√r in real time

Scientific references

[1]T. Young, "The Bakerian Lecture: Experiments and calculations relative to physical optics", Philosophical Transactions of the Royal Society 94 (1804)
[2]E. Hecht, Optics, 5th ed., Pearson (2017)
[3]A. Taflove & S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Artech House (2005)
[4]B. Engquist & A. Majda, "Absorbing boundary conditions for the numerical simulation of waves", Mathematics of Computation 31 (1977)