Circular Wavefronts: Ripples in a Bath

A disturbance and a wave equation

Touch the surface of still water at one point and the disturbance spreads outward as a circle, governed by the same 2D wave equation that describes drumheads, seismic waves and sound in a plane:

∂²u/∂t² = c² * ( ∂²u/∂x² + ∂²u/∂y² )
                (c = wave speed on the surface)

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Why the wavefront stays a perfect circle

In a still bath the wave speed c is the same in every direction, so a disturbance released at one instant from one point reaches every direction at the same rate — a circle is simply the set of points at equal distance travelled in equal time. This is also exactly what Huygens's principle says more generally: every point that the wavefront reaches becomes a new source of tiny secondary wavelets, and their combined envelope is what carries the wavefront onward, circle after circle.

Simulating it on a grid

A discretised version of the wave equation updates the surface height at each grid point using its two previous values and its neighbours, the same finite-difference family used for FDTD electromagnetic waves and for seismic simulations:

u_new[i,j] = 2*u[i,j] - u_old[i,j]
           + (c*dt/dx)² * ( u[i+1,j]+u[i-1,j]+u[i,j+1]+u[i,j-1] - 4*u[i,j] )

Just like FDTD, this scheme is only stable below a Courant limit on the time step relative to the grid spacing and wave speed; push it too far and the ripples blow up into numerical noise instead of decaying gracefully.

Why the ripple shrinks as it spreads

Even with no friction at all, a circular wave's amplitude falls as it travels, simply because the same initial energy is spread over an ever-larger circumference: for a 2D circular wave the amplitude falls roughly as 1/√r, where r is the radius already travelled — geometric spreading, not damping. Real bathwater also loses energy to viscosity and surface tension, which adds true damping on top of the geometric fall-off, so ripples in an actual bath die out even faster than the ideal undamped equation predicts.

Interference when two ripples meet

Tap the water twice at different points and you get two expanding circles that cross. Because the wave equation is linear, the surface height where they overlap is simply the sum of both individual displacements — the superposition principle. Where two crests coincide the water rises higher than either wave alone (constructive interference); where a crest meets a trough they partially or fully cancel (destructive interference). Once the wavefronts pass through the overlap region, both continue on completely undisturbed by the encounter, exactly as if the other wave had never been there.

Frequently asked questions

Why does the ripple's height shrink even before it hits the walls of the bath?

Because of geometric spreading: the same amount of energy released at the initial disturbance has to be distributed over an ever-growing circle as the wavefront expands, and for a two-dimensional circular wave the amplitude falls off roughly as one over the square root of the distance travelled, independent of any friction or damping.

What happens exactly where two ripples cross?

Because the wave equation is linear, the water's displacement at any overlapping point is simply the sum of what each wave alone would produce there, so crests meeting crests add up and crests meeting troughs partially cancel. Once the two wavefronts have passed through each other, each continues on exactly as if the crossing never happened.

Is this the same equation used for sound or light?

It's the same broad family, a linear second-order wave equation, though not the identical equation: sound is a pressure wave in a compressible medium and light obeys Maxwell's equations, while real water-surface waves also involve gravity and surface tension in a fuller treatment than the simplified version simulated here.

Try it live

Everything above runs in your browser — open Bath Waves and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

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Ripples in a Bath: Circular Wavefronts, Spreading and Interference