We decipher the rich and complex two-dimensional wave transition and evolution
dynamics on a falling film both theoretically and numerically. Small-amplitude white
noise at the inlet is filtered by the classical linear instability into a narrow Gaussian
band of primary frequency harmonics centred about ωm. Weakly nonlinear zero-mode
excitation and a secondary modulation instability then introduce a distinct
characteristic modulation frequency Δ [Lt ] ωm. The primary wave field evolves into
trains of solitary pulses with an average wave period of 2π/ωm. Abnormally large
‘excited’ pulses appear within this train at a relative frequency of Δ/ωm due to the
modulation. The excited pulses travel faster than the equilibrium ones and eliminate
them via coalescence to coarsen the pulse field downstream. The linear coarsening
of wave period downstream is a universal (0.015/〈u〉) s cm−1 and the final wave
frequency is the modulation frequency Δ for 0.1 < δ < 0.4 where 〈u〉 is the flat-film
average Nusselt velocity, δ = (3R2/W)1/3/15 is a normalized Reynolds number, R is
the flat-film Reynolds number and W the Weber number.