Large-Time Series Expansion of the Wave Front Length in the Euclidean Disk

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In the paper [4], the second author proves that the length | S t | of the wave front St at time t of a wave propagating in an Euclidean disk D of radius 1, starting from a source q, admits a linear asymptotics as t → + ∞ : | S t | = λ ( q ) t + o ( t ) with λ ( q ) = 2 arcsin a and a = d ( 0 , q ) . We will give a more direct proof and compute the oscillating corrections to this linear asymptotics. The proof is based on the “stationary phase” approximation.