We study the wave equation on an unbounded network of N, N ∈ N * , finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges. The dissipation is given by a damping constant α > 0 via the condition N j=0 ∂xu j (0, t) = α∂tu 0 (0, t). We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for α = N + 1 we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.

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