Source-Induced Reflection in Balanced Shallow-Water Networks

Width-balance conditions at a junction are often associated with reflectionless transmission, or transparency, in some one-dimensional wave models on networks. We show that, on cyclic networks, this identification is incomplete: local balance is a vertex-level condition, while transparency also requires synchronization of the path travel times. To make this precise, we consider a shallow-water wave model and derive the channel-width-weighted scattering law for a vertex of arbitrary degree and introduce a source-relative definition of balance, reflecting the fact that the same junction may be reflective or reflectionless depending on which edges carry incoming waves. Under this definition, a balanced vertex is transparent only to the synchronized incoming-amplitude direction. For an $N$-path island, the resulting first-generation upstream reflection is governed in frequency space by the width-weighted distribution of path delays. Exact broadband cancellation requires all path travel times to agree; when they do not, commensurability of the path-length differences produces a periodic comb of frequency-selective zeros, independent of channel widths. At $N\ge 4$ we further exhibit a hybrid regime in which the reflection factor vanishes without commensurability among the path lengths.