Passive acoustics navigation of an autonomous underwater vehicle (AUV) can reduce localization errors while being relatively low-cost. A promising approach for navigation in shallow water is to exploit the waveguide invariant, which makes it possible to estimate the range between an acoustic source with a known position and an acoustic receiver on the AUV. This range estimate can then be fused in a sequential Bayes framework to improve the navigation information of the AUV. While previously proposed methods consider a range-independent environment with a single acoustic source and a single receiver, we investigate a more realistic scenario with multiple sources in a potentially range-dependent environment. In particular, a shallow water environment with a constant slope and sound speed is considered. To resolve multiple sources, it is assumed that the AUV tows an acoustic line array. Simulated acoustic data provided by a normal mode program are used to study the feasibility of the AUV navigation in shallow water by exploiting the waveguide invariant in this setting.