In the low-Froude number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Previous studies have considered linearized steady flow past a submerged source in infinite-depth fluids, in which exponential asymptotics were used to determine the behaviour of downstream longitudinal and transverse free-surface gravity waves. Here, unsteady flow past a submerged source in an infinite-depth fluid is investigated, with the free surface taken to be initially waveless. The source is taken to be weak, and the flow is linearized about the undisturbed solution. Exponential asymptotics are applied to determine the wave behaviour on the free surface in terms of the two-dimensional plan-view, in order to show how the free surface waves evolve over time and eventually tend to the steady solution.