This paper proposes a new mathematical formulation for flow measurement based
on the inverse source problem for wave equations with partial boundary
measurement. Inspired by the design of acoustic Doppler current profilers
(ADCPs), we formulate an inverse source problem that can recover the flow field
from the observation data on a few boundary receivers. To our knowledge, this
is the first mathematical model of flow measurement using partial differential
equations. This model is proved well-posed, and the corresponding algorithm is
derived to compute the velocity field efficiently. Extensive numerical
simulations are performed to demonstrate the accuracy and robustness of our
model. Our formulation is capable of simulating a variety of practical
measurement scenarios