Comparison of slowly varying phases through a uniform (Eq. (8)) and a laminar (Eq. (4)) flow velocity profile along the no-flow acoustic axis (x = y = 0) up to z = d = 10 m, for frequencies of 150 kHz (ka = 23.6, row 1) and 500 kHz (ka = 78.5, row 2). The red dot indicates the reception point at z = d for each pipe diameter. v _avg = 8 m/s (column 1), v _avg = 24 m/s (column 2), and v _avg = 40 m/s (column 3).