An apparatus for measuring instantaneous velocity or acceleration of blood in a blood vessel, the apparatus comprises means (32, 52, 54, 56, 60) for obtaining, from ultrasonic energy, a Doppler signal having a frequency spectrum determined by relative movement of the components of the blood in a blood vessel, the apparatus being characterised by having means (110 or 302 to 316) for tracking a predetermined portion of the spectrum, provided that when the predetermined portion is a high frequency edge of the spectrum, the Doppler signal is modulated up to a frequency above 100 kHz, to obtain a tracking signal representative of velocity, and optionally means (152 or 320) for differentiating the tracking signal to obtain a signal representative of acceleration.