Doppler shift equation and measurement errors affected by spatial variation of the speed of sound in sea water

Underwater ultrasonic devices using the Doppler effect generally presuppose that the speed of sound is uniform in a propagation medium. Actually, however, the speed of sound in the sea varies with water depth, so that the assumption of such uniform speed has the potential to cause measurement errors. The present study is then involved in theoretically improving the conventional Doppler shift equation by taking into account the fact that the speed of sound is dependent on the propagation path. The study also evaluates measurement errors caused by spatial variation in the speed of sound. Interestingly, the theory predicts that only the speeds of sound at a sound source position and the target position affect the Doppler shift, and the error of the target speed, which is the value that is ultimately measured in ultrasonic Doppler devices, is inversely proportional to the ratio of the speed of sound near the target to that near the source. In order to validate the improved Doppler equation, experiments are conducted using a water tank. The measured Doppler shift data are in agreement with the theoretical predictions within the order of a few tens of hertz.