Using a tank with the top filled with water and the bottom filled with a gel which is slightly denser than the water. Inside the gel is a tube tilted 60 degrees above the horizontal axis.

The tube has blood-like material flowing inside. Given my setup, I am trying to calculate the flow rate using the ultrasound Doppler formula: $$ v=(f_{\text{shift}}\cdot c)/(2f_\text{source}\cos(\theta)).$$

The only thing confusing is the velocity $c$. I know the velocity of sound in water and the velocity of sound in the gel. The ultrasound is only slight submerged in water. I don't know if I should use the velocity of the gel or the velocity of the water or something else to get a tube flow reading?


1 Answer 1


In all the following I'll drop the factor 2 due to pulse-echo and assume the scatterer- emitter is moving with velocity component $v_{\theta}$ in the direction of the transducer. I will assume the transducer is directly in contact with the gel, for simplicity.

Let us isolate a single wavelength (for instance between two wave pressure maxima) and follow it as it propagates to the transducer.$ T_{0} $ is the period the of emitted wave:

In water: $ \lambda_{w} = (c_{w}-v_{\theta}) T_{0}$

In the gel: $\lambda_{g} =\frac{c_{g}}{c_{w}} \lambda_{w} $

At the receiver, the measured time between two maxima, defining the period of signal $T$ is : $ T=\lambda_{g} /c_{g} = \lambda_{w} /c_{w} =(1- \frac{v_{\theta}}{c_{w}}) T_{0}$

So it seems that the only velocity that matters is that of the medium in which the source moves.

This is assuming you use the "true" Doppler effect and you can safely ignore the rest of the answer if this is the case. If however you were to use the time shift (phase shift) $\Delta t$ experienced by the received signals between two or more pulse transmits (as is done in modern ultrasound scanners) beware that a similar formula to the one you gave is sometimes used in Doppler ultrasound, (sometimes referred to as Doppler shift, which is very confusing) . Suppose $\Delta t$ is the time shift and $T_{r}$ is the time period between pulse transmits. Because the emitter has moved $v_{\theta} T_{r}$ in the direction of the emitter between pulse transmits and reduced the propagation distance by this amount , $\Delta t$ is given by :

$\Delta t = \frac{v}{c} T_{r}$.

If we write the phase shift $2\pi f_{0} \Delta t$ in terms of a new frequency $2 \pi f T_{r}$ we find that : $f=\frac{v}{c} f_{0}$, looking very similar to a Doppler shift... Anyway with this type of technique, you should use a weighted velocity based on the distance travelled in each medium.


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