Question

A 5 MHz ultrasound transducer is being used to measure the velocity of blood. When the transducer is placed at an angle of \(45^\circ\) to the direction of blood flow, a frequency shift of 200 Hz is observed in the echo. Assume that the velocity of sound is \(1500\ \text{m/s}\). What is the velocity (in cm/s) of the blood flow? (Round off the answer to one decimal place.)

Hint:

For pulsed/continuous-wave Doppler with reflections, always use the \textbf{double} Doppler shift: \(\Delta f = 2 f_0 v \cos\theta / c\).

Answer 1

Correct Answer: 4.1

Step 1: For Doppler ultrasound with backscattered echo, the frequency shift is \[ \Delta f = \frac{2 f_0 v \cos\theta}{c}, \] where \(f_0\) is the transmitted frequency, \(v\) the blood speed, \(\theta\) the angle between beam and flow, and \(c\) the sound speed. Step 2: Solve for \(v\): \[ v=\frac{\Delta f\, c}{2 f_0 \cos\theta}. \] Step 3: Substitute values \(f_0=5\times10^6\ \text{Hz}\), \(\Delta f=200\ \text{Hz}\), \(c=1500\ \text{m/s}\), \(\theta=45^\circ\) (\(\cos45^\circ=\tfrac{\sqrt2}{2}\)): \[ v=\frac{200\times 1500}{2\times 5\times10^6 \times \cos45^\circ} =\frac{300000}{10^7 \times 0.7071}\approx 0.0424\ \text{m/s} =4.24\ \text{cm/s}. \] Rounded to one decimal place: \(\boxed{4.2\ \text{cm/s}}\).