State the general Doppler equation used in ultrasound for the Doppler shift fD given transmitter frequency f0, target velocity v, incidence angle θ, and tissue speed c.

• A. fD = f0 v / (c cos θ)
• B. fD = 2 f0 v cos θ / c.
• C. fD = 2 f0 sin θ / c
• D. fD = f0 / (2 v cos θ c)

Answer: (B)

In ultrasound Doppler, the frequency shift you measure comes from moving scatterers (like red blood cells) that scatter the transmitted wave and then re-radiate it as if they were moving mirrors. Because the wave has to travel to the target and back, the shift is effectively doubled. Only the component of the velocity along the beam matters, which is v cos θ, where θ is the angle between the flow and the beam. The shift scales with the transmitted frequency f0 and is inversely proportional to the speed of sound in tissue c, giving the standard form f_D = 2 f0 v cos θ / c. This means the shift is largest when the flow is aligned with the beam (cos θ ≈ 1) and zero when the flow is perpendicular to the beam (cos θ = 0). The other forms miss one or more of these essential factors: they omit the factor of 2, use sin instead of cos, or omit the velocity term, or mix up the dependencies, so they don’t match the physics of double Doppler shifts in backscattered ultrasound.