Doppler shift equation for velocity along the beam?

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

Answer: (D)

The Doppler shift for ultrasound comes from the round-trip travel to moving scatterers and back, and only the component of the scatterer’s velocity along the beam contributes to the shift. The beam sees a velocity component v cos θ, where θ is the angle between flow direction and the beam. Because the wave travels to the scatterer and back, the shift is doubled. For speeds much less than the speed of sound, the change in frequency is approximated by Δf ≈ (2 f0 v cos θ)/c. This is why the correct expression includes the factor 2, the frequency f0, the velocity component along the beam v cos θ, and the speed of sound c. If θ = 0, the shift is maximal; if θ = 90°, the shift is zero.