A cardiologist uses an ultrasound scanner with an operating frequency of 3.5 MHz that can detect Doppler frequency shifts as small as 0.1 kHz. What is the smallest flow velocity detectable with this device?
Two frequencies are given so I know I have to convert the units but im thinking if I have to find the wavelength and then velocity. I'm looking at all the formulas and i'm just having trouble with this problem
I'm thinking about using this formula, and stating that vs = vair= 343 m/s
f^' = fo (1 + V/vs)
Hmmmm. I am uncertain of what your text has presented, so let me try this...
doppler=fo-f=2fo(v/vsound) for head on doppler, if one is observing at an angle, cosTheta should be a multiplier.
.1khz=2*3.5M*v/vs
now what is vs: near speed in water, 1496m/s . calculate v
I get v approximately equal to
.1k*1496/2*3.5M = 0.021m/s or about 2cm/s which is a reasonalble flow rate for blood.
yes that's right, thank you very much.
To find the smallest flow velocity detectable with this ultrasound scanner, we can use the Doppler frequency shift formula:
Δf = 2f * (V / c) * cos(θ)
Where:
Δf = Doppler frequency shift
f = operating frequency of the ultrasound scanner (3.5 MHz = 3.5 * 10^6 Hz)
V = flow velocity
c = speed of sound in the medium (assumed to be the same as the speed of sound in air, ~343 m/s)
θ = angle between the direction of flow and the direction of ultrasound waves
In this case, we want to find the smallest flow velocity detectable, which means the smallest Doppler frequency shift. The question states that the scanner can detect Doppler frequency shifts as small as 0.1 kHz (0.1 * 10^3 Hz). Substituting this value for Δf, we can solve for V.
0.1 * 10^3 = 2 * 3.5 * 10^6 * (V / 343) * cos(θ)
Simplifying:
0.1 = 7 * (V / 343) * cos(θ)
Now, we need to estimate the value of cos(θ). Since we don't have any information about the angle, we can assume the most conservative scenario where cos(θ) = 1.
0.1 = 7 * (V / 343)
0.1 * 343 = 7V
34.3 = 7V
V = 34.3 / 7 ≈ 4.9 m/s
Therefore, the smallest flow velocity detectable with this ultrasound scanner is approximately 4.9 m/s.
To find the smallest flow velocity detectable with the given ultrasound scanner, we can use the Doppler frequency shift formula:
Δf = (2 * fo * V * cosθ) / c
Where:
Δf is the Doppler frequency shift
fo is the operating frequency of the ultrasound scanner (3.5 MHz, which can be converted to 3.5 * 10^6 Hz)
V is the flow velocity
θ is the angle between the ultrasound beam and the direction of flow (assume 0 degrees)
c is the speed of sound in the medium (in this case, assume it is the same as the speed of sound in air, which is approximately 343 m/s)
Rearranging the formula to solve for V:
V = (Δf * c) / (2 * fo * cosθ)
Given that the smallest detectable Doppler frequency shift is 0.1 kHz (which can be converted to 0.1 * 10^3 Hz), plugging in the values:
V = (0.1 * 10^3 Hz * 343 m/s) / (2 * 3.5 * 10^6 Hz * cos0°)
Simplifying:
V = 4.9 cm/s
Therefore, the smallest flow velocity detectable with this ultrasound scanner is approximately 4.9 cm/s.