Model ultrasound attenuation in tissue
Use an exponential model to calculate how ultrasound intensity falls as it travels through matter.
A Level · Cambridge International Physics 9702, 2025–2027 syllabus. Topic 24 was checked against Cambridge’s 2028–2030 syllabus; the other remaining syllabus-year changes are still being reconciled. Online study supports the theory but cannot replace supervised practical work, full timed assessment or human subject-teacher review.
Before you begin
- Use exponential functions, wave speed, energy conservation and SI units; convert between joules and electronvolts when required.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What mathematical form describes attenuation in matter?
What you will learn
- Use I = I₀e⁻μˣ for ultrasound attenuation
- Interpret the attenuation coefficient and path length
- Compare transmitted intensities without assuming a linear decrease
Intensity decreases exponentially with distance
The attenuation model is I = I₀e⁻μˣ, where I₀ is the incident intensity, I is the intensity after travelling distance x through matter, and μ is the attenuation coefficient for that material and wave conditions.
The exponent μx must be dimensionless. If x is in metres, use μ in m⁻¹; if x is in centimetres, use μ in cm⁻¹. Intensity falls by a constant fraction over equal path lengths, not by a constant absolute amount.
The model helps explain depth-dependent echoes
A pulse loses intensity as it travels into tissue. Its returning echo also travels back through matter, so a deep reflector can produce a weaker detected signal than a similar shallow reflector.
The simple equation models attenuation over one path of length x. If the question asks for the outgoing and return paths, use the total distance travelled in the attenuating medium.
Pause and explain
What unit relationship makes the exponent μx valid?
Worked example
Ultrasound enters tissue with intensity I₀. If μ = 0.20 cm⁻¹ and the wave travels 5.0 cm, calculate I/I₀.
Show the worked solution
- The exponent is μx = 0.20 × 5.0 = 1.0.
- I/I₀ = e⁻¹.
- e⁻¹ ≈ 0.368.
Answer I/I₀ ≈ 0.368, so about 36.8% of the initial intensity remains.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For ultrasound, μ = 0.10 cm⁻¹ and x = 8.0 cm. Find I/I₀.
Give me a hint
Calculate μx before using the exponential.
Compare my reasoning
- μx = 0.10 × 8.0 = 0.80.
- I/I₀ = e⁻⁰·⁸⁰.
- e⁻⁰·⁸⁰ ≈ 0.449.
I/I₀ ≈ 0.449, or 44.9%.
Look for these in your work
- Finds the dimensionless exponent
- Uses a negative exponent
- Interprets the remaining intensity
A returning echo has passed through 4.0 cm of tissue on its way out and 4.0 cm on its way back. If μ = 0.15 cm⁻¹, calculate the fraction of its original intensity remaining after attenuation.
Give me a hint
Use the total path through the tissue, not just the boundary depth.
Compare my reasoning
- Total distance in tissue is 4.0 + 4.0 = 8.0 cm.
- μx = 0.15 × 8.0 = 1.20.
- I/I₀ = e⁻¹·²⁰ ≈ 0.301.
About 0.301, or 30.1% of the original intensity remains.
Look for these in your work
- Adds the outward and return path lengths
- Uses the total path in μx
- Evaluates the exponential ratio
At fixed μ, how does increasing the ultrasound path length affect intensity? Explain why doubling the path does not generally halve I.
Give me a hint
Compare the exponential expressions at x and 2x.
Compare my reasoning
- At distance x, I/I₀ = e⁻μˣ.
- At distance 2x, I/I₀ = e⁻²μˣ = (e⁻μˣ)².
- The intensity is multiplied by the same fractional factor for each added equal path, so the decrease is exponential rather than linear.
Doubling x squares the remaining fraction at x; attenuation is exponential, not a fixed linear loss.
Look for these in your work
- Writes the expression for 2x
- Relates e⁻²μˣ to the square of e⁻μˣ
- Distinguishes fractional from absolute loss
Common mistakes
- Replacing exponential attenuation with a linear subtraction
- Using μ and x in incompatible units
- Forgetting that a reflected pulse may travel out and back through the tissue
What can you explain now?
What mathematical form describes attenuation in matter?
Compare with the explanation
An exponential decrease with distance
The relation is I = I₀e⁻μˣ.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
If an echo travels through tissue both ways, which distance belongs in the attenuation exponent?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.