recall and use I = I0e–μx for the attenuation of ultrasound in matter

What is Ultrasound?

Ultrasound refers to sound waves with frequencies higher than the upper audible limit of humans (~20 kHz). Think of it as the “invisible music” that can travel through solids, liquids, and gases but is too high‑pitched for us to hear. 🎶

In physics, we treat ultrasound like any other wave: it has a wavelength, frequency, speed, and intensity. The key formula we’ll use is the attenuation equation:

$$ I = I_0\,e^{-\mu x} $$

Where:

• $I$ = intensity after travelling distance $x$
• $I_0$ = initial intensity at the source
• $\mu$ = attenuation coefficient (depends on the material)
• $x$ = distance travelled through the material (in cm)

The exponential term shows that intensity drops rapidly as the wave travels further. It’s like shouting in a crowded room: the louder you shout (higher $I_0$), the more the sound is muffled by people (higher $\mu$) and distance ($x$).

Exam Tip: Remember the Units!

• $I$ and $I_0$ are measured in W m⁻² (watts per square metre). • $\mu$ is in cm⁻¹ (per centimetre). • $x$ is in cm.

When you plug numbers into the equation, keep the units consistent. If you accidentally mix cm and m, the result will be wrong.

👉 Practice: Convert 0.5 m to cm before using it in the formula.

A medical ultrasound probe emits a sound with an initial intensity of $I_0 = 2.0\times10^4\;\text{W m}^{-2}$. In soft tissue, the attenuation coefficient is $\mu = 0.5\;\text{cm}^{-1}$. What is the intensity after the wave has travelled 10 cm?

1. Identify the known values: $I_0 = 2.0\times10^4$, $\mu = 0.5$, $x = 10$.
2. Insert into the formula: $$ I = 2.0\times10^4\,e^{-0.5\times10} $$
3. Compute the exponent: $-0.5\times10 = -5$.
4. Calculate $e^{-5} \approx 0.0067$.
5. Multiply: $I \approx 2.0\times10^4 \times 0.0067 \approx 134\;\text{W m}^{-2}$.

So after 10 cm, the intensity drops to about $134\;\text{W m}^{-2}$—a huge reduction! This explains why ultrasound imaging requires powerful probes and why deeper tissues are harder to image. 🩺

• The attenuation equation $I = I_0\,e^{-\mu x}$ tells us how intensity decreases with distance.
• Higher $\mu$ (e.g., bone) means faster attenuation than lower $\mu$ (e.g., water).
• Always check units and keep them consistent.
• Use the equation to solve exam problems involving intensity after a given distance.

A diagnostic ultrasound probe emits a sound with $I_0 = 3.0\times10^4\;\text{W m}^{-2}$. The attenuation coefficient for the patient's breast tissue is $\mu = 0.3\;\text{cm}^{-1}$. Calculate the intensity after the wave has travelled 15 cm. Show all steps and give your answer in W m⁻².

Tip: Write the formula first, then plug in the numbers. Remember $e^{-x}$ can be approximated using a calculator or a table of $e$ values.