As ultrasound travels through tissue, its intensity decreases because energy is lost through absorption and scattering. This attenuation is one of the most important practical limits in medical imaging, because it affects how deeply an ultrasound beam can penetrate while still producing a usable echo signal.
In clinical ultrasound, attenuation is commonly modeled in decibels rather than directly in exponential form. A standard empirical relation is
\[
\Delta \text{dB} = \alpha f z,
\]
where \(\alpha\) is the attenuation coefficient in dB/cm/MHz, \(f\) is the frequency in MHz, and \(z\) is the propagation depth in cm. A typical soft-tissue value is around \(0.5\ \text{dB/cm/MHz}\), although real tissues vary.
Once the dB loss is known, it can be converted into an intensity ratio using
\[
\frac{I}{I_0} = 10^{-\Delta \text{dB}/10},
\]
so the remaining intensity is
\[
I = I_0 10^{-\Delta \text{dB}/10}.
\]
This is the intensity form most consistent with decibel attenuation. It is also possible to write the same physics as an exponential decay,
\[
I = I_0 e^{-kz},
\]
where the exponential coefficient \(k\) is related to the dB attenuation law by
\[
k = \frac{\alpha f \ln 10}{10}.
\]
The sample values in this calculator are \(I_0 = 1\ \text{W/cm}^2\), \(f = 5\ \text{MHz}\), \(z = 10\ \text{cm}\), and \(\alpha = 0.5\ \text{dB/cm/MHz}\). The attenuation in decibels is
\[
\Delta \text{dB} = (0.5)(5)(10) = 25\ \text{dB}.
\]
That means the remaining intensity ratio is
\[
\frac{I}{I_0}=10^{-25/10}=10^{-2.5}\approx 0.00316,
\]
so the final intensity is
\[
I \approx 0.00316\ \text{W/cm}^2.
\]
This means the sample output quoted as \(0.082\ \text{W/cm}^2\) and \(-21\ \text{dB}\) does not match the standard attenuation law with the listed values. With \(\alpha=0.5\), \(f=5\), and \(z=10\), the mathematically consistent result is a \(25\ \text{dB}\) drop and an intensity near \(0.00316\ \text{W/cm}^2\). The calculator above follows the standard dB-based model.
This behavior explains the major imaging trade-off in ultrasound. Higher frequency improves spatial resolution, but attenuation grows roughly in proportion to frequency, so penetration becomes worse. Lower-frequency probes penetrate more deeply but usually give less detail. That is why abdominal imaging often uses lower frequencies than superficial structures such as vessels, thyroid tissue, or musculoskeletal targets.
This calculator uses a simple linear attenuation model. More advanced treatments may include nonlinear propagation, frequency-dependent spectral changes, beam focusing, and tissue heterogeneity. Even so, the basic attenuation relation here is one of the most useful first formulas for understanding why ultrasound fades with depth and why probe frequency matters so much in practice.