The time delay between transmission and the arrival of the reflected

wave of a signal using ultrasound traveling through a piece of fat tissue was 0.13ms. At what depth did this reflection occur?

A simple speed formula can be given as

\[v = \frac{{\Delta d}}{{\Delta t}}\]

Here, \[v\] is the speed of sound, \[\Delta d\] which is the distance between the transmission and reflecting surface which is called depth. And \[\Delta t\] represents the time when ultrasound waves just fall on the reflecting surface of a tissue.

Consider the given data as below.

The speed of sound in fat tissue, \[v = 1450m/s\]

Time delay, \[\Delta t = 0.13{\rm{ }}ms = 0.13 \times {10^{ - 3}}{\rm{ }}s\]

This is the time of round trip of sound wave

Here the first trip is completed when the wave starts from the transmission (at the start) surface and ends on the reflected surface.

And the second trip is completed when this wave again comes back to the transmission surface.

Then time to reach the reflective surface will be \[\frac{{\Delta t}}{2}\].

Then actual formula from depth will be

\[v = \frac{{2\Delta d}}{{\Delta t}}\]

Depth \[\Delta d\] will be as follows

\begin{aligned}2\Delta d &= v\Delta t\\\Delta d &= \frac{{v\Delta t}}{2}\end{aligned}

Now substitute the values in the above equation, and you have

\begin{aligned}\Delta d &= \frac{{1450 \times 0.13 \times {{10}^{ - 3}}}}{2}\\ &= 0.094{\rm{ }}m\end{aligned}

Hence, the required depth of fat tissue is 0.094m.