Pulsed wave equation

Question

I came across the following pulsed wave plot in a paper:

In the paper, there is no scale as this plot is meant to visualize how a pulsed wave response would look like for ultrasound waves in soft tissues.

Now, I would like to obtain the corresponding equation in order to reproduce that plot. Thank you very much in advance.

Answer 1

You mention that it might be the ultrasound response through a soft tissue.

Actually a tissue can be modelled as a continuous 3D network of spings, masses and dampers, which is the analog of an electrical network with distributed resistance, capacity and inductance.

Considering an input and an output point (small areas) of application and detection of the signal, the influence of the whole body is practically limited to a small cylinder (better a Fresnel's ellipsoid) around the input-output segment.

That means that we can simulate the propagation as the mechanical analogue of the Telegrapher's equation, or as the motion propagating along an elastic, and massive and dissipative, line.

The signal that you depicted might well be the response to a sinusoidal impulse, as in this sketch.

The signal that you depicted might well be the response to a sinusoidal impulse, as in this sketch.

Answer 2

You can take

$$y = e^{-ax^2}\sin(bx + c).$$

If you play around with the constants $a$, $b$ & $c$, then you'll get what you're looking for. Note - try starting with $a$ as a fraction, $b$ as 1 and $c$ will move the sine wave forwards or backwards but not the "pulse" part.