Ultrasound Derivation

How can the pulse be modelled in ultrasound?

The pulse is modelled as:

$n(t) = n_e(t) e^{-j(2\pi f_0 t - \phi)}$

This is a modulated signal, where:

• $n_e(t)$: the envelope (shape) of the pulse over time (original freq).

• $f_0$: the new frequency of the ultrasound wave.

• $\phi$: the phase of the signal.

• $e^{-j(2\pi f_0 t - \phi)}$: represents a cosine-like oscillation inside the envelope.

This is a common way to represent ultrasound pulses mathematically—a high-frequency wave (carrier) inside a shaped pulse.

Why is it a carrier wave?

We send short pulses of ultrasound into the body.

But those pulses aren’t just "clicks" or single spikes. They are typically bursts of a sinusoidal wave (like a few cycles of a sine wave) modulated in amplitude to form a pulse.

• Each transducer point acts as an acoustic dipole. This means each point on the transducer is treated as producing waves.

Calculating Pressure at Transducer - Part 1

Geometry: Fig. 7.2

• The transducer sits in the x–y plane (at $z = 0$).

• The point $(x_0, y_0, 0)$ is a specific location on the transducer face.

• $(x, y, z)$ is the point in space where we want to calculate the sound wave.

• The distance between these points is $r_0$, calculated using geometry.

Equation (7.3): Contribution from One Point

$p(x, y, z, t) = \frac{z}{r_0^2} n\left(t - \frac{r_0}{c}\right)$

This gives the pressure wave (i.e., the sound signal) at point $(x, y, z)$ due to one small area on the transducer:

• $r_0$:
  The straight-line distance from the source point on the transducer to the observation point $(x, y, z)$.

  $r_0 = \sqrt{(x - x_0)^2 + (y - y_0)^2 + z^2}$

• $t - \frac{r_0}{c}$:
  Accounts for the time delay—it takes time for the wave to travel a distance $r_0$ at speed $c$ (speed of sound).

  At $t=0$, there should be no pressure experienced at that point as the wave hasn’t reached yet. This is why we shift it by this amount.

• $\frac{z}{r_0^2}$: This term accounts for the wave weakening with distance and with angle

  • $z=\cos\left(\theta\right)r_0$

  • Therefore, $z / r_0^2 = \frac{\cos(\theta)}{r_0}$

  • Wave weakens with distance and with angle

🔹 Equation (7.4): Total Sound from Whole Transducer

$p(x, y, z, t) = \iint_{A_0} \frac{z}{r_0^2} n\left(t - \frac{r_0}{c}\right) \, dx_0 dy_0$

This is the total sound at point $(x, y, z)$, which is the sum of all contributions from every point on the transducer’s surface $A_0$:

• We integrate over the entire face of the transducer.

• Each point contributes with its own delay and strength.

🧠 In Simple Terms:

• The flat transducer is made of many tiny sources working together.

• Each one sends out a delayed and scaled version of the pulse.

• The sound at any point in space is the sum of all these delayed pulses.

• This is how realistic ultrasound beams are formed—not just along a line, but spread in space with depth-dependent structure.