1. Momentum balance along the tube

Newton's second law on the fluid slug (inertia + viscous friction):

\[ P_{source}(t) - P_{sensor}(t) = I\frac{dQ}{dt} + R\,Q(t) \]

2. Mass conservation at the sensor cavity

Flow into the fixed cavity volume shows up as a pressure rise:

\[ Q(t) = C\frac{dP_{sensor}}{dt} \]

3. Eliminate the flow variable

Differentiate step 2 and substitute into step 1:

\[ P_{source} - P_{sensor} = IC\,\ddot P_{sensor} + RC\,\dot P_{sensor} \] \[ IC\,\ddot P_{sensor} + RC\,\dot P_{sensor} + P_{sensor} = P_{source} \]

4. Laplace transform (zero initial conditions)

\[ \big(IC\,s^2 + RC\,s + 1\big)P_{sensor}(s) = P_{source}(s) \]

5. Transfer function

\[ H(s) = \frac{P_{sensor}(s)}{P_{source}(s)} = \frac{1}{IC\,s^2 + RC\,s + 1} \]

6. Standard second-order form

Divide through by IC and match coefficients against \( H(s)=\dfrac{1}{s^2/\omega_n^2+(2\zeta/\omega_n)s+1} \):

\[ \omega_n = \frac{1}{\sqrt{IC}} \qquad\qquad \zeta = \frac{R}{2}\sqrt{\frac{C}{I}} \]

7. Lumped elements from geometry

\[ I=\frac{\rho L}{A} \qquad R=\frac{8\mu L}{\pi r^4} \qquad C=\frac{V}{\rho c^2} \]