Fluid dynamical equations

Flow through the slit is the sum of two terms:

$u_f= u + S_r\dot{x}_r$

• $u$ is the flow at the mouthpiece, $u=(p^+ -p^-)/Z_0$.
• $S_r\dot{x}_r$ is an additional term due to reed motion.
• $S_r$ is held constant.

From Bernoulli equation:

$u_f=A\cdot \mbox{sgn}(\Delta p)\vert\Delta p\vert^\mu x_r^\nu$

• For an ideal fluid .
• For the clarinet reed (Backus) .
• Air inertance neglected.

$\Downarrow$

$\Delta p = A^{-3/2}\cdot \mbox{sgn}(u_f) \vert u_f\vert^{3/2}/x_r^2:=f(u_f, x_r)$