Impedance of a blowing instrument

Question

I'm trying to reproduce the model described in this paper https://hal.archives-ouvertes.fr/file/index/docid/683477/filename/clarinette-logique-8.pdf.

Here is a method that returns the transfer matrix for a cylinder of radius a and length L.

# density of air
rho = 1.2250

# viscosity of air
eta = 18.5

# transfer matrix for a cylinder
def Tcyl(a, L):
    rv = a * math.sqrt(rho*omega/eta)
    Z0 = (rho * constants.c) / (constants.pi * a**2)
    Zc = Z0 * complex(1 + 0.369/rv, -(0.369/rv + 1.149/rv**2))
    gamma = k * complex(1.045/rv + 1.080/rv**2, 1 + 1.045/rv)
    return
       np.matrix([[cmath.cosh(gamma*L),Zc*cmath.sinh(gamma*L)],
       [(1.0/Zc)*cmath.sinh(gamma*L),cmath.cosh(gamma*L)]], dtype=complex)

The input impedance is calculated for different frequencies as follow.

for f in range(1,4000):
    
    # angular frequency
    omega = 2*constants.pi*f

    # wave number
    k = omega/constants.c

    # transfer matrix for a cylinder of radius 7.5mm and length 100mm
    T = Tcyl(7.5, 100.0)

    # radiation impedance
    Zr = complex(0,0)

    # [P,U] vector (pression and velocity)
    T = np.dot(T,np.matrix([[Zr],[complex(1,0)]], dtype=complex))

    # input impedance (Z = P/U)
    Z = T.item(0)/T.item(1)

The playing frequency satisfy the equation Im[Z]=0. When plotting the imaginary part of Z I get the following figure : wrong impedance

This is clearly wrong as the expected output should be something like this : correct impedance

What am I doing wrong? Thank you.

(1) Show in the question the code how you plot this. (2) If variables "omega" and "k" are used in "Tcyl" they should also be parameters of the function. — Michael Butscher, Mar 21 '21 at 03:05
You are right omega and k should be parameters of the function — pamplemousse, Mar 21 '21 at 18:27

score 0 · Accepted Answer · answered Mar 21 '21 at 03:10

0

You have

    Z0 = (rho * constants.c) / (constants.pi * a**2)

The impedance of a clarinet depends on the speed of SOUND, not the speed of LIGHT. Replace constants.c with 343 and your results will be closer. I'm still not sure it's quite right, but closer.

As a clarinetist, I try to make people think my fingers move as fast as light, but it ain't so.

answered Mar 21 '21 at 03:10

Tim Roberts

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Haha this is so embarrassing. It works perfectly now ! Thank you !!! – pamplemousse Mar 21 '21 at 18:28
Some other corrections : air viscosity must be 0.0000181, radius and length must be expressed in meters. – pamplemousse Mar 21 '21 at 20:04

Impedance of a blowing instrument

1 Answers1