Addendum 2

Acoustic Time Constant

The transmission of a signal by means of fluid motion in an acoustic system ie, air in tubing, cups, etc, can be analyzed mathematically by a set of differential equations quite similar to those that apply to electrical signals.

If we consider Pressure, p, dynes per square centimeter, to correspond to voltage, and fluid velocity, v, cubic centimeters per second, to correspond to current, we can define acoustic resistance, inductance, and capacitance analogous to the electrical elements. Ohm's law then applies:

p = Rv

The physical significance of these elements is that resistance corresponds to frictional losses, such as that due to the viscosity or turbulance of a vibrating fluid. Inductance corresponds to the mass or inertia of the moving fluid, and capacitance corresponds to the elasticity or compressibility of the fluid. The magnitudes of these acoustic circuit elements may be calculated from the physical dimensions of the structures involved. Thus a long thin tube has primarily resistance R_a and inductance L_a which may be calculated by the formulas:

See Equations in PDF file.

Where1 = length of tube, cm

r = diameter of tube, cm

v = viscosity coefficient (1.86 x 10^-4 dynes/cm²/cm/sec for air

p = density (1.20 x 10^-3 gms/cm³ for air at 20°^c)

A short hollow space is essentially a capacitance C_a whose value is given by the relation.

See Equation in pdf file.

Where V = volume, cm³

c = velocity of sound (3.44 x 10⁴ cm/sec in air)

In the acoustic circuit of the Sanborn transducer, consisting of the bell placed against the patient's chest, the flexible tube coupling and the piezoelectric detector, the bell and the diaphragm space constitute capacitance elements. A bell with a volume of 1.8 cm³ corresponds to 1.3 acoustic microfarads. The diaphragm air space is about 6 cm³ or about 4 acoustic microfarads. At the frequencies with which we are concerned, the coupling tube presents only a negligibly small resistance which is in series with the two capacitors. However, the small air leak which was deliberately introduced to test the acoustic system (a tube 0.3 mm in diameter and 5 mm long), represents a resistance of 3,300 ohms. Since it leads from the air system to the outside space, it is in effect a resistance to ground, R₁ as shown in Figure 10. Thus, we have an RC circuit with capacitance of 5 microfarads and resistance of 3,300 ohms or a time constant of 16 msec. This extremely short time constant completely overrides any of the electrical time constants of the transducer or of the amplifier and is responsible for the very sharp, differentiating action that is seen in the output signal of leaky systems.

To obtain the overall time constant for a series of components each of which has its own time constant, such as is illustrated in Figure 10, a calculation is performed like that of resistance in parallel.

See Equation in pdf file.

Where T_e = overall; t_a = acoustic, t_x =transducer, and t_g = the amplifier time constant.

Thus the shortest time constant is the controlling one.