UI.TRASONIC METHOD OF FLOW MEASUREMENT 559 '• V2 l o = •,Cnl These functions are plotted in Fig. 1. For constant n, these graphs can be used to determine the value of l required for simulation of an un- bounded or infinite medium. For large values of l, corresponding to the unbounded case, the components of the impedance (which is Z0 by defini- tion) become, Ro = X'o = (wpn/2) which equals unity on the normalized ordinate scale employed. For very small values of/. R0 = 0 Of greater importance in practice, however, is the case in which l is constant but the properties of the materials vary. Figure 2 illustrates ,this Gr assumedl =0.01 cm., p = 1.0 and w =2v(30 X 10a). Case I/I: Zo of Fiscoe/astic MateriM Following a similar procedure we find 1 sin2y x)• ' [cosey + sinh g + sinh g sinh 2x x) + 4.49 ( sin 2y X0 = • + sinh • 'cos • • • •nh•x) 3 where and x = (0.02220 cop '• Y = X,•n] (0.446/) These functions are plotted in Figs. 3 and 4 as functions of Ol and viscosity, respectively. ULTRASONIC METHODS Mason (3) developed an ultrasonic instrument in which a torsionally vibrating cylinder is employed to measure the real and imaginary com- ponents of the driving point impedance. This technique has been re- peated in a number of laboratories and is extremely useful when both the shear rigidity and viscosity of a material are to be measured. By con- structing a number of torsion vibrators each having a particular frequency of operation, the frequency sensitivity of both the rigidity and the vis-

o •.8. 411- •ry $rr ioyr •7"1' 1411, • rl' 1817' R.o rr Figure &--Driving point impedance of viscoelastic film rs. thickness in skin depth: g/•o• = 10. /6 14 4 Figure 4.--Driving point impedance of viscoelastic liquid rs. viscosity: l • 0.01 cm, p • 1.0 g/cc, •oo = 1.88 10 • rad/sec, g/•Om = 10.