PULSE WAVES IN VISCO-ELASTIC TUBINGS I I : 



- = a2 = l2V(ro^ + u2C2)-"2-[i + v„ni;!' + m^C^)-"^] (3.15)'' 

 ^ = J^c'ftv" + a>2C2)-"2 • [l - VoHvo' + a,2C2)->«] (3.16) 



These two equations tell us how the speed of propa- 

 gation and the damping constant 13 depend on the 

 frequency. Nevertheless, it must be kept in mind 

 that the ''constants" z'o and C are not true constants 

 and that these may also be functions of frequency, 

 which must be determined experimentally. 



It remains for us now to express the frictional 

 constant Ri contained in the constant C in terms of 

 the viscosity of the wall and the geometrical data 

 of the conduit. For this purpose we reason as follows: 

 according to equation 3.2 an excess pressure p — po 

 in the tube brings about the hoop tension 

 Ear[{i/ro) — (i/'')] due to the elastic restoring 

 force plus a tension Ri-r-dr/dt (force per unit length). 

 This additional wall tension is, by definition, equal 

 to the tension on a strip of wall material of length 

 / = 27rr, thickness a, and breadth b, stretched at the 

 speed dl/dt = 2Trdr/dt. According to section 2, the 

 force R dl/dt needed to overcome the friction en- 

 countered in stretching a band of length / = 2Tr, 

 breadth b, and thickness a{q = ab) is 



dl 



It ' 



I 



(3.'7) 



The corresponding hoop tension per unit length will 

 therefore be 



dr rj-a dr 

 dl r dl 



(3-i8) 



and we obtain thus 



Ri =-'— and C = -!^— 

 r* 2rp 



We shall now compare these theoretical considera- 



* These equations can also be deri\ed from Ranke's theory 

 (16) if we replace the undamped spring of his model by a rigid 

 bar; that is to say, if we write according to Ranke's notation 

 £2 — =»,K2= <x e = GO, then Ranke's formula for propaga- 

 tion and damping (16, page 192) are identical with equations 

 3.15 and 3.16 in this text for V = a7)/(2 r). Ranke's equation 

 I a (16, page igi) leads likewise to this value for V if we trans- 

 form it in our notation by replacing volume elasticity by 

 Young's modulus, but is inconsistent with the preceding equa- 

 tion for a where F stands for 7; in our notation. That the second 

 term oj- C under the root sign should only depend on the specific 

 constant and not on the dimensions of the tube can indeed 

 not be understood. 



100-10' 



50 



FIG. 8. Dynamic modulus of elasticity versus frequency. 

 Inside picture: velocity of propagation in cm/sec versus fre- 

 quency. 



tions and results with experimental findings. Measure- 

 ments of the speed of propagation of sinusoidal pulse 

 waves in tubes of soft India rubber show, as a rule, 

 a slight rise with increasing frequencies (12). Ac- 

 cording to equation 315 a dependence on frequency 

 can occur only if either vd- = Ea/ (arp) or the product 

 coC and therefore cot; or both quantities, change with 

 frequency. The determination of wq is rather difficult 

 when we use a method which permits continuous 

 v-ariation of the frequency. As indicated by figure 5 

 the product oj?/ was, at least from frequency i on up, 

 practically constant for the kind of rubber used. 

 Similar results have been obtained by other investi- 

 gators (10). The frequency dependence of v in such 

 cases is therefore determined by the frequency de- 

 pendence of the dynamic modulus of elasticity. A 

 typical frequency curve for £dyn of India rubber is 

 plotted in figure 8, where the extrapolation to the 

 lower frequencies is, of course, somewhat arbitrary. 

 The increase of isdyn up to frequency 3 amounts to 

 about 35 per cent for the given example. The velocity 

 of propagation must therefore rise at the rate of i : 

 1.35"-; that is 1 : 1. 1 6. In the same figure, values of r 

 obtained on a tube of the same material at about the 

 same time are also plotted. From i/ = o up to v = 3 

 V increases by about 15 per cent, which agrees well 

 with expectation. 



If we calculate the quantity [Ea/(2rp)yi' from 

 the measured values £dyn , ", a, r, and p we find that 

 it fits, as a rule, quite well with the obtained values 

 of v extrapolated for zero frequency, if we use instead 

 of E the quantity suggested by Korteweg (6), 



