PULSE WAVES IN VISCO-ELASTIC TUBINGS 



109 



and the time t; that is to say, r = r(z, t) and Q^ = 

 Qlz, 0- Let us consider as an element of volume a 

 small disc of thickness dz and radius r (see fig. i). 

 The driving force acting on it in the c-direction is 

 then given by 



dF, = 



{Q.-P)^+,, + (Q.-P). = - -^^^^ dz 



dZ 



(1.6) 



A further important relationship is furnished h\- the 

 so-called equation of continuity. For our case, this is 

 identical with the statement that the intake of volume 

 on the front side is equal to the sum of the outflow 

 from the back and the increase in volume of the disc 

 being considered. In mathematical language tliis is 

 stated as: 



d,Az t) = ^^ dz 



dz 



dlr 



(I. 12) 



where ?V denotes a radial current. We need still 

 another relationship between the increase of pressure 

 and the corresponding increase in radius. If the 

 pressure on the outside of the tube is taken to be 

 zero, the pressure/) in the tube will create a tangential 

 wall tension pr (force per unit length). According to 

 Hooke's law, a strip of wall material of thickness a, 

 width dz, and length 27rr will be stretched under the 

 influence of a pressure increase dp to the amount 



or 



- dF, = 



Q. 



dp 

 dz 



dQ_ 



■dz 



(..7) 



If the relative change dQ^/Q^ is sufficiently small, 

 dQJdz will also be small and we may drop the second 

 term in equation 1.7. Thus we will have 



dp 

 JF, = 0- - -dz 

 dz 



(1.8) 



The small disc contains the mass dm = pQ^ dz- Appli- 

 cation of Newton's law (force = mass times accelera- 

 tion) gives the equation the following form 



dt^ 



dj> 

 dz 



(1.9) 



Now, instead of displacement f let us use flow volume, 

 which is often used in theoretical acoustics. It is de- 

 fined as the product of the cross section and the mean 

 particle velocity d^/dt. As we have assumed the 

 velocity to be constant over the entire cross section 

 of the tube and the variations in the cross section 

 to be small, we now obtain for the flow volume in 

 ^-direction 



'. = Q. 



dt 



and Newton's equation takes the form 



dl, 



dt 



9: . ^ 

 p dz 



(1. 10) 



(i.n) 



27rrfr = 



E a-dz 



d(p-r)-dz 



(1.13) 



where E is Young's modulus of elasticity. Considering 



as negligible /) dr as against r dp} we obtain finally 



dp = (Ea/r''-)-dr 



(1.14) 



Introducing the radial current di^ = dr/dt awr dz = 

 — di, we obtain 



dp 

 dt 



I di, . 



- - (E-a/r^) 

 27r dz 



Diff'erentiation with respect to z gives 



dz \dt J 



E-a d-i, 

 27rr3 a<2 



J,- E-a dii dr 



27rr' 



dz 



(■■>5) 



(>.i6) 



For small relative changes in r the second term on 

 the right side can again be dropped, and we obtain 

 finally the equation 



dtdZ 



E-a d-i 

 ■iTir' dz^ 



(1.17)' 



We can eliminate the current from equations i . 1 1 

 and 1 . 1 7 if we differentiate the former twice with 

 respect to ^ and the latter once with respect to t. 

 This leads to 



gyp 



P dz- 



2«-3 d-p 



E-a a/2 



' We always postulate the relative variation in radius dr/r 

 to be sufficiently small. 



' The index Z is omitted in this and the following equations. 



