CLEARINGHOUSE FOR FEDERAL SCIgCTIFIJ AND TECHNICAL INFORMATION, CFSTI DOCUMENT KANAGEWEirr BRANCH UO.ll LIMITATIONS IN REPRODUCTION QUALITY

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CLEARINGHOUSE FOR FEDERAL SCIgCTIFIJ AND TECHNICAL INFORMATION, CFSTI DOCUMENT KANAGEWEirr BRANCH UO.ll LIMITATIONS IN REPRODUCTION QUALITY Aession # /^."V 1.

I / 2, A portion of the original doment ontpins fine detail whih may make reading of photoopy diffilt, / ' 7 3.

1 CLEARINGHOUSE FOR FEDERAL SCIgCTIFIJ AND TECHNICAL INFORMATION, CFSTI DOCUMENT KANAGEWEirr BRANCH UO.ll LIMITATIONS IN REPRODUCTION QUALITY Aession # /^."V 1. We regret that legibility of this dov-nent is in part nsatisfatory,'. Reprodtion has been made from best available opy. I / 2, A portion of the original doment ontpins fine detail whih may make reading of photoopy diffilt, / ' 7 3. The original doment ontains olor, bt distribtion opies a*' available in blak-and-white reprodtion only, / / U» The original distribtion opies ontain olor whih will be shown in blak-and-white when it is neessary to reprint, [ ] 5. The proessing opy is available on loan at CFSTI.

$CD."V> 4 7 J <o -V \ /' CORSTAIT STRAH WAVES II STRIKS ^ J. D. Cole C.$ $- 1 12 Jaaoary 1953 P-359 V b { [VV'^W '** <JT^ f^- 15 *- "OOPY.. '/ -.$

2 CD."V> 4 7 J <o -V \ /' CORSTAIT STRAH WAVES II STRIKS ^ J. D. Cole C. B. Doogborty J. H. Both Jaaoary 1953 P-359 V b { [VV'^W '** <JT^ f^- 15 *- "OOPY.. '/ -. OF..J " HARD COPY $. A MICROFICHE %. * D D C A DCXMRA C -Tf ßflHD &*&**&*,«M 110*»OW«TM IT. SANTA MOMICA C*ltfORMIA<

$P-359-2- SUMMARY \. A non-linear theory Is developed for onstant-strain waves in elasti strings. The speed of longitdinal and transverse waves is related to the strair and tension.$ where INTRODUCTION: BQUATIONS OP MOTION If it is assned that the tension T in i string ie given by T = initial tension = onstant o o strain and F(a) = E a T = T o + F(o) (1) then soltions to the

where INTRODUCTION: BQUATIONS OP MOTION If it is assned that the tension T in i string ie given by T = initial tension = onstant o o strain and F(a) = E a T = T o + F(o) (1) then soltions to the

3 P SUMMARY \. A non-linear theory Is developed for onstant-strain waves in elasti strings. The speed of longitdinal and transverse waves is related to the strair and tension. The reslts an be sed to allate tension de to impat and ths breakxng loads. Some generalizations are sgg.sted. ) ^~. where INTRODUCTION: BQUATIONS OP MOTION If it is assned that the tension T in i string ie given by T = initial tension = onstant o o strain and F(a) = E a T = T o + F(o) (1) then soltions to the eqations of motion an be; given for the propa- gation of waves of onstant strain and tension. Gh soltions are not restrited to small displaements or slopes and are hene of speial interest. The vaves whih propagate onsist of straight line segments whih move with onstant veloity. The aleratione are ths zero exept at points where the veloity Jmps. At these points the aeleration is infinite. A onvenient way to disss the speed of propagation is by onsidering the differential, aqations of motion. The differential eqations an be onsidered to ontain the ImpJse- momentm laws. Another approah is to apply the implfe-monentm laws diretly. Newton's laws of motion applied to an infinitesimal element of string, are p tt = (T os e) x (2) pv tt» (T sin e) x (.)

P-359-3- where (see Figre l), v -» horizontal and vertial dleplaementa rospetlvely of point loated at x In nstrained string.

The horizontal projetion of an element originally of length A x 1B (1 + ) Ax; the vertial pro- + id Its lengtl l A & = V(i > a2 X sin 6» V X.2 + x) + Vd 2 V X V 2 X x.

4 P where (see Figre l), v -» horizontal and vertial dleplaementa rospetlvely of point loated at x In nstrained string. 9 «Inlination of string p mass per length onstant Ho external fores are onsidered. The horizontal projetion of an element originally of length A x 1B (1 + ) Ax; the vertial pro- + id Its lengtl l A & = V(i > a2 X sin 6» V X.2 + x) + Vd 2 V X V 2 X x. Therefore 00 COB 6 «^(1 (1 + ) x' 5 V X (5) A_»-Ax A x jil + x ) + 2 V X. 1 (6) The harateristi lines, vhlh represent the los of possible disontinities or wave fronts, an be fond for the system of eqations (1) - (6). Two seti haraterlstls appear representing different wave veloities. These speeds are 2' rs 11 o h i; a (8) C, Is the speed of lo^ltdlnal waves In a solid bar of the same material. Cp, for small strains, beomes the speed of transverse waves In a string. For most appliations C. > C 2. It shold be noted that one speed of propagation Is variable and depends on the strain o. The details

P-559-5- of the harateristis are not shown here sine the same reslt abot propagation speeds Is derived by physial onsiderations in paragraph 2.

(6) that Ö, a, T are onstants and that the differential eqations (2) and O) are satisfied trivially, exept where the veloity Jmps. The sioplest example leading to waves of this type is disssed below.

5 P of the harateristis are not shown here sine the same reslt abot propagation speeds Is derived by physial onsiderations in paragraph 2. The general soltion for onstant strain waves an be written in the form v (x, t) «linear fntion of (x, t) (x, t) = linear fntion of (x, t) Then sine v,, v, are onstants it Is easily seen from (l) - (6) that Ö, a, T are onstants and that the differential eqations (2) and O) are satisfied trivially, exept where the veloity Jmps. The sioplest example leading to waves of this type is disssed below. RADIATION PROBLEM: SEMI-IHFIHITE STRIHG Conslier the string at rest initially with zero displaement x = v «v = D t = 0 (9) Also, onsider at first an infinite string -o < x < oo. Let a onstant vertial veloity be imported at the point where x => 0 when t = 0. Then by siimnetry =» 0 at that point for all time and hene x = 0 there. The bondary onditions are ths, for t>0. v (0, t) = V = onstant. (10) (0, t)» 0 It is now sffiient to onsider x > 0, The soltion of the orret / fom Is V (t v - ) t > V V v - (11) 0 t <~ ^ V

It an be shown, and is verified below, that the momentm eqations an- not be satisfied if is assmed greater than. The ndetermined onstants in (11) and (12) are Ü,,.

6 -6- ( Ox V V 0 (t 1 --I X X < t < U V (12) t < vhare - veloity wave«, and = veloity J of v waves, ^ * Tm vorioa regions in the (x, t) diagram and the shape of the waves are shown in Figre 2. The shape of the waves was detemlned by the fat that < and thk.t = 0 a x - 0 and far x > t. It an be shown, and is verified below, that the momentm eqations an- not be satisfied if is assmed greater than. The ndetermined onstants in (11) and (12) are Ü,,. These are ompted by ap- plying the horizontal implse-momentm law along x <= t and the norizontal Mid vertial laws along x = t. Before doing this it is onvenient to ompte the slope and the strains whih are onstant in the varios regions. We have U_ U_ «2 1 --I v 2-0 X (15) ir U,2 V 2 5 I V V sin, V V.fx

P-559-8- 0«- Ö- = 0-0 COB Ö, = * The ebsrlpts denote the regions shown In Figre 2.

0 0 i <0 pv tt - - i- ^ sin j^ B (t - f-) (15) where 5 delta f-nation.

7 P «- Ö- = 0-0 COB Ö, = * The ebsrlpts denote the regions shown In Figre 2. (l > -) * -5 V V Consider now the vertial motion, rormally, ve may write where HJUS (3) beomes T sin 6 - T, sin 8^^ H (t - -) (lu) v H (z) => nit step fntion = 1 z ^ 0 0 i <0 pv tt - - i- ^ sin j^ B (t - f-) (15) where 5 delta f-nation. Integration of (l r with respet to t gives the veloity Jmp aross x - t Similarly - - T^ sin e 1 = pv whih by (11) = pv (16) v t T os - T 1 os ^ H (t - ~) + T 0 <E (t - ^-) - H (t - -)> and TJI - H (t - i-)> (17) + T,<] x i- (T - T ) = p (^ - VL ) vhih by (Ik) * -AH- (18) t "'t - V ^- T T, os p (n^ - ) whih by (ih) V V t t ^-^H_ (19)

' P-559-9- Rov sine o., o«are known In terms of (U,, ), T, and T 2 are known ff fntions of U,, and th*» Eqs. (16), (18) and (19) determine U,, v.

^ 1-^ from whih may be fond as I p 1 (21) It follows that (16) and (19) are two eqations fr U and, and these beome or T 1 sin e i = - Q V v T, os, = T - Kp U 1 1 o V_ - p V ' tan 0,» a.

8 ' P Rov sine o., o«are known In terms of (U,, ), T, and T 2 are known ff fntions of U,, and th*» Eqs. (16), (18) and (19) determine U,, v. i'kat is, T, and T 0 an be expressed as '12 T, e T + E 1 o o,. T 0. E J (1.2.)%4-4 T 2 = T 0 + E o 2 = T 0 - E 1--^ (20) Ths (18) beomes Ä 1 T^ = T 55 " ^ U_ E-^ 1.^ 1-^ from whih may be fond as I p 1 (21) It follows that (16) and (19) are two eqations fr U and, and these beome or T 1 sin e i = - Q V v T, os, = T - Kp U 1 1 o V_ - p V ' tan 0,» a. whih by (l 7^) = - 1 "»T - p K l! o Tl - (P V v ) 2 + (T o - p ü ) 2 = (T 0 + E a^2 V (22) (23) (2U) (2^) an be solved for U to yield U V + U V T 2 o where»= = speed of v waves in a small displßement theory. Using (25), {2h) is the eqation for. (25)

C - C U V ' 1 U V A aparlson of (29) and (20) yields the Interesting reslt that o 1 = a 2 and T 1 = T 2 (^0) The strain and tension have the same onstant vale over the entire distrbed part of the

9 -10- Before solving for It Is «eefl to find exprssione for the strain. Using (a^) also Hene (2^) beomes V v T - p U = p 0 U K V + U V + U V I V (27) whih an be solved for a. T T l 2 = P 2 v 2 ^v 2 ( )2 > = ( T o + S 0 )2 (28) rs 2 -,. ^/ + 0. C - C U V ' 1 U V A aparlson of (29) and (20) yields the Interesting reslt that o 1 = a 2 and T 1 = T 2 (^0) The strain and tension have the same onstant vale over the entire distrbed part of the string. Now sing (27) and (20), eqation (2^) yields the following qarti eqation for (V,, ). 2 2 ( v - v ) ( + v )2. V 2 ( 2-2 ) 2 v ' U V ' P p p o v ( " C) W When is fond from this eqation the strain Is given by (29). Two Uniting ases are of Interest (1) V Is small; ("31) shows that as v - 0 a soltion exists Inde- pendent of V, namely ". The strain Is zero and this theory goes over to the sal linearized theory. o (11) 0; waves are sent ot in a string whose Initial tension is zero. Writing V «a = \ V \ > (32)

$P-559-11- (55) beones \ (1 + a ) - 2 v - 2 a \ + a^ - 0 (55) For a, \ «1 an approxloate soltion to (55) an b«obtained by ne- {leting the X. term in (55).$ The orresponding expression for strain is h 1. 1 /V x 5 '!- r^r-^^y t' 6 ' (Ä y 2' -1 Now T, = Tp E o, so that (56) on be sed to ompte the veloity of motion whih wold break a given string.

The orresponding expression for strain is h 1. 1 /V x 5 '!- r^r-^^y t' 6 ' (Ä y 2' -1 Now T, = Tp E o, so that (56) on be sed to ompte the veloity of motion whih wold break a given string.

10 P (55) beones \ (1 + a ) - 2 v - 2 a \ + a^ - 0 (55) For a, \ «1 an approxloate soltion to (55) an b«obtained by ne- {leting the X. term in (55). Solving the reslting bi eqation yields x-^ + M«2 ) W Hene, approximately, «v = (^) T v T (55) The last reslt learly shows the non-linearity dependene of trans- verse wave speed on inoming veloity. The orresponding expression for strain is h 1. 1 /V x 5 '!- r^r-^^y t' 6 ' (Ä y 2' -1 Now T, = Tp E o, so that (56) on be sed to ompte the veloity of motion whih wold break a given string. It an be noted that the stress and strain are Independent of the area of string. The inlination of the string is given by 1 V 5/! ' / 1 ^ 1 1,V 1/5 eind the onstant fore reqired to prode the wave is r i = E V 5^ - T i Elne i^:i7t^ w Gome typial nmerial valer are those for 5/16" arbon steel ^able where E - 'K? x 10 5 lbs, > 17,0^0 ft/se; if V 690 ft/se T 1 = 5900 lbs 09)

-12- Reflet on problems an b«treated in the same way. It an be ehovb that the refletions of the longitdinal wave at an end travels bak with the speed C. f relative to the original oordinate x.

o The refletion of the retrning longitdinal wave with the on- oming transverse vave an also be worked ot in the same way.

11 -12- Reflet on problems an b«treated in the same way. It an be ehovb that the refletions of the longitdinal wave at an end travels bak with the speed C. f relative to the original oordinate x. Upon refletion from a fixed end ( =» 0) the strain and hene that part of the tension de to strain is dobled. Upon refletion from a free end the strain beomes zero and the tension is then T. o The refletion of the retrning longitdinal wave with the on- oming transverse vave an also be worked ot in the same way. COBCLUDHIQ REMARKS It has been shown how the problem of propagation and refletion of onstant-strain waves in an idealized elasti string an be treated. The reslts are espeially interesting sine they are not limited to small defletions. Longitdinal and transverse waves or whih travel with different speeds relative to the string. One limiting ase shows how a wave propagates bease of elastiity in a string with zero initial tension. The reslts have a pratial appliation in estimating the sd- denly applied onstant fore whih will break a rope. An example of this was presented for zero initial tension whih learly shows the no- linear dependene of tension on fore (Eqs. (')8) - C+O). It shold be remarked that the reslts for the idealized string are independent of the ross-setion area, so long as the string b< haves in a similar manner. The ase of varying veloity applied to a string an be treated by replaing the veloity rve by segmnte of om ^ant veloity. Then the atal motion an be approximated by varios onstant-ptraln waves, simi- lar problems or where a load moves as a string.

The Simple Solutions of Four Actual Problems. of General Theory of Relativity.

The Simple Solutions of Four Actual Problems. of General Theory of Relativity. The Simple Soltions of For Atal Problems of General Theory of Relativity. H Changwei Room 81, No.17,Lane 1769, Pdong Wlian Road, 19 Shanghai China,1-8818, hhangwei5@yahoo.om.n Abstrat: It is qite ompliated