# May Contents. III. Discussion of the values found and theoretical considerations 214'

Save this PDF as:

Size: px
Start display at page:

Download "May Contents. III. Discussion of the values found and theoretical considerations 214'"

## Transcription

1 On te formation of Caldera s BY Dr. B.G. Escer Publised on te occasion ol' te Fourt Pacific Science Congress Batavia Bandoeng (Java). May Contents Page. Acknowledgment 184 I. Teory 184 II. Calculation Deduction of te general equation Example: Te Tengger-caldera 188. General solution Tabl&s for 4r and 194, III. Discussion of te values found and teoretical considerations 214'. 1. On te applicability of te new teory. Te value of x.. 214, 2. On te dept of te cored-out cylinders. Te value of?> On te degree of force of volcanic eruptions 215! 4. On te most probable combination of x and ;! ;. 216' 5. On te duration of te paroxysms and of te periods of quiescence in caldera-forming volcanoes 217 IV. Synopsis of te teory of te formation of ealdora's 218 Bibliograpy 219

2 ACKNOWLEDGMENT. In te following study more matematics are used tan is usual in dealing wit problems of dynamical geology. Altoug te matematics tat are applied are simple, it is only by te elp of some colleagues tat tis study could be carried troug. Prof. W. v. d. Woude elped me in te solution of te equations of te tird degree. To Prof. Dr. W. de Sitter I owe te elegant metod for determining te middle root of te 60 equations of te tird degree by a simpler metod. Tese values were calculated in is institute. To bot tese colleagues as well as to Prof. Erenfest I tender my sincere tanks for te elp tey gave me. LEIDEN, Februari I. THEOY. In two previous publications (bibl. 1 and 2) I ave brougt te formation of calderas into relation wit te gas pase, observed by Perret during te eruption of Vesuvius in 1906 (bibl. ). In tese papers I arrived at te conclusion tat during te gas pase a cylinder is cored out, and tat tis may be te cause of caldera formation. In te first te paper subject was treated geometrically, wile in te second calculations were made of a particular case (te Krakatoa eruption of 188) to see if tey would bear out tis teory. Tis caldera-formation, owever, is not a typical case, as tere must previously ave been an older Krakatoa-caldera, and in Aug. 188 it was not a large portion of te volcanic cone tat disappeared, but only an island wic projected little above sealevel; te nortern part of te ancient island akata, wit te volcanoes Perboewatan and Danan. How a caldera migt be formed from a cored-out cylinder I ave tried to explain in two different ways. In te case of te Tengger-caldera I assumed, in analogy wit wat appened in Vesuvius after 1906 (bibl. and 4) tat te uppermost part of te cylinder was transformed into a funnel-sape by crumbling away of te walls, and tat rising lava, as in Vesuvius , formed a flat bottom wic continually reaced iger levels. Tis explanation does not apply to te caldera of Krakatoa, as after te great eruption of Aug. 26 t to 28 t 188 no furter signs of eruption were observed, until in Dec a new pase began in tis famous

3 185 volcano. In te case of Krakatoa in 188, terefore, I tougt it justifiable to apply te penomena, known to occur in coal mining, of recent subsidences wic are caused by te working of coal seams lower down. Goldreic (bibl. 5) gives 45 as te angle of inclination of te slides in loose, detrital formations, and a. larger angle, as great as 85 for more compact formations (fig. 1). Fig. 1. Subsidences due to coalmining,according to A. H. Goldreic (bibl. 5, p. 88, fig. 59). If we imagine a sufficiently large disc-saped mass to be removed from te coal seam, te sliding plane will become funnel-saped. Te applications of tis penomenon to te caldera-formation brougt me to te following ypotesis: if not only te cylindrical disc in te dept is removed, but a complete cylinder from te bottom to te surface, a forteriori a funnel-saped sliding plane 1 ) will form, at least if te radius of te eruption cylinder is not too small. Te collapsed material will ten presumably ave a sallow basin-saped surface. Tis basin sape migt be transformed into a more or less orizontal plane by lava-streams or volcanic ases or bot. In tis article te line of reasoning is resumed, and te last mentioned views on caldera formation from cored-out cylinders are treated as generally as possible. Prom tis, owever, it must not be concluded tat te first point of view is entirely abandoned, as narrow eruption cylinders will not subside tso easily, and te new teory is especially meant for larger eruption cylinders. If tis teory is applied to te normal volcanic cone we arrive at te following results (see fig. 2). As generating line of te external slope of te volcanic cone a straigt line is taken, forming an angle ß wit te orizontal. Te angle of te slide, is also measured wit regard to a orizontal line. a, Wen te cylinder (AA'B'B) is expelled from te volcano, te volume ') Te term "sliding plane" is used ere for te sake of simplicity. It first forms as a funnel saped fault along wic subsequently sliding takes place.

4 186 witin te surface of revolution (ABC) will collapse into te volume (A A'DP). Te contents of tese two bodies must e equal. Fig. 2. Diagram of te formation of a caldera from an eruption cylinder by a funnelsaped slide. = 1000, d = 50, α = 70, ß = 15, x = 246, = 2021 (See table B 5). Of a given caldera te radius of te upper rim is known as well as d te dept below te rim of te caldera bottom, wile ß and a may vary witin certain limits and can be taken as te known variables. Te dept E K =, of te former eruption cylinder, after te collapse, is a function of a, of te radius of te eruption cylinder x, and in a lesser degree of ß also. In tis paper te general equation will be deduced from wic x can be found and afterwards,. Ten a particular case will be treated, and finally a general solution will be given, to wic teoretical considerations will be attaced. Of course in reality neiter A C not B C will be straigt lines and in te calculations, for te sake of simplification, in bot cases a mean slope as been taken, wile D F as also been taken as straigt.

5 ( x) x) )* II. CALCULATION. 1. Deduction of te general equation. Known:, d, ß and a. Fig.. Te measurements used in te calculation. Te drawing was made wit: = 40, d =, α = 6 26, (tan α = 2), ß = 26 4 (tan ß = ½). Unknown: x, wen: Contents (AB C) = Contents (AKEF)., = (, = tan ß tan x r = tan a. =t H( -ti)- x S w Cont. ( t + 2 ) ( 2 +x + x 2) -jr (4 + 2 ) x 2 = =\ tt (tan ß + tan *) ( x ) ( 2 +x - x 2) (II). Cont. (A KEF) J = % (r 2 +rx + x 2) = ó = I-srtanajlß x! (III). ' ( \ tan a, ) )

6 x) 1080x - 60x tan = (II) = (III) terefore: (tan ß 4- tan *) ( - ( 2 + x x 2 ) = 8 >. (V tan x > ) = K ir tan x \ ( ) (2 tan js + tan x) x (tan ß + tan x) x (tan ß + tan x) / d \ x V tan a, i (IV). 2. Example: Te Tengger Caldera. Te Sandsea is ere taken, wereby te straigt boundery, te Tjemora Lawang, is left out of consideration; = 4000 m., d = 00 m. Wat te mean angle of incline ß is for te part of te volcanic cone tat as disappeared, could only e approximated by a special investigation. Te larger te angle is, te larger te volume of te eruption cylinder must be under oterwise equal conditions of te d and. About 25 would be te maximum tat can be assumed for te mean angle of incline. To simplify te calculations tan ß = y 2 is taken ß = 26 4' terefore. For a was taken 45, 66 19', 6 25', 75 58'' and 84 17' successively, sotat tan«becomes successively: 1, / 2, 2, 4, 10. Te general equation (IV) ten gives te following five equations, togeter wit wic te approximated roots are listed, everyting being expressed in ectometers. TABEL A (ectometers). ß tan ß oc tan x Equation. Approximate roots. x(i) *(.) x( ) ( 2 ) 26 4' Vi x = » '»/. llx - 480x = Î) ' 2 14x 8-600x = ÏÎ ' 4 26x' = n ii 84 17' 10 62x x = Tese five groups of tree roots are grapically represented in fig 4.- Te negative and te smallest positive root approac zero, wic is reaced at a = 90. Te greatest positive root lies close to, te value of 40 being reaced at a=90.

7 189 Te negative root lias no real significance for our problem, nor as te largest positive root any volcaiiological importance for our problem, Fig. 4. Grap of te tree roots of te equations of table A. as te diameter of te eruption cylinder would ten e almost equal to tat of te caldera and at te same time te dept of te eruption cylinder would be very sligt (fig. 5). It is only te middle root tat is of importance, tat is te smallest of te two positive roots.

8 190 Fig. 5. epresentation of an application of te two positive roots of te equation. 14x³ 600x ² = 0. = 40, d =, α = 6 26 (tan α = 2), ß = 26 4 ( tan ß = 4). Te two positive values of x are 15.5 and 8.4, te values of belonging to tese are 46 and 0.2. In fig. 6 te equation for te Tengger caldera for 26 4' and = /? a = 6 26': y 14 = x x is represented grapically. Tis sows te general course of te functions ere dealt wit. Fig. 6. Grap of te equation y = 14 x³ 600 x² Tis equation olds for = 40, d =, α = 6 26 (tan α = 2),ß = 26 4 (tan ß = ½). Te tree roots for x lie on te point of section of te curve wit te x-axis (y = 0).

9 191. General solution. Te solution for te smallest positive root tat ere follows I owe to Prof. dr. W. de Sitter, wo also ad te accompanying 10 tables calculated for me. As variable knowns besides a and ß we ave It and d. It is more convenient to take S ( = -=r- ) a.s known instead of and d. By te measurements of a few calderas it was found tat 8 will usually lie between 0.01 and Te following caldera's were consulted: a. Tengger caldera (Java) (bibl. 1). Te "Zandzeo", leaving out of consideration te straigt boundery of te Tjemoro Lawang. = 4000 M., d = 00 M.. terefore Batoer Caldera I (Bali) (bibl. 6 and 7). Kemmerunc; (bibl. 6, atlas, plates V and VI) assumes tat tere must ave been an earlier large Batoer caldera, ere called I for wic I ave taken = 6000 M. and d = 250 M. = =0^ ' o. Batoer caldera II (bibl. b' and 7). Te second Batoer caldera is bounded by te terrace of Kintamani. Following Stens map (bibl. 7) for te diameter of tis caldera and te same map for te difference in eigt between caldera rim and caldera bottom, I take = 600, d = 00 terefore = = Sflfl 600 d. Idjen caldera 's (Java) (bibl. 8 and 9). Kemmerling as discovered a terrace on te inner slope of te Goenoeng Kendeng (bibl. 8, p. 57) from wic e concludes tat in tis caldera also, at least two collapses ave taken place. Moreover e is of opinion tat te elliptic form of tis caldera, wit a long axis of 20 k.m. from E. to W. and a sort axis of 16 k.m. from N to S., indicates a sifting of te volcanic axis. He considers te Idjen to be a composite caldera. For te two Kendeng caldera 's Kemmerling estimates te amount of te two collapses (bibl. 8, p. 140) at 00 m. and 500 m. Hero I ave taken for te first caldera formation E = 6000 m., d = 00 and d = 500 m. for te second. For te first caldera we find S = = second S = = If we divide te general equation (IV) : tan x 2 tan ( tan x + 2 tan ß) x (tan x -+ ß) 4- (tan x + ß) t d \ - tan a, = 0. \ tan x' by tan a, it is canged to:

10 ( s s) I 192 (»-«S-Äi)-»f 1 + i)'+(> +S5) - ( -^r» If we now put: x = e, K = 8 cot a, y = tan ß cot a and d=8k. ten te equation: ( ) -(l + y)t.2 +(l + y )-'(&IZ- 8^Hf=0. is arrived at, Z ( + 2y)«; -(l+y) *+l + y -(l-k) =0 (V). If we now put: A = 1 K ) = K (1 k -f I K 2). We arrive at te equation: A+ (l_ )y (e 2 e a )=0 (VI). Tis equation is linear in y and for given s( =-=--1. By letting e vary between 0 te equation in A and y is expressed in straigt lines. Prom te nomogram tus found wit straigt lines, on te oter and e can be read for different y's and A's. By letting a and ß increase by 5 different y's occur, for one particular 8 different k's are ten found, and terefore A's also. After aving found for 10 different 8's (0.01, 0.02, ), 9 different «'s (45, 50, ) and 7 different js's (0, 5, 10-0 ), te 60 corresponding values for -gwere calculated. In (I) we found: -A = tan*j(t 6)-xj. If. we substitute te values found above for d = 8 and x= e te following equation is arrived at:, = tan # ( 6 V tan«1 ' 1)., = tan x (1 5 1 terefore: = tan a (1 e) (VII).

11 must first be ascertained from te topograpical map, from wic 8 p Values of α. = S for 19 Te ten tables for te different 8's follow ere. (Tables B 1 B 10 ). If we now wis to apply te reasoning of te new teory to any particular caldera, to see wat conclusions we are brougt to, and d follows and so sows wic of te ten tables must be used. Corresponding to te tables graps ave been made (fig. 7 16) x from wic and can be read for all a's between 45 and 85 and all ß's between 0 and 0. Values of ß. From te incline of te remaining foot of a volcano tat as been canged into a caldera by collapsing, a mean angle of incline ß can be deduced by comparison wit te inclination of large volcanic cones tat ave not been attacked by caldera formation. All /S's tat belong to a particular angle of a lie upon a straigt line in te graps. For eac angle a tere is terefore a straigt line of /-points; te /8-lines x for all angles of a bisect eacoter at te point 1 and -~ just above te 0-abscisse terefore, tat lies at te top of te graps. Prof. W. v. d. Woude was so kind as to provide me wit tis important addition. It enables us to read for one interpolated «-value x and ~- te different /-values. Te coice between te different a's seems more difficult. In III tis point will be furter considered. After a a- and a /?-value as been x are read from te graps or te tables. By multiplying by te particular value for wic we ave read from te map, x (= te radius of te cored-out cylinder) and are found.

12 194 TABLE B1. σ = /\ T 0,10 0,094 0,085 0,078 0,071 0,06 0,055 0,045 0,02 0 0,887 1, ,587 1,98 2,564, ,05 0,214 0,19 0,178 0,160 0,14 0,126 0,111 0,086 0, ,776 0,952 1,164 1,445 1,828 2,91, ,71 0,282 0,258 0,25 0,216 0,191 0,168 0,144 0,114 0, ,708 0,874 1,082 1,48 1,725 2,276,18 5,01 10,51 0,48 0,09 0,282 0,258 0,22 0,206 0,174 0,142 0, ,642 0,812 1,015 1,275 1,67 2,171,07 4,85 10,28 0,85 0,52 0,2 0,294 0,268 0,26 0,202 0,160 0,112 20" 0,605 0,762 0,957 1,21 1,560 2,089 2,97 4,75 10,14 0,427 0,9 0,60 0,28 0,297 0,26 0,228 0,182 0, ,56 0,712 0,904 1,154 1,498 2,015 2,87 4,6 9,96 ~ 0,466 0, ,62 0,27 0,290 0,250 0,202 0, ,524 0,669 0,851 1,095 1,44 1,940 2,79 4,51 9,80

13 Fig. 7. Grap of table B1.

14 196 TABLE B. σ = \ ' 70 D ,15 0,140 0,126 0,112 0,100 0,087 0,07 0,058 0,046 0 e., 0,827 1,00 1,228 1,518 1,910 2,488,44 5,2 10,88 0, ,184 0,160 0,141 0,120 0,098 0, ,76 0,906 1,121 1,9 1,782 2,40,27 5,09 10,61 0,07 0,281 0,254 0,22 0,204 0,180 0,152 0,127 0, ,67 0,87 1,045 1,10 1,687 2,2,15 4,9 10,42 IT 0,59 0,29 0,00 0,277 0,24 0,21 0,181 0,149 0,10 15 K 0,621 0,780 0,980 1,22 1,604 2,142,04 4,81 10,2 0,405 0,71 0,9 0,09 0,275 0,244 0,208 0,168 0, , ,924 1,177 1,55 2,057 2,94 4,70 10,08 0,446 0,411 0,74 0,42 0,06 0,271 0,21 0,189 0, ,54 0,682 0,874 1,120 1,469 1,98 2,86 4,58 9,91 " 0,484 0,446 0,410 0,74 0,5 0,297 0,256 0,209 0, ,496 0,640 0,82 1,064 1,406 1,911 2,77 4,46 9,74

15 Fig. 8. Grap of table B2.

16 198 TABLE B. σ = 0.0. \ " ,191 0,172 0,158 0,140 0,125 0,110 0,091 0,07 0,057 0, 0,779 0,957 1,172 1,460 1,847 2,415,6 5,2 10,75 0,270 0,246 0,225 0,200 0,179 0,158 0,1 0,104 0, ,700 0,869 1,077 1,56 1,71 2,28,20 5,05 10,51 0,29 0,00 0,272 0,246 0,220 0,194 0,164 0,11 0, ,641 0,804 1,010 1,276 1,64 2,184,09 4,90 10,4 ~ 0,80 0,46 0,16 0,285 0,255 0,227 0,192 0,152 0, ,590 0,750 0,947 1,208 1,568 2,09 2,98 4,78 10,16 0,42 0,88 0,56 0,21 0,288 0,254 0,218 0,172 0, ,547 0,700 0,890 1,146 1,497 2,019 2,89 4,66 10,02 0,46 0,426 0,90 0,5 0,17 0,282 0,240 0,191 0, ,507 0,654 0,841 1,091 1,45 1,942 2,80 4,56 9,85 0,501 0,461 0,422 0,85 0,46 0,06 0,26 0,211 0, ,469 0,612 0,795 1,05 1,7 1,876 2,72 4,44 9,69

17 Fig. 9. Grap of table B.

18 ' 200 TABLE B 4. σ = " / \ 0,224 0,205 0,186 0,166 0,149 0,11 0,112 0,087 0, ,76 0,908 1,122 1,404 1,785 2,47,27 5,14 10,64 IT 0,294 0,270 0,245 0,220 0,197 0,172 0,148 0,116 0,084 5, 0,666 0,80 1,08 1,11 1,682 2,25,14 4,97 10,4 0,52 0,21 0,291 0,262 0,24 0,206 0,176 0,141 0, ,608 0,769 0,972 1,28 1,60 2,141,0 4,8 10,27 "" 0,401 0,66 0,2 0,299 0,269 0,26 0,202 0,161 0, ,559 0,716 0,914 1,174 1,528 2,059 2,94 4,72 10,09 T 0,442 0,407 0,69 0, 0,00 0,26 0,227 0,180 0, ,518 0,667 0,861 1,115 1,462 1,985 2,84 4,61 9,94 0,482 0,44 0,40. 0,66 0,29 0,290 0,248 0,199 0, K 0,478 0,624 0,81 1,058 1,99 1,910 2,76 4,50 9,78 0 0,518 0,478 0,47 0,97 0,57 0,15 0,271 0,218 0,15 0,442 0,582 0,764 1,004 1,9 1,842 2,68 4,9 9,64

19 Fig. 10. Grap of table B 4.

20 202 TABLE B 5. σ = J0\ c ~ 0,252 0,229 0,208 0,189 0,166 0,146 0,120 0,099 0, ,698 0,869 1,081 1,55 1,79 2,296,2 5,06 10,64 0,21 0,291 0,262 0,27 0,209 0,18 0,154 0,127 0, ,629 0,795 1,004 1,272 1,647 2,194,11 4,90 10,41 0,74 0,40 0,07 0,279 0,246 0,216 0,181 0,149 0,098 10" 0,576 0,77 0,940 1,199 1,567 2,101,00 4,78 10,26 ~~ 0,421 0,82 0,48 0,14 0,279 0,246 0,207 0,168 0, ,529 0,687 0,881 1,18 1,497 2,021 2,91 4,67 10,10 0,46 0,422 0,8 0,47 0,09 0,272 0,21 0,187 0, r" 0,487 0,69 0,81 1,018 1,4 1,950 2,82 4,56 9,9 0,501 0,458 0,418 0,78 0,-8 0,298 0,252 0,206 0, ,449 0,596 0,781 1,027 1,70 1,878 2,74 4,45 9,78 0 ~ 0,57 0,492 0,450 0,409 0,64 0,22 0,272 0,22 0,154 0,41 0,556 0,75 0,974 1,14 1,812 2,67 4,6 9,62

21 Fig. 11. Grap of table B 5.

22 204 TABLE B6. σ = ~ 0,280 0,252 0,229 0,208 0,185 0,162 0,14 0,112 0,07 0 0,660 0,82 1,041 1,12 1,688 2,242,17 4,97 10,54 0,42 0,11 0,282 0,254 0,224 0,198 0,167 0,17 0, ,598 0,761 0,965 1,22 1,605 2,14,05 4,8 10, 0,94 0,59 0,24 0,292 0, ,19 0,158 0, ,546 0,704 0,905 1,166 1,527 2,058 2,95 4,71 10,18 0,441 0,400 0,64 0,28 0,291 0,257 0,217 0,177 0, ,499 0,655 0,848 1,104 1,461 1,891 2,86 4,61 10,00 0,48 0,48 0,99 0,60 0,22 0,282 0,240 0,194 0,12 20 b 0,457 0,610 0,798 1,048 1,94 1,912 2,77 4,51 9,86 0,521 0,474 0,42 0,90 0,48 0,07 0,261 0,212 0, ,419 0,567 0,751 0,997 1,9 1,844 2,70 4,41 9,71 ~ 0,555 0,507 0,46 0,420 0,77 0,2 0,282 0,20 0, ,85 0,528 0,707 0,945 1,276 1,775 2,62 4,1 9,56

23 Fig. 12. Grap of table B6.

24 206 TABLE B7. σ = * ,07 0,278 0,249 0,222 0,195 0,172 0,149 0,117 0, ,62 0,791 1,002 1,277 1,665 2,205,10 4,94 10,45 BT 0,68 0, 0,298 0,268 0,26 0,207 0,180 0,140 0, ,562 0,725 0,92 1,198 1,570 2,108 2,99 4,81 10,25 T 0,418 0,79 0,41 0,0 0,272 0,28 0,20 0,160 0, ,512 0,670 0,871 1,17 1,492 2,02 2,90 4,69 10,10 0,461 0,420 0,79 0,9 0,0 0,26 0,227 0,180 0, ,469 0,621 0,817 1,075 1,425 1,955 2,81 4,58 9,94 0,499 0,455 0,412 0,72 0,2 0,289 0,248 0,198 0, ,41 0,580 0,770 1,018 1,6 1,88 2,7 4,48 9,80 0,540 0,492 0,445 0,401 0,60 0,14 0,270 0,215 0, ,90 0,516 0,72 0,967 1,0 1,814 2,65 4,8 9,67 0 0,574 0,525 0,477 0,40 0,86 0,8 0,291 0,22 0,161 0,56 0,496 0,677 0,917 1,247 1,749 2,57 4,28 9,52

25 Fig. 1. Grap of table B7.

26 208 TABLE B8. σ = ,1 0,00 0,271 0,240 0,21 0,188 0,158 0,126 0, ,589 0,754 0,961 1,26 1,608 2,151,06 4,88 10,6 0,90 0,52 0,18 0,285 0,251 0,220 0,186 0,148 0,10 5 0,50 0,692 0,894 1,158 1,527 2,06 2,96 4,75 10,17 0,48 0,97 0,57 0,20 0,284 0,249 0,210 0,167 0, ,482 0,69 0,88 1,098 1,456 1,98 2,87 4,64 10,02 0,48 0,47 0,9 0,54 0,14 0,276 0,28 0,187 0, ,47 0,591 0,787 1,09 1,91 1,909 2,78 4,5 9,86 0,52 0,47 0,427 0,85 0,42 0,299 0,25 0,20 0, ,97 0,548 0,78 0,985 1,1 1,846 2,71 4,44 9,74 0,559 0,508 0,459 0,41 0,68 0,25 0,274 0,220 0, ,61 0,506 0,69 0,97 1,276 1,774 2,6 4,4 9,62 0,59 0,541 0,492 0,442 0,95 0,48 0,296 0,28 0, ,27 0,467 0,645 0,886 1,218 1,711 2,55 4,24 9,46

27 Fig. 14. Grap of table B 8.

28 210 TABLE B 9. σ = / \ ,57 0,21 0,288 0,258 0,229 0,200 0,170 0,15 0,09 0 0,55 0,719 0,927 1,195 1,564 2,108,01 4,81 10,28 0,412 0,72 0, 0,298 0,266 0,22 0,195 0,155 0, ,498 0,659 0,862 1,126 1,484 2,020 2,91 4,70 10,11 0,460 0,414 0,72 0,4 0,296 0,260 0,219 0,175 0, ,450 0,609 0,807 1,064 1,420 1,94 2,82 4,59 9,97 0,504 0,454 0,408 0,67 0,25 0,285 0,240 0,192 0, ,406 0,561 0,755 1,006 1,58 1,874 2,74 4,49 9,8C 0,541 0,492 0,441 0,97 0,5 0,09 0,261 0,209 0, a 0,69 0,516 0,708 0,954 1,298 1,808 2,67 4,9 9,68 0,580 0,525 0,472 0,424 0,78 0,2 0,282 0,225 0, ,0 0,476 0,664 0,908 1,244 1,745 2,59 4,0 9,56 "F 0,615 0,558 0,50 0,454 0,405 0,56 0,02 0,242 0, ,295 0,47 0,620 0,856 1,186 1,679 2,51 4,21 9,42

29 Fig. 15. Grap of table B9.

30 212 TABLE B 10. σ = / \ ,82 0,40 0,07 0,272 0,242 0,210 0,181 0,145 0, ,518 0,687 0,890 1,161 1,526 2,070 2,95 4,75 10,19 0,47 0,91 0,51 0,1 0,279 0,240 0,206 0,164 0, ,46 0,626 0,827 1,090 1,447 1,988 2,86 4,64 10,0 ~~ 0,485 0,42 0,88 0,47 0,07 0,267 0,228 0,18 0, ,415 0,577 0,774 1,01 1,86 1,914 2,78 4,5 9,90 0,527 0,472 0,42 0,80 0,8 0,292 0,250 0,200 0, ,7 0,529 0,724 0,974 1,20 1,845 2,70 4,44 9,7 0,567 0,507 0,456 0,409 0,62 0,15 0,270 0,216 0, , 0,488 0,677 0,924 1,269 1,782 2,62 4,5 9,6 0,604 0,541 0,488 0,48 0,89 0,9 0,289 0,22 0, ,296 0,447 0,6 0,87 1,211 1,716 2,55 4,25 9,50 0,640 0,57 0,519 0,465 0,415 0,62 0,09 0,249 0,17 0 s 0,260 0,409 0,587 0,827 1,155 1,65 2,48 4,16 9,5

31 Fig. 16. Grap of table B 10.

32 III. DISCUSSION OF THE VALUES FOUND, AND THEOETICAL CONSIDEATIONS. 1. On te applicability of te new teory. Te values of x. Te teory of te formation of ealdera's discussed ere is based upon te formation of a cored-out cylinder caused by te gas pase of Perret (bibl. ). Secondarily from tis cylinder, a caldera is formed by te walls subsiding along a funnel-saped surface. Te formation of a caldera (a negative form), tat is to say a large flat crater bottom wit steep sides, is a logical consequence of te fact tat te angle ß is always smaller tan te angle a. Te walls of te cored-out cylinder form a vault wit a vertical axis wic can support radial It is clear tat wit uniform pressure. material te radius of te cored-out cylinder will decide te amount of pressure tat can be supported. Te smaller te radius, te greater te pressure tat can be born by te vertical arc. It sould be possible to make estimations of te relation between tese two, but ere we will do no more tan point out tat te radius of te cored-out cylinder of Vesuvius, according to MaltiAdra's profile (bibl. 4) is about 100 m., wile from Frank Perret 's work (bibl., p. 67 and fig. 1) te conclusion migt be drawn tat x lay between 100 m. and 200 m. After te Vesuvius eruption in 1906 tere did not follow any caldera formation, in te manner discussed. Te "collapse" of wic Perret (bibl., p. 94) speaks and wic e sows in poto's 7 and 56, migt be considered te as beginning of a sliding in of te walls. Te '' external collapse" took place on te sout east flank of te cone. It is tougt to ave been due to a less resistant part of te cored-out cylinder, but I may point out tat Perret explains it in anoter It way. seems, in fact, tat wit te Vesuvius-rock te radius x = 100 m. was too small to occasion a caldera formation from collapse. Notwitstanding, te Vesuvius crater gradually acquired a caldera-like sape, caused by te walls of te cylinder crumbling away from above, were te material was loose, and later on by rising lava filling up te vent (bibl. 1, and 4). Large caldera's, wic migt also be called true caldera's are formed terefore, only wen x is large, as it is only ten, tat te vent can be pressed in. We must not terefore be afraid of assuming a somewat large radius of te cylinder. Tis is one of te necessary deductions of te teory developed ere: it applies only to spacious cored-out cylinders. If we take te angle ß 15, = x for S = 0.5, will be between 0.1 and 0.4 tat is for = 5000 m. between 500 and 2000 m. 2. On te dept of te cored-out cylinder. Te value of. Te dept of te part of te eored-out cylinder tat is ere called, as also tat of te entire original cored-out cylinder, depends primarily

33 te dept of 50 km te initial pressure would terefore be kg/cm upon te angle a. As soon as a becomes more acute tan 70, rapidly increases. If we take ß = 15, s for 8= 0.5 varies between 0.5 li and 10. As in comparatively sallow coal mines slides of 85 ave been registered in ard rock, it seems to me probable tat in our case te mean angle of sliding will be 70 or larger. For ß = 15 and 8 = 0.5, tis means tat, as a value between 2 and 10. For =5000 m., would lie between 10 km and 50 km. At tese depts we approac te magma camber. For some geologists tese depts may seem outrageous, and I must confess tat at Bret sigt I found difficulty in accepting suc a conclusion, but we sall see tat furter considerations lead us in te same direction. To make tis clear, we must first examine te degree of force of a volcanic eruption.. On te degree of force of a volcanic eruption. Te motive force of a volcanic eruption is te magmatic gas. As in a violent eruption magma is always trown out, it cannot e asserted in general tat in te magma camber before te eruption tere as l>een a separation between gas and magma. Before te-eruption te gas is in solution under pressure in te magma. According to te present state of penological knowledge, I believe, we must suppose tat by crystallization of te magma in te dept te gas is eld in solution in te residuary magma under a constantly increasing pressure. Te increase of pressure can only go as far as te counter-pressure allows. In oter words te maximum tension of te gasses dissolved in te residual magma is equal to te resistance of te rock above te camber. Te pressure of gas at te moment of eruption, tat is te initial pressure of te eruption, just exceeds te counter-pressure wic is exerted upon "te magma camber. It may be taken tat up to depts of 100 km te pressure will increase wit a constant ratio to te dept. Assuming to tese depts a mean specific gravity of te rock of, te increase of pressure per 10 km will be 000 kg/cm 2. At a dept of 50 km te pressure would be kg/om 2. Wit an eruption from Tere is anoter factor besides te gas pressure wic determines te force of te eruption. Tis is te lengt of time during wic large quantities of gas per time unite escape. A certain pressure being given by te dept, te force of an eruption will depend upon te total amount of dissolved gas and terefore upon te volume of te magma in te camber, from wic te gas escapes. It is terefore logical to assume, tat violent eruptions can only take place, were te magma camber is large. Te force of a volcanic eruption, terefore, can be pysically defined as te product of gas pressure and te amount of gas, and geologically expressed as te function of te dept of te camber and of its volume. In 1927 I was able to prove by experiments (bibl. 1), tat te radius of a cored-out cylinder is a function of te velocity of te eroding gas.

34 216 Te velocity itself is a function of te amount per time unite of escaping gas. Furter te experiments seemed to sow, tat at a given pressure a certain amount of time is needed for coring out te cylinder. Te widening begins below and works upwards by eddies to te top, provided tere is a sufficient amount of available. gas Caldera-formation as ere described will only take place wen te camber lies deep down, as it is only ten tat te tension of te gas can be great, and wen te volume of te camber is great, as only ten te quantity of gas is sufficient to core out a cylinder. It sould be possible to find te dept of te magma camber from te initial pressure. So far tere are no data known on tis subject, but is ougt to be possible to determine te initial pressure approximately from te gas column of wic Perket gives a representation (bibl., p. 45, fig. 1), as it rose, up during te intermediary gas pase in Vesuvius and wic only spread out orizontally at a eigt of 10 km above te vent. '' Let te reader tink of globular masses of compressed vapour almost exploding in te air at more tan 10 kilometers above te vent, and ten try to imagine te original tension of te. gas and te degree of its acceleration 'witin te saft of te volcano" (bibl., p ). If it prove leasable to arrive at an idea of te initial pressure in tis way and tat tis is found to be very ig, wc must not forget tat te force of te eruption of Vesuvius in 1906 during te intermediate gas pase was small compared to te force tat is needed to form a true caldera via a cored-out cylinder. In te Krakatoa eruption in 188 te gas column was probably 50 km ig, and terefore te minimuim dept of te magma camber muc greater in tan in te Vesuvius eruption 4. On te most probable combination of x and. eturning to wat we said on p. 19 and 215 about te angle of a, we must point out in te first place tat a given size of te caldera can lie arrived at matematically by various combinations of x wit tat differ greatly. For S = 0.5 and ß = 15 te combinations vary between x=0.4 wit = 0.5 wit a = 45 and x= 0.1 wit = 10 wit a = 85. Te point now is to ascertain geologically wic combination is te most probable, tat wit a large or wit a small. Te experiments already described sowed tat te greater te pressure is in te duct of te gas, te greater te radius of te cored-out saft becomes. Te pressure in te gas duct was measured in a reduction valve. A similar reduction valve will be present in nature, te opening of te magma camber. But in te natural state te opening will be constantly enlarged by te continuous erosion of te gas-stream. Only wen te magma camber is very large te initial pressure of te escaping gas will be comparatively slowly reduced. Only in tat case a spacious cored-out vent can be formed. Wit a low pressure it is a priori impossible for te vent to be cored-out. It seems to me, terefore, tat a combination of great dept and great volume of te magma camber

35 217 are necessary for te formation of a caldera in te way suggested. It follows tus tat a must be large. A steep funnel is te most probable, not only for te reasons given above, but also for te following. Te deeper te cause of te sliding down is, te greater must be rigidity of te rock, terefore te greater te angle of slide «must be. Te combination of a large wit a small x seems to me, terefore, te most probable. For a caldera wit = 5000 m. and S= 0.5 and ß t= 15 te most probable combinations would lie between: x= 0.1 wit 10, 500 c= tereforex= m wit = 50 km (<* 85 ). = and x= 0.2 wit =, terefore 1000 x= m wit = 15 km (a 75 ). = 5. On te duration of te paroxysms and te periods of quiescence in caldera forming volcanoes. In bibl. 2 I ave endeavoured to sow tat te cored-out cylinder of Krakatoa in 188 wic led to te caldera being formed, did not lie at te point of te preliminary eruptions wic took place from May 20t to Aug. 26t from te Perboewatan volcano amongst oters, but about 2 km to te sout.. D. M. Verbeek ad already ascertained (bibl. 10) tat te caldera formation of 188 was not te first, but ad been preceeded by a larger one. After te first caldera formation.according to te teory ere developed, te cored-out cylinder must ave been coked up by te sliding down material. It is probable tat a funnel-saped vent closed in tis way by loose material, will sut off te magma camber less effectively tan te former narrow vent filled wit congealed magma. It is also natural tat wit te increasing gas pressure in te magma camber te gas will sometimes find an outlet along a weak line in te sliding plane. I regard te formation of te Perboewatan and Danan volcanoes as due to te escape of gas along a sliding plane, just as I regard te present activity of Krakatoa as escape of gas along te sliding plane of 188. Wit te leaking out of magmatic gas, magma is carried along and tus a volcano is built up. Tis stage may continue for a long time. In te case of Krakatoa it probably lasted for centuries. Later on te leakage was coked (1680), so tat te pressure in te magma camber increased more and more during 20 years. On May te weak place was once more opened, te activity gradually increased until on Aug. 26 t 188 te paroxysm took place. Te actual eruption from te central vent wic formed te cored-out cylinder, began on Aug. 26t 188, reaced its eigt on Aug. 27* and died out on Aug. 28 t - at 6 o 'el. in te morning. Te duration of' te actual eruption, wic caused te caldera formation, was tus very sort. Tis is quite natural, because as soon as a wide cannel as been bored by te eroding current of gas, te tension of te gasses dissolved in te magma, quickly decreases. Ten te wide eruption cannel is coked up and te eruption is over. Tere was no activity at Krakatoa after Aug. 28* 188.

36 218 It was 44 years, up to Dec. 1927, before te gasses ad again acquired sufficient tension to leak out along weak lines in te sliding plane. Presumably, terefore, we are now witnessing te formation of a new series of small secondai-y Krakatoa volcanoes, again lying excentric wit respect to te principle axis. (In oter caldera's, suc as in Batoer on Bali (bibl. 7) te secondary volcanoes are formed centrally). To sum up, I tink I may conclude, tat te paroxysmal eruption wic primarily creates a cored-out cylinder and secondarily a caldera, lasts only a very sort time, some days, but tat te period of rest between two caldera forming eruptions is very long (centuries), tat moreover between two paroxysms of te first order a lengty period of subdued constructive volcanic activity intervenes wic may last for centuries. Te great dept of te volcanic magma camber tat is ere postulated, demands a great lengt of time to bring about te ig gas-pressure necessary to overcome te counter-pressure of te overlying rock. A small leakage as little influence upon te increasing gaspressure. Several volcanoes are known in wic a caldera formation must ave taken place more tan once, fe. g. Krakat.oa, Batoer and Idjen. Te raritywit wic caldera forming eruptions ave taken place in istorical time is in itself owever a proof tat te period of rest between two paroxysms of te first order must e very great. IV. SYNOPSIS OF THE THEOY OF THE FOMATION OF CALDEAS. Hypotetical premises. 1. It is possible for Pkrret's gas pase to be more violent tan in te eruption of Vesuvius in (Perret type). 2. Te primary result of te gas pase is te formation of a cored-out cylinder and secondarily te collapse of te cylinder along a funnelsaped sliding plane, formes a caldera. Conclusions. 1. For tis caldera formation a large quantity of gas under ig pressure is necessary. 2. A matematical treatment of te problem leads to te conclusion tat te dept of te camber magma must be great (probably of te order of km).. Only violent eruptions (paroxysms of te first order) can form a large cored-out cylinder. Te degree of force of a volcanic eruption is a function of te dept and te volume of te magma camber. 4. For collapses along sliding planes a wide eruption cylinder is necessary probably of te order of m).

37 Te duration of a paroxysm of te first order (destructive activity) is very sort (some days) as compared to te period of quiescence before te next paroxysm of te first order (some centuries). 6. During tese centuries te gas pressure must rise to equal te weigt of te overlying rocks. 7. During te quiescent period secondary volcanoes may be formed by leakage of gas (constructive activity). BIBLIOGAPHY. 1. B. G. Escer, Vesuvius, te Tenggcr Mountains and te Problem of Caldera 's. Leidsce Geolog. Mededeolingen, II, pp , (Wit topograpical map of te Tengger-caldera 1: ). 2. B. G. Escer, Krakatau in 188 en in Tijdscr. Kon. Nederlandsc Aardrijkskundig Gen., 2e Ser., dl. LV, pp Frank A. Perret, Te Vesuvius Eruption of Study of a Volcanic Cycle. Carnegie Institution of Wasington, Publication No. 9, Juli A. Malladra, Sul graduale riempimento del cratere del Vesuvio. Communicazione tenuta all' VIII Congresso Geografico Italiano, Vol. II, degl "Atti", deren Einfluss 5. A. H. Goldkeicii, Die Bodenbewegungen im Kolenrevier und auf die Tagesoberfläcc. Berlin, b". G. L. L. Kemmerlino, De Vulkanen Goenoeng Batoer en Goenoeng Agoeng op Bali. Jaarboek v.. Mijnwezen in Ned. Oost-Indië. 48e Jaarg., 1927, Verandelingen Ie Gedeelte, Batavia 1918, pp (Wit Atlas). 7. C. E. Steun, De Batoer op Bali en zijn eruptie in Vulkanologisce en Seismologisee Mededeelingen, No. 9. Dienst v. d. Mijnbouw in Ned. Indië, Bandoeng (Wit map of te Batoer-caldera 1: ). 8. G. L. L. Kemmerlino, Het Idjen-Hoogland, Monografie II. De Geologie en Geomorpologie van den Idjen. Batavia (Witout date). (Wit geological map 1: of te Idjen-caldera). Topografisce kaart 1: Idjen-Hoogland. Toopgrafisce Inricting, Batavia B. D. M. Verbeek, Krakatau, Batavia 1885.

### INTRODUCTION AND MATHEMATICAL CONCEPTS

INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,

### Section 2: The Derivative Definition of the Derivative

Capter 2 Te Derivative Applied Calculus 80 Section 2: Te Derivative Definition of te Derivative Suppose we drop a tomato from te top of a 00 foot building and time its fall. Time (sec) Heigt (ft) 0.0 00

### Combining functions: algebraic methods

Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

### A Reconsideration of Matter Waves

A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,

### Continuity and Differentiability Worksheet

Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;

### SECTION 1.10: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES

(Section.0: Difference Quotients).0. SECTION.0: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES Define average rate of cange (and average velocity) algebraically and grapically. Be able to identify, construct,

### The Basics of Vacuum Technology

Te Basics of Vacuum Tecnology Grolik Benno, Kopp Joacim January 2, 2003 Basics Many scientific and industrial processes are so sensitive tat is is necessary to omit te disturbing influence of air. For

### Derivation Of The Schwarzschild Radius Without General Relativity

Derivation Of Te Scwarzscild Radius Witout General Relativity In tis paper I present an alternative metod of deriving te Scwarzscild radius of a black ole. Te metod uses tree of te Planck units formulas:

### The Laws of Thermodynamics

1 Te Laws of Termodynamics CLICKER QUESTIONS Question J.01 Description: Relating termodynamic processes to PV curves: isobar. Question A quantity of ideal gas undergoes a termodynamic process. Wic curve

### Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.

Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions

### Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006

Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2

### Section 15.6 Directional Derivatives and the Gradient Vector

Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,

### THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:

### 2.3. Applying Newton s Laws of Motion. Objects in Equilibrium

Appling Newton s Laws of Motion As ou read in Section 2.2, Newton s laws of motion describe ow objects move as a result of different forces. In tis section, ou will appl Newton s laws to objects subjected

### Copyright c 2008 Kevin Long

Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula

### Differential Calculus (The basics) Prepared by Mr. C. Hull

Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit

### Work and Energy. Introduction. Work. PHY energy - J. Hedberg

Work and Energy PHY 207 - energy - J. Hedberg - 2017 1. Introduction 2. Work 3. Kinetic Energy 4. Potential Energy 5. Conservation of Mecanical Energy 6. Ex: Te Loop te Loop 7. Conservative and Non-conservative

### LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION

LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y

### Section 3: The Derivative Definition of the Derivative

Capter 2 Te Derivative Business Calculus 85 Section 3: Te Derivative Definition of te Derivative Returning to te tangent slope problem from te first section, let's look at te problem of finding te slope

### 1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.

Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of

### Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative

Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x

### Chapter 2 Ising Model for Ferromagnetism

Capter Ising Model for Ferromagnetism Abstract Tis capter presents te Ising model for ferromagnetism, wic is a standard simple model of a pase transition. Using te approximation of mean-field teory, te

### Part 2: Introduction to Open-Channel Flow SPRING 2005

Part : Introduction to Open-Cannel Flow SPRING 005. Te Froude number. Total ead and specific energy 3. Hydraulic jump. Te Froude Number Te main caracteristics of flows in open cannels are tat: tere is

### 3.1 Extreme Values of a Function

.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find

### The Krewe of Caesar Problem. David Gurney. Southeastern Louisiana University. SLU 10541, 500 Western Avenue. Hammond, LA

Te Krewe of Caesar Problem David Gurney Souteastern Louisiana University SLU 10541, 500 Western Avenue Hammond, LA 7040 June 19, 00 Krewe of Caesar 1 ABSTRACT Tis paper provides an alternative to te usual

### CHAPTER (A) When x = 2, y = 6, so f( 2) = 6. (B) When y = 4, x can equal 6, 2, or 4.

SECTION 3-1 101 CHAPTER 3 Section 3-1 1. No. A correspondence between two sets is a function only if eactly one element of te second set corresponds to eac element of te first set. 3. Te domain of a function

### Numerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1

Numerical Analysis MTH60 PREDICTOR CORRECTOR METHOD Te metods presented so far are called single-step metods, were we ave seen tat te computation of y at t n+ tat is y n+ requires te knowledge of y n only.

### Calculus I Homework: The Derivative as a Function Page 1

Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap.

### E p = mgh (if h i=0) E k = ½ mv 2 Ek is measured in Joules (J); m is measured in kg; v is measured in m/s. Energy Continued (E)

nergy Continued () Gravitational Potential nergy: - e energy stored in an object due to its distance above te surface of te art. - e energy stored depends on te mass of te object, te eigt above te surface,

### Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

### CHAPTER 3: Derivatives

CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Cange 3.2: Derivative Functions and Differentiability 3.3: Tecniques of Differentiation 3.4: Derivatives of Trigonometric Functions

### MTH-112 Quiz 1 Name: # :

MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation

### Problem Set 7: Potential Energy and Conservation of Energy AP Physics C Supplementary Problems

Proble Set 7: Potential Energy and Conservation of Energy AP Pysics C Suppleentary Probles 1. Approxiately 5.5 x 10 6 kg of water drops 50 over Niagara Falls every second. (a) Calculate te aount of potential

### A general articulation angle stability model for non-slewing articulated mobile cranes on slopes *

tecnical note 3 general articulation angle stability model for non-slewing articulated mobile cranes on slopes * J Wu, L uzzomi and M Hodkiewicz Scool of Mecanical and Cemical Engineering, University of

### 11-19 PROGRESSION. A level Mathematics. Pure Mathematics

SSaa m m pplle e UCa ni p t ter DD iff if erfe enren tiatia tiotio nn - 9 RGRSSIN decel Slevel andmatematics level Matematics ure Matematics NW FR 07 Year/S Year decel S and level Matematics Sample material

### AN IMPROVED WEIGHTED TOTAL HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES

AN IMPROVED WEIGHTED TOTA HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES Tomas A. IPO University of Wisconsin, 45 Engineering Drive, Madison WI, USA P: -(608)-6-087, Fax: -(608)-6-5559, lipo@engr.wisc.edu

### Mathematics 123.3: Solutions to Lab Assignment #5

Matematics 3.3: Solutions to Lab Assignment #5 Find te derivative of te given function using te definition of derivative. State te domain of te function and te domain of its derivative..: f(x) 6 x Solution:

### Reflection Symmetries of q-bernoulli Polynomials

Journal of Nonlinear Matematical Pysics Volume 1, Supplement 1 005, 41 4 Birtday Issue Reflection Symmetries of q-bernoulli Polynomials Boris A KUPERSHMIDT Te University of Tennessee Space Institute Tullaoma,

### Lesson 4 - Limits & Instantaneous Rates of Change

Lesson Objectives Lesson 4 - Limits & Instantaneous Rates of Cange SL Topic 6 Calculus - Santowski 1. Calculate an instantaneous rate of cange using difference quotients and limits. Calculate instantaneous

### Neutron transmission probability through a revolving slit for a continuous

Neutron transmission probability troug a revolving slit for a continuous source and a divergent neutron beam J. Peters*, Han-Meitner-Institut Berlin, Glienicer Str. 00, D 409 Berlin Abstract: Here an analytical

### Derivatives of trigonometric functions

Derivatives of trigonometric functions 2 October 207 Introuction Toay we will ten iscuss te erivates of te si stanar trigonometric functions. Of tese, te most important are sine an cosine; te erivatives

### MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

### Extracting Atomic and Molecular Parameters From the de Broglie Bohr Model of the Atom

Extracting Atomic and Molecular Parameters From te de Broglie Bor Model of te Atom Frank ioux Te 93 Bor model of te ydrogen atom was replaced by Scrödingerʹs wave mecanical model in 96. However, Borʹs

### Derivative as Instantaneous Rate of Change

43 Derivative as Instantaneous Rate of Cange Consider a function tat describes te position of a racecar moving in a straigt line away from some starting point Let y s t suc tat t represents te time in

### 2.3 Product and Quotient Rules

.3. PRODUCT AND QUOTIENT RULES 75.3 Product and Quotient Rules.3.1 Product rule Suppose tat f and g are two di erentiable functions. Ten ( g (x)) 0 = f 0 (x) g (x) + g 0 (x) See.3.5 on page 77 for a proof.

### Homework 1. Problem 1 Browse the 331 website to answer: When you should use data symbols on a graph. (Hint check out lab reports...

Homework 1 Problem 1 Browse te 331 website to answer: Wen you sould use data symbols on a grap. (Hint ceck out lab reports...) Solution 1 Use data symbols to sow data points unless tere is so muc data

### nucleus orbital electron wave 2/27/2008 Quantum ( F.Robilliard) 1

r nucleus orbital electron wave λ /7/008 Quantum ( F.Robilliard) 1 Wat is a Quantum? A quantum is a discrete amount of some quantity. For example, an atom is a mass quantum of a cemical element te mass

### June : 2016 (CBCS) Body. Load

Engineering Mecanics st Semester : Common to all rances Note : Max. marks : 6 (i) ttempt an five questions (ii) ll questions carr equal marks. (iii) nswer sould be precise and to te point onl (iv) ssume

### Solution for the Homework 4

Solution for te Homework 4 Problem 6.5: In tis section we computed te single-particle translational partition function, tr, by summing over all definite-energy wavefunctions. An alternative approac, owever,

### Problem Set 4: Whither, thou turbid wave SOLUTIONS

PH 253 / LeClair Spring 2013 Problem Set 4: Witer, tou turbid wave SOLUTIONS Question zero is probably were te name of te problem set came from: Witer, tou turbid wave? It is from a Longfellow poem, Te

### Molecular symmetry. An introduction to symmetry analysis

Molecular symmetry 6 Symmetry governs te bonding and ence te pysical and spectroscopic properties of molecules In tis capter we explore some of te consequences of molecular symmetry and introduce te systematic

### Phase space in classical physics

Pase space in classical pysics Quantum mecanically, we can actually COU te number of microstates consistent wit a given macrostate, specified (for example) by te total energy. In general, eac microstate

### AMS 147 Computational Methods and Applications Lecture 09 Copyright by Hongyun Wang, UCSC. Exact value. Effect of round-off error.

Lecture 09 Copyrigt by Hongyun Wang, UCSC Recap: Te total error in numerical differentiation fl( f ( x + fl( f ( x E T ( = f ( x Numerical result from a computer Exact value = e + f x+ Discretization error

### The structure of the atoms

Te structure of te atoms Atomos = indivisible University of Pécs, Medical Scool, Dept. Biopysics All tat exists are atoms and empty space; everyting else is merely tougt to exist. Democritus, 415 B.C.

### Chapter Seven The Quantum Mechanical Simple Harmonic Oscillator

Capter Seven Te Quantum Mecanical Simple Harmonic Oscillator Introduction Te potential energy function for a classical, simple armonic oscillator is given by ZÐBÑ œ 5B were 5 is te spring constant. Suc

### A PHYSICAL MODEL STUDY OF SCOURING EFFECTS ON UPSTREAM/DOWNSTREAM OF THE BRIDGE

A PHYSICA MODE STUDY OF SCOURING EFFECTS ON UPSTREAM/DOWNSTREAM OF THE BRIDGE JIHN-SUNG AI Hydrotec Researc Institute, National Taiwan University Taipei, 1617, Taiwan HO-CHENG IEN National Center for Hig-Performance

### Velocity distribution in non-uniform/unsteady flows and the validity of log law

University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 3 Velocity distribution in non-uniform/unsteady

### Digital Filter Structures

Digital Filter Structures Te convolution sum description of an LTI discrete-time system can, in principle, be used to implement te system For an IIR finite-dimensional system tis approac is not practical

### Order of Accuracy. ũ h u Ch p, (1)

Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical

### THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA EXAMINATION MODULE 5

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA EXAMINATION NEW MODULAR SCHEME introduced from te examinations in 009 MODULE 5 SOLUTIONS FOR SPECIMEN PAPER B THE QUESTIONS ARE CONTAINED IN A SEPARATE FILE

### Chemistry. Slide 1 / 63 Slide 2 / 63. Slide 4 / 63. Slide 3 / 63. Slide 6 / 63. Slide 5 / 63. Optional Review Light and Matter.

Slide 1 / 63 Slide 2 / 63 emistry Optional Review Ligt and Matter 2015-10-27 www.njctl.org Slide 3 / 63 Slide 4 / 63 Ligt and Sound Ligt and Sound In 1905 Einstein derived an equation relating mass and

### 1 Solutions to the in class part

NAME: Solutions to te in class part. Te grap of a function f is given. Calculus wit Analytic Geometry I Exam, Friday, August 30, 0 SOLUTIONS (a) State te value of f(). (b) Estimate te value of f( ). (c)

### Diffraction. S.M.Lea. Fall 1998

Diffraction.M.Lea Fall 1998 Diffraction occurs wen EM waves approac an aperture (or an obstacle) wit dimension d > λ. We sall refer to te region containing te source of te waves as region I and te region

### FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukherji WALRASIAN AND NON-WALRASIAN MICROECONOMICS

FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji WALRASIAN AND NON-WALRASIAN MICROECONOMICS Anjan Mukerji Center for Economic Studies and Planning, Jawaarlal Neru

### y = x 5 ( ) + 22 at the rate of 3 units per second.

C 4-6,8 Review Pre-Calculus Name Period THEME 1 Parametric Equations Refreser notes: 1) Maggie moves at constant speed of 6 m/s. Se starts at te point (12,3) and eads towards te y-axis along te line y

### Exam 1 Solutions. x(x 2) (x + 1)(x 2) = x

Eam Solutions Question (0%) Consider f() = 2 2 2 2. (a) By calculating relevant its, determine te equations of all vertical asymptotes of te grap of f(). If tere are none, say so. f() = ( 2) ( + )( 2)

### Forces, centre of gravity, reactions and stability

Forces, centre of gravity, reactions and stability Topic areas Mecanical engineering: Centre of gravity Forces Moments Reactions Resolving forces on an inclined plane. Matematics: Angles Trigonometric

### Chapter 4: Numerical Methods for Common Mathematical Problems

1 Capter 4: Numerical Metods for Common Matematical Problems Interpolation Problem: Suppose we ave data defined at a discrete set of points (x i, y i ), i = 0, 1,..., N. Often it is useful to ave a smoot

### A Numerical Scheme for Particle-Laden Thin Film Flow in Two Dimensions

A Numerical Sceme for Particle-Laden Tin Film Flow in Two Dimensions Mattew R. Mata a,, Andrea L. Bertozzi a a Department of Matematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles,

### Investigation of Tangent Polynomials with a Computer Algebra System The AMATYC Review, Vol. 14, No. 1, Fall 1992, pp

Investigation of Tangent Polynomials wit a Computer Algebra System Te AMATYC Review, Vol. 14, No. 1, Fall 199, pp. -7. Jon H. Matews California State University Fullerton Fullerton, CA 9834 By Russell

### Quantum Numbers and Rules

OpenStax-CNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.

### Stepped-Impedance Low-Pass Filters

4/23/27 Stepped Impedance Low Pass Filters 1/14 Stepped-Impedance Low-Pass Filters Say we know te impedance matrix of a symmetric two-port device: 11 21 = 21 11 Regardless of te construction of tis two

### Nonlinear correction to the bending stiffness of a damaged composite beam

Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Nonlinear correction to te bending stiffness of a damaged composite beam W.

### MATH 111 CHAPTER 2 (sec )

MATH CHAPTER (sec -0) Terms to know: function, te domain and range of te function, vertical line test, even and odd functions, rational power function, vertical and orizontal sifts of a function, reflection

### Flavius Guiaş. X(t + h) = X(t) + F (X(s)) ds.

Numerical solvers for large systems of ordinary differential equations based on te stocastic direct simulation metod improved by te and Runge Kutta principles Flavius Guiaş Abstract We present a numerical

### The Electron in a Potential

Te Electron in a Potential Edwin F. Taylor July, 2000 1. Stopwatc rotation for an electron in a potential For a poton we found tat te and of te quantum stopwatc rotates wit frequency f given by te equation:

### An Elementary Proof of a Generalization of Bernoulli s Formula

Revision: August 7, 009 An Elementary Proof of a Generalization of Bernoulli s Formula Kevin J. McGown Harold R. Pars Department of Matematics Department of Matematics University of California at San Diego

### WJEC FP1. Identities Complex Numbers, Polynomials, Differentiation. Identities. About the Further Mathematics. Ideas came from.

Furter Matematics Support Programme bout te Furter Matematics Support Programme Wales Wales WJEC FP Sofya Lyakova FMSP Wales Te Furter Matematics Support Programme (FMSP) Wales started in July 00 and follows

### Chapter 1D - Rational Expressions

- Capter 1D Capter 1D - Rational Expressions Definition of a Rational Expression A rational expression is te quotient of two polynomials. (Recall: A function px is a polynomial in x of degree n, if tere

### arxiv: v1 [physics.flu-dyn] 3 Jun 2015

A Convective-like Energy-Stable Open Boundary Condition for Simulations of Incompressible Flows arxiv:156.132v1 [pysics.flu-dyn] 3 Jun 215 S. Dong Center for Computational & Applied Matematics Department

### Example: f(x) = x 3. 1, x > 0 0, x 0. Example: g(x) =

2.1 Instantaneous rate of cange, or, an informal introduction to derivatives Let a, b be two different values in te domain of f. Te average rate of cange of f between a and b is f(b) f(a) b a. Geometrically,

Experimental procedure and uncertainty analysis of a guarded otbox metod to determine te termal transmission coefficient of skyligts and sloped glazing Elmady, A.H.; Haddad, K. NRCC-43378 A version of

### Functions of the Complex Variable z

Capter 2 Functions of te Complex Variable z Introduction We wis to examine te notion of a function of z were z is a complex variable. To be sure, a complex variable can be viewed as noting but a pair of

### Key Concepts. Important Techniques. 1. Average rate of change slope of a secant line. You will need two points ( a, the formula: to find value

AB Calculus Unit Review Key Concepts Average and Instantaneous Speed Definition of Limit Properties of Limits One-sided and Two-sided Limits Sandwic Teorem Limits as x ± End Beaviour Models Continuity

### M12/4/PHYSI/HPM/ENG/TZ1/XX. Physics Higher level Paper 1. Thursday 10 May 2012 (afternoon) 1 hour INSTRUCTIONS TO CANDIDATES

M12/4/PHYSI/HPM/ENG/TZ1/XX 22126507 Pysics Higer level Paper 1 Tursday 10 May 2012 (afternoon) 1 our INSTRUCTIONS TO CANDIDATES Do not open tis examination paper until instructed to do so. Answer all te

### the first derivative with respect to time is obtained by carefully applying the chain rule ( surf init ) T Tinit

.005 ermal Fluids Engineering I Fall`08 roblem Set 8 Solutions roblem ( ( a e -D eat equation is α t x d erfc( u du π x, 4αt te first derivative wit respect to time is obtained by carefully applying te

### Part C : Quantum Physics

Part C : Quantum Pysics 1 Particle-wave duality 1.1 Te Bor model for te atom We begin our discussion of quantum pysics by discussing an early idea for atomic structure, te Bor model. Wile tis relies on

### Chapter 3 Thermoelectric Coolers

3- Capter 3 ermoelectric Coolers Contents Capter 3 ermoelectric Coolers... 3- Contents... 3-3. deal Equations... 3-3. Maximum Parameters... 3-7 3.3 Normalized Parameters... 3-8 Example 3. ermoelectric

### RECOGNITION of online handwriting aims at finding the

SUBMITTED ON SEPTEMBER 2017 1 A General Framework for te Recognition of Online Handwritten Grapics Frank Julca-Aguilar, Harold Moucère, Cristian Viard-Gaudin, and Nina S. T. Hirata arxiv:1709.06389v1 [cs.cv]

### Dedicated to the 70th birthday of Professor Lin Qun

Journal of Computational Matematics, Vol.4, No.3, 6, 4 44. ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS Guang-wei Yuan Xu-deng Hang Laboratory of Computational Pysics,

### Parameter Fitted Scheme for Singularly Perturbed Delay Differential Equations

International Journal of Applied Science and Engineering 2013. 11, 4: 361-373 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Awoke Andargiea* and Y. N. Reddyb a b Department

### On the drag-out problem in liquid film theory

J. Fluid Mec. (28), vol. 617, pp. 28 299. c 28 Cambridge University Press doi:1.117/s22112841x Printed in te United Kingdom 28 On te drag-out problem in liquid film teory E. S. BENILOV AND V. S. ZUBKOV

### Scaling Network Verification using Symmetry and Surgery

Scaling Network Verification using Symmetry and Surgery Gordon D. Plotkin Nikolaj Bjørner Nuno P. Lopes Andrey Rybalcenko George Vargese LFCS, University of Edinburg, UK Microsoft Researc, UK Microsoft

### Technology-Independent Design of Neurocomputers: The Universal Field Computer 1

Tecnology-Independent Design of Neurocomputers: Te Universal Field Computer 1 Abstract Bruce J. MacLennan Computer Science Department Naval Postgraduate Scool Monterey, CA 9393 We argue tat AI is moving

### The Cardinality of Sumsets: Different Summands

Te Cardinality of Sumsets: Different Summands Brendan Murpy, Eyvindur Ari Palsson and Giorgis Petridis arxiv:1309.2191v3 [mat.co] 17 Feb 2017 Abstract We offer a compete answer to te following question

### 7 Semiparametric Methods and Partially Linear Regression

7 Semiparametric Metods and Partially Linear Regression 7. Overview A model is called semiparametric if it is described by and were is nite-dimensional (e.g. parametric) and is in nite-dimensional (nonparametric).

### Kernel Density Estimation

Kernel Density Estimation Univariate Density Estimation Suppose tat we ave a random sample of data X 1,..., X n from an unknown continuous distribution wit probability density function (pdf) f(x) and cumulative

### Solve exponential equations in one variable using a variety of strategies. LEARN ABOUT the Math. What is the half-life of radon?

8.5 Solving Exponential Equations GOAL Solve exponential equations in one variable using a variety of strategies. LEARN ABOUT te Mat All radioactive substances decrease in mass over time. Jamie works in