water surface Profile methods

7/24/2019 water surface Profile methods

1/16

Cristobal, Emmanuel, D

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8/16

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14%7*+ $%+73+ 14%(* $%$$11 $%77+ !$%$$$#+( 1%$**$7+ 1%$3+*7 1(%+(+ 14%7# $%+7$*# 14%(#47 $%$$111 $%7+**7 !$%$$$3(

14%77( $%+7*3 14%(7 $%$$11 $%777 !$%$$$#*4 1%$*7$++7 1%$*$$33 1(%+4+ 14%74 $%+71* 14%(4$( $%$$111 $%7+3$( !$%$$$314%777 $%+77( 14%(7 $%$$11 $%77* !$%$$$#4+ 1%$*+$3$4 1%$*1#* 1(%+*7 14%73 $%+7(31 14%(434 $%$$11 $%7+#4* !$%$$$47


10/16

14%7+1 $%+77 14%(+ $%$$11 $%774 !$%$$$##( 1%$*+*77 1%$*(3 1(%++4 14%73 $%+7##+ 14%(3$4 $%$$11 $%7+1 !$%$$$43

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14%7( $%+7* 14%( $%$$11 $%77( !$%$$$(+ 1%$7147+# 1%$*3* 1(%#( 14%77 $%+7374 14%(*#+ $%$$1$ $%77+$( !$%$$$#

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14%+$( $%++11 14%( $%$$11 $%7* !$%$$$(3# 1%$7#*713 1%$*++*# 1(%74 14%7+ $%+7777 14%(733 $%$$1$+ $%774*1 !$%$$$##14%+$3 $%++1* 14%# $%$$11 $%7*+ !$%$$$(4( 1%$74#4$7 1%$*74# 1(%+* 14%7+ $%+7+# 14%(7 $%$$1$7 $%77#3+ !$%$$$#(

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14%+13 $%++## 14%#1 $%$$1 $%7*3 !$%$$$($4 1%$7*74+1 1%$7(++ 1#%$#1 14%+ $%++$3 14%(1* $%$$1$* $%7*( !$%$$$(7

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14%+(7 $%++3 14%#( $%$$1 $%7*# !$%$$$1*7 1%$7(4#7 1%$7*11* 1#%$77 14%+1 $%++(+3 14%#$43 $%$$1$3 $%7**1 !$%$$$(114%+( $%++3# 14%#( $%$$1 $%7*( !$%$$$1* 1%$7*+4* 1%$7**+# 1#%$+3 14%+( $%++#(4 14%#$*7 $%$$1$3 $%7*334 !$%$$$(1

14%+# $%++3* 14%#( $%$$1 $%7*( !$%$$$134 1%$+$1$+( 1%$77((7 1#%$# 14%+( $%++#*( 14%#$+ $%$$1$3 $%7*4( !$%$$$(

14%+#( $%++3 14%#( $%$$1 $%7*1 !$%$$$14+ 1%$+$313( 1%$7774 1#%1$1 14%+( $%++# 14%#11 $%$$1$4 $%7*4#( !$%$$$1

14%+#4 $%++*( 14%#( $%$$1 $%7*1 !$%$$$14( 1%$+$$*# 1%$7+(4 1#%1$+ 14%+( $%++4#4 14%#1# $%$$1$4 $%7*#73 !$%$$$1+

14%+#* $%++*3 14%## $%$$1 $%7* !$%$$$1#* 1%$+1(+(# 1%$7+7# 1#%113 14%+( $%++4*7 14%#14 $%$$1$4 $%7*#( !$%$$$17

14%+#7 $%++*7 14%## $%$$1 $%7* !$%$$$1#1 1%$+1*4# 1%$711 1#%1(1 14%+# $%++3 14%#1*+ $%$$1$4 $%7*(*7 !$%$$$17

14%+# $%++7 14%## $%$$1 $%73 !$%$$$1(* 1%$+113 1%$7*#4 1#%1(+ 14%+# $%++3#1 14%#1+3 $%$$1$4 $%7*(1* !$%$$$1*

14%+4 $%++7( 14%## $%$$1 $%73 !$%$$$1(1 1%$+(#(3 1%$+$$3 1#%1#4 14%+# $%++3* 14%#($( $%$$1$4 $%7*1*+ !$%$$$13

14%+4( $%++74 14%## $%$$1 $%73 !$%$$$117 1%$+(*47* 1%$+$4*+ 1#%14 14%+# $%++3+ 14%#(1 $%$$1$4 $%7*1(1 !$%$$$13

14%+4# $%++7* 14%## $%$$1 $%73+ !$%$$$11( 1%$+(37 1%$+$+*1 1#%143 14%+# $%++*1* 14%#(#4 $%$$1$# $%7*$77 !$%$$$14

14%+44 $%++7+ 14%## $%$$1 $%73+ !$%$$$1$+ 1%$+#(34 1%$+1(# 1#%131 14%+4 $%++*4# 14%#(3 $%$$1$# $%7*$#4 !$%$$$14

14%+4* $%+++ 14%## $%$$1 $%73+ !$%$$$1$4 1%$+#3413 1%$+1*$( 1#%13* 14%+4 $%++**+ 14%#(*4 $%$$1$# $%73( !$%$$$1#

14%+47 $%+++( 14%#4 $%$$1 $%737 !$%$$$1 1%$+#+174 1%$+131 1#%1*1 14%+4 $%++*( 14%#(7+ $%$$1$# $%733# !$%$$$1#

14%+4+ $%+++4 14%#4 $%$$1 $%737 !%*E!$$3 1%$+4$+#1 1%$+((+7 1#%1** 14%+4 $%++71* 14%#(1 $%$$1$# $%7313 !$%$$$1(

14%+4 $%+++* 14%#4 $%$$1 $%737 !%#E!$$3 1%$+4##+ 1%$+(*1 1#%17 14%+4 $%++7#+ 14%##$4 $%$$1$# $%73+7+ !$%$$$1(

14%+3 $%++++ 14%#4 $%$$1 $%737 !+%E!$$3 1%$+43+3( 1%$+((1 1#%173 14%+4 $%++7* 14%##17 $%$$1$# $%73+4# !$%$$$11

14%+31 $%+++ 14%#4 $%$$1 $%73* !+%*E!$$3 1%$+4+((4 1%$+#((1 1#%17 14%+4 $%++7+1 14%##( $%$$1$# $%73+$ !$%$$$11

14%+3# $%++1 14%#4 $%$$1 $%73* !+%#E!$$3 1%$+3$3$ 1%$+#3$+ 1#%1+# 14%+3 $%+++$1 14%##4 $%$$1$# $%73777 !$%$$$1

14%+3# $%++( 14%#4 $%$$1 $%73* !+%$E!$$3 1%$+3(71 1%$+#7+* 1#%1+7 14%+3 $%+++(1 14%##31 $%$$1$( $%73743 !$%$$$1

14%+34 $%++4 14%#4 $%$$1 $%73* !7%7E!$$3 1%$+34+# 1%$+4$3( 1#%11 14%+3 $%+++# 14%##*( $%$$1$( $%73713 !1E!$$4

14%+33 $%++3 14%#4 $%$$1 $%733 !7%4E!$$3 1%$+3*+74 1%$+4#$ 1#%13 14%+3 $%+++37 14%##7( $%$$1$( $%73*+* !E!$$314%+3* $%++7 14%#4 $%$$1 $%733 !7%(E!$$3 1%$+3++4# 1%$+4337 1#%1+ 14%+3 $%+++74 14%##+( $%$$1$( $%73*3+ !E!$$3


11/16

14%+37 $%+++ 14%#4 $%$$1 $%733 !*%E!$$3 1%$+*$74 1%$+473 1#%($( 14%+3 $%+++1 14%##( $%$$1$( $%73*#1 !E!$$3

14%+3+ $%++ 14%#3 $%$$1 $%733 !*%*E!$$3 1%$+*(3* 1%$+3$(4 1#%($3 14%+3 $%++$7 14%#4$1 $%$$1$( $%73*$* !+E!$$3

14%+3 $%+$1 14%#3 $%$$1 $%734 !*%4E!$$3 1%$+*4##( 1%$+3(4* 1#%($+ 14%+3 $%++(( 14%#41 $%$$1$( $%733+1 !+E!$$3

14%+3 $%+$( 14%#3 $%$$1 $%734 !*%(E!$$3 1%$+**$#1 1%$+343 1#%(11 14%+3 $%++#7 14%#41+ $%$$1$( $%73337 !+E!$$3

14%+* $%+$# 14%#3 $%$$1 $%734 !*%$E!$$3 1%$+*7** 1%$+3**4 1#%(14 14%+* $%++3( 14%#4(7 $%$$1$( $%733#4 !7E!$$3

14%+*1 $%+$4 14%#3 $%$$1 $%734 !3%7E!$$3 1%$+*(4 1%$+3+*1 1#%(17 14%+* $%++*3 14%#4#4 $%$$1$( $%73311 !7E!$$3

14%+*( $%+$3 14%#3 $%$$1 $%734 !3%3E!$$3 1%$+7$77( 1%$+*$3( 1#%(( 14%+* $%++7 14%#44( $%$$1$( $%734 !7E!$$314%+*( $%+$* 14%#3 $%$$1 $%734 !3%#E!$$3 1%$+7((4 1%$+*(#* 1#%((( 14%+* $%++1 14%#44 $%$$1$( $%734* !7E!$$3

14%+*# $%+$7 14%#3 $%$$1 $%73# !3%1E!$$3 1%$+7#*3* 1%$+*41# 1#%((3 14%+* $%+$$4 14%#437 $%$$1$( $%7344 !*E!$$3

14%+*# $%+$+ 14%#3 $%$$1 $%73# !3%$E!$$3 1%$+73$(( 1%$+*3+# 1#%((7 14%+* $%+$1* 14%#4*# $%$$1$( $%734# !*E!$$3

14%+*4 $%+$ 14%#3 $%$$1 $%73# !4%+E!$$3 1%$+7*## 1%$+*74+ 1#%(# 14%+* $%+$(7 14%#47 $%$$1$1 $%7341( !*E!$$3

14%+*3 $%+1 14%#3 $%$$1 $%73# !4%*E!$$3 1%$+77*1 1%$+*$7 1#%(#( 14%+* $%+$#+ 14%#47* $%$$1$1 $%73#4 !*E!$$3

14%+*3 $%+11 14%#3 $%$$1 $%73# !4%3E!$$3 1%$+7++#3 1%$+7$* 1#%(#4 14%+* $%+$4 14%#4+( $%$$1$1 $%73#77 !*E!$$3

14%+** $%+11 14%#3 $%$$1 $%73# !4%#E!$$3 1%$++$$1+ 1%$+7($7 1#%(#* 14%+* $%+$3 14%#4++ $%$$1$1 $%73#* !3E!$$3

14%+** $%+1( 14%#3 $%$$1 $%73# !4%1E!$$3 1%$++113+ 1%$+7#4 1#%(#+ 14%+* $%+$* 14%#44 $%$$1$1 $%73#44 !3E!$$3

14%+*7 $%+1# 14%#3 $%$$1 $%73( !4%$E!$$3 1%$++((3+ 1%$+74+* 1#%(4 14%+* $%+$7 14%#4 $%$$1$1 $%73#( !3E!$$3

14%+*7 $%+14 14%#3 $%$$1 $%73( !#%E!$$3 1%$++##( 1%$+7*1 1#%(4( 14%+* $%+$++ 14%#3$3 $%$$1$1 $%73#14 !3E!$$314%+*+ $%+14 14%#3 $%$$1 $%73( !#%7E!$$3 1%$++4#44 1%$+774* 1#%(44 14%+* $%+$7 14%#31 $%$$1$1 $%73# !3E!$$3

14%+*+ $%+13 14%#3 $%$$1 $%73( !#%*E!$$3 1%$++3##( 1%$+7+* 1#%(4* 14%+7 $%+1$3 14%#313 $%$$1$1 $%73(+* !4E!$$3

14%+* $%+1* 14%#3 $%$$1 $%73( !#%3E!$$3 1%$++*(+* 1%$+7++ 1#%(47 14%+7 $%+114 14%#3( $%$$1$1 $%73(7# !4E!$$3

14%+* $%+1* 14%#3 $%$$1 $%73( !#%#E!$$3 1%$++7($* 1%$++1$# 1#%(4 14%+7 $%+1(( 14%#3(4 $%$$1$1 $%73(* !4E!$$3

14%+* $%+17 14%#* $%$$1 $%73( !#%(E!$$3 1%$+++$4 1%$++(1# 1#%(31 14%+7 $%+1( 14%#3( $%$$1$1 $%73(4+ !4E!$$3

14%+7 $%+1+ 14%#* $%$$1 $%73( !#%1E!$$3 1%$+++31 1%$++#( 1#%(3( 14%+7 $%+1#7 14%#3## $%$$1$1 $%73(#* !4E!$$3

14%+7 $%+1+ 14%#* $%$$1 $%73( !#%$E!$$3 1%$++77+ 1%$++4(# 1#%(34 14%+7 $%+144 14%#3#7 $%$$1$1 $%73((4 !4E!$$3

14%+7 $%+1 14%#* $%$$1 $%73( !(%E!$$3 1%$+$37* 1%$++3(( 1#%(33 14%+7 $%+131 14%#341 $%$$1$1 $%73(1# !4E!$$3

14%+71 $%+1 14%#* $%$$1 $%731 !(%+E!$$3 1%$+1#47 1%$++*1+ 1#%(3* 14%+7 $%+13+ 14%#343 $%$$1$1 $%73($# !#E!$$3

14%+71 $%+( 14%#* $%$$1 $%731 !(%7E!$$3 1%$+($1 1%$++71 1#%(3+ 14%+7 $%+1*4 14%#34+ $%$$1$1 $%731( !#E!$$3

14%+71 $%+( 14%#* $%$$1 $%731 !(%*E!$$3 1%$+(+$+ 1%$++7 1#%(3 14%+7 $%+17 14%#33( $%$$1$1 $%731+( !#E!$$3

14%+7( $%+(1 14%#* $%$$1 $%731 !(%3E!$$3 1%$+#3$1 1%$++++3 1#%(* 14%+7 $%+17* 14%#333 $%$$1$1 $%7317# !#E!$$3

14%+7( $%+(1 14%#* $%$$1 $%731 !(%4E!$$3 1%$+417 1%$++*+ 1#%(*1 14%+7 $%+1+( 14%#33 $%$$1$1 $%731*# !#E!$$3

14%+7( $%+(( 14%#* $%$$1 $%731 !(%#E!$$3 1%$+4+1* 1%$+$4+ 1#%(*# 14%+7 $%+1++ 14%#3*( $%$$1$1 $%73133 !#E!$$3

14%+7# $%+(( 14%#* $%$$1 $%731 !(%#E!$$3 1%$+34# 1%$+1(* 1#%(*4 14%+7 $%+1# 14%#3*3 $%$$1$1 $%7314* !#E!$$3

14%+7# $%+(# 14%#* $%$$1 $%731 !(%(E!$$3 1%$+*$41 1%$+( 1#%(*3 14%+7 $%+1+ 14%#3*+ $%$$1$1 $%731#+ !#E!$$3

14%+7# $%+(# 14%#* $%$$1 $%731 !(%1E!$$3 1%$+**(( 1%$+(7# 1#%(** 14%+7 $%+($# 14%#371 $%$$1$1 $%731# !#E!$$3

14%+7# $%+(# 14%#* $%$$1 $%731 !(%$E!$$3 1%$+71+( 1%$+#4( 1#%(*7 14%+7 $%+($+ 14%#374 $%$$1$1 $%731(( !#E!$$3

14%+74 $%+(4 14%#* $%$$1 $%731 !(%$E!$$3 1%$+77(4 1%$+4$ 1#%(*+ 14%+7 $%+(1# 14%#37* $%$$1$1 $%73114 !(E!$$3

14%+74 $%+(4 14%#* $%$$1 $%731 !1%E!$$3 1%$++(47 1%$+474 1#%(* 14%+7 $%+(17 14%#37 $%$$1$1 $%731$7 !(E!$$3

14%+74 $%+(3 14%#* $%$$1 $%731 !1%+E!$$3 1%$++731 1%$+3#7 1#%(7 14%+7 $%+((( 14%#3+1 $%$$1$1 $%731 !(E!$$314%+74 $%+(3 14%#* $%$$1 $%731 !1%+E!$$3 1%$+(# 1%$+37 1#%(7 14%+7 $%+((* 14%#3+4 $%$$1$1 $%73$# !(E!$$3


12/16

14%+73 $%+(3 14%#* $%$$1 $%731 !1%7E!$$3 1%$+7$ 1%$+*33 1#%(71 14%+7 $%+(# 14%#3+* $%$$1$1 $%73$+7 !(E!$$3

14%+73 $%+(* 14%#* $%$$1 $%73 !1%7E!$$3 1%$$$1*4 1%$+71( 1#%(7( 14%+7 $%+(#4 14%#3++ $%$$1$1 $%73$+1 !(E!$$3

14%+73 $%+(* 14%#* $%$$1 $%73 !1%*E!$$3 1%$$$*$( 1%$+7** 1#%(7# 14%+7 $%+(#+ 14%#31 $%$$1 $%73$73 !(E!$$3

14%+73 $%+(* 14%#* $%$$1 $%73 !1%3E!$$3 1%$$1$(* 1%$++1 1#%(74 14%+7 $%+(41 14%#3# $%$$1 $%73$* !(E!$$3


13/16

MOD;;ED EULER'S METHOD

-5i6$%3 -5i61

9: -5i6$%3 A . R T S/ r2( 5&i6$%3'

!(%7 $%*1( 11%437 14%434+ $%+(*4 1#%+4 $%$$1#7 $%+*77# !$%$$(+# $%(7*4(1

!(%7 $%*31* 11%3147 14%#1*# $%+$4# 1#%+*1 $%$$13# $%1(+( !$%$$*1(1 $%4417$1#

!(%7 $%*$4( 11%44 14%(31 $%+$$ 1#%+4( $%$$13* $%($$* !$%$$7$$7 $%*#$++*

!(%7 $%71+3 11%*$73 14%#4*( $%+$1 1#%++7 $%$$13 $%$(7+ !$%$$3113 $%7*$$74

!(%7 $%+(711%7*(1 14%#* $%+17 1#%#( $%$$144 $%++*44 !$%$$#+* $%+7#4*3

!(%7 $%($3 11%++++ 14%4#** $%+(#3 1#%*+ $%$$1# $%+7#43 !$%$$#114 $%3734+4

!(%7 $%*1 11%43 14%47$4 $%+(+ 1#%+ $%$$1#* $%+*(+* !$%$$(*$ 1%$$(7+**!(%7 1%$$*$7 1(%$+3 14%4# $%+##3 14%$(4 $%$$1## $%+3#7 !$%$$((44 1%$$++37*3

!(%7 1%$117 1(%1*4 14%3(44 $%+#73 14%$47 $%$$1# $%+4*#3 !$%$$1*7 1%$141*7*(

!(%7 1%$1**7 1(%(#4 14%34*7 $%+41 14%$*7 $%$$1(+ $%+#* !$%$$174+ 1%$1+++3*

!(%7 1%$(11( 1(%(** 14%3*** $%+44( 14%$+4 $%$$1(* $%+##+1 !$%$$13* 1%$(#1(#4

!(%7 1%$(314 1(%#3#( 14%3+4* $%+47 14%1$1 $%$$1(4 $%+(+3* !$%$$14(( 1%$(**133

!(%7 1%$(+7 1(%4$4+ 14%*$$ $%+4* 14%113 $%$$1(# $%+(#+( !$%$$1(* 1%$#$4*1+4

!(%7 1%$#(14 1(%43( 14%*13 $%+3( 14%1( $%$$1(1 $%+13# !$%$$11+ 1%$##*7(#*

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