Heat Conduction vs. One-Dimensional Wave Equation Conduction and...Microsoft Word - Heat Conduction...
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Heat Conduction and One-Dimensional Wave Equations β !! = ! vs. Ξ± !! = !! Heat Conduction: β ! !! = ! Boundary conditions: (0, ) = 0, (, ) = 0 Case: Bar with both ends kept at 0 degree General Solution: , = ! ! !!! !β ! ! ! ! ! !/! ! !"# ! Steady State Solution: () = 0 Other info: ! = ! = 2 ! ! Heat Conduction: β ! !! = ! Boundary conditions: ! (0, ) = 0, ! (, ) = 0 Case: Bar with both ends perfectly insulated General Solution: , = ! + ! ! !!! !β ! ! ! ! ! !/! ! !"# ! Steady State Solution: () = ! Other info: ! = ! ! ! ! = ! = 2 ! !
Transcript of Heat Conduction vs. One-Dimensional Wave Equation Conduction and...Microsoft Word - Heat Conduction...
Heat Conduction and One-Dimensional Wave Equations βπ π’!! = π’! vs. Ξ±ππ’!! = π’!!
Heat Conduction: β! π’!! = π’!
Boundary conditions: π’(0, π‘) = 0,π’(πΏ, π‘) = 0
Case: Bar with both ends kept at 0 degree General Solution: π’ π₯, π‘ = πΆ!!
!!! π!β!!!!!!/!!π ππ !"#
!
Steady State Solution: π£(π₯) = 0 Other info:
πΆ! = π! =2πΏ
π π₯ π πππππ₯πΏππ₯
!
!
Heat Conduction: β! π’!! = π’!
Boundary conditions: π’!(0, π‘) = 0,π’!(πΏ, π‘) = 0
Case: Bar with both ends perfectly insulated General Solution: π’ π₯, π‘ = πΆ! + πΆ!!
!!! π!β!!!!!!/!!πππ !"#
!
Steady State Solution: π£(π₯) = πΆ! Other info: πΆ! =
!!!
πΆ! = π! =2πΏ
π π₯ πππ πππ₯πΏππ₯
!
!
Heat Conduction: β! π’!! = π’!
Boundary conditions: π’ 0, π‘ = π!,π’ πΏ, π‘ = π!
Case: Bar with π! degrees at the left end, and π!degrees at the right end General Solution: π’ π₯, π‘ = !!!!!
!π₯ + π! + πΆ!!
!!! π!β!!!!!!/!!π ππ !"#
!
Steady State Solution: π£ π₯ = !!!!!
!π₯ + π!
Other info: π£(π₯) = π΄π₯ + π΅ , and π€ π₯, 0 = π π₯ β π£(π₯)
πΆ! = π! =2πΏ
(π π₯ β π£ π₯ )π πππππ₯πΏππ₯
!
!
One-Dimensional Wave Equations: Ξ±!π’!! = π’!!
Boundary conditions: π’ 0, π‘ = 0,π’ πΏ, π‘ = 0
Initial conditions: π’(π₯, 0) = π(π₯),π’!(π₯, 0) = π(π₯)
Case: Undamped One-dimensional Wave Equation General Solution: π’ π₯, π‘ = (π΄!!
!!! πππ !"#$!+ π΅!π ππ
!"#$!)π ππ !"#
!
Other info: ** See Special Cases Below **
π΄! = π! =2πΏ
π π₯ π πππππ₯πΏππ₯
!
!
π΅! =πΏπππ
π! =2πππ
π π₯ π πππππ₯πΏππ₯
!
!
One-Dimensional Wave Equations: Ξ±!π’!! = π’!!
Boundary conditions: π’ 0, π‘ = 0,π’ πΏ, π‘ = 0
Initial conditions: π’ π₯, 0 = 0 ,π’!(π₯, 0) = π(π₯)
Case: Special Case of Undamped One-dimensional Wave Equation ππππ‘πππ πππ πππππππππ‘ = 0 General Solution: π’ π₯, π‘ = (π΅!π ππ
!"#$!)π ππ !"#
!
Other info: π΄! = 0
π΅! =πΏπππ
π! =2πππ
π π₯ π πππππ₯πΏππ₯
!
!
One-Dimensional Wave Equations: Ξ±!π’!! = π’!!
Boundary conditions: π’ 0, π‘ = 0,π’ πΏ, π‘ = 0
Initial conditions: π’(π₯, 0) = π(π₯),π’!(π₯, 0) = 0
Case: Special Case of Undamped One-dimensional Wave Equation ππππ‘πππ π£ππππππ‘π¦ = 0 General Solution: π’ π₯, π‘ = (π΄!!
!!! πππ !"#$!)π ππ !"#
!
Other info: π΅! = 0
π΄! = π! =2πΏ
π π₯ π πππππ₯πΏππ₯
!
!
π΅! = 0
