Heat Conduction vs. One-Dimensional Wave Equation Conduction and … · Heat Conduction and...
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Heat Conduction and One-Dimensional Wave Equations ∝𝟐 𝑢!! = 𝑢! vs. α𝟐𝑢!! = 𝑢!!
Heat Conduction: ∝! 𝑢!! = 𝑢!
Boundary conditions: 𝑢(0, 𝑡) = 0,𝑢(𝐿, 𝑡) = 0
Case: Bar with both ends kept at 0 degree General Solution: 𝑢 𝑥, 𝑡 = 𝐶!!
!!! 𝑒!∝!!!!!!/!!𝑠𝑖𝑛 !"#
!
Steady State Solution: 𝑣(𝑥) = 0 Other info:
𝐶! = 𝑏! =2𝐿
𝑓 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥
!
!
Heat Conduction: ∝! 𝑢!! = 𝑢!
Boundary conditions: 𝑢!(0, 𝑡) = 0,𝑢!(𝐿, 𝑡) = 0
Case: Bar with both ends perfectly insulated General Solution: 𝑢 𝑥, 𝑡 = 𝐶! + 𝐶!!
!!! 𝑒!∝!!!!!!/!!𝑐𝑜𝑠 !"#
!
Steady State Solution: 𝑣(𝑥) = 𝐶! Other info: 𝐶! =
!!!
𝐶! = 𝑎! =2𝐿
𝑓 𝑥 𝑐𝑜𝑠𝑛𝜋𝑥𝐿𝑑𝑥
!
!
Heat Conduction: ∝! 𝑢!! = 𝑢!
Boundary conditions: 𝑢 0, 𝑡 = 𝑇!,𝑢 𝐿, 𝑡 = 𝑇!
Case: Bar with 𝑇! degrees at the left end, and 𝑇!degrees at the right end General Solution: 𝑢 𝑥, 𝑡 = !!!!!
!𝑥 + 𝑇! + 𝐶!!
!!! 𝑒!∝!!!!!!/!!𝑠𝑖𝑛 !"#
!
Steady State Solution: 𝑣 𝑥 = !!!!!
!𝑥 + 𝑇!
Other info: 𝑣(𝑥) = 𝐴𝑥 + 𝐵 , and 𝑤 𝑥, 0 = 𝑓 𝑥 − 𝑣(𝑥)
𝐶! = 𝑏! =2𝐿
(𝑓 𝑥 − 𝑣 𝑥 )𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥
!
!
One-Dimensional Wave Equations: α!𝑢!! = 𝑢!!
Boundary conditions: 𝑢 0, 𝑡 = 0,𝑢 𝐿, 𝑡 = 0
Initial conditions: 𝑢(𝑥, 0) = 𝑓(𝑥),𝑢!(𝑥, 0) = 𝑔(𝑥)
Case: Undamped One-dimensional Wave Equation General Solution: 𝑢 𝑥, 𝑡 = (𝐴!!
!!! 𝑐𝑜𝑠 !"#$!+ 𝐵!𝑠𝑖𝑛
!"#$!)𝑠𝑖𝑛 !"#
!
Other info: ** See Special Cases Below **
𝐴! = 𝑏! =2𝐿
𝑓 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥
!
!
𝐵! =𝐿𝑎𝑛𝜋
𝑏! =2𝑎𝑛𝜋
𝑔 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥
!
!
One-Dimensional Wave Equations: α!𝑢!! = 𝑢!!
Boundary conditions: 𝑢 0, 𝑡 = 0,𝑢 𝐿, 𝑡 = 0
Initial conditions: 𝑢 𝑥, 0 = 0 ,𝑢!(𝑥, 0) = 𝑔(𝑥)
Case: Special Case of Undamped One-dimensional Wave Equation 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 = 0 General Solution: 𝑢 𝑥, 𝑡 = (𝐵!𝑠𝑖𝑛
!"#$!)𝑠𝑖𝑛 !"#
!
Other info: 𝐴! = 0
𝐵! =𝐿𝑎𝑛𝜋
𝑏! =2𝑎𝑛𝜋
𝑔 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥
!
!
One-Dimensional Wave Equations: α!𝑢!! = 𝑢!!
Boundary conditions: 𝑢 0, 𝑡 = 0,𝑢 𝐿, 𝑡 = 0
Initial conditions: 𝑢(𝑥, 0) = 𝑓(𝑥),𝑢!(𝑥, 0) = 0
Case: Special Case of Undamped One-dimensional Wave Equation 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 0 General Solution: 𝑢 𝑥, 𝑡 = (𝐴!!
!!! 𝑐𝑜𝑠 !"#$!)𝑠𝑖𝑛 !"#
!
Other info: 𝐵! = 0
𝐴! = 𝑏! =2𝐿
𝑓 𝑥 𝑠𝑖𝑛𝑛𝜋𝑥𝐿𝑑𝑥
!
!
𝐵! = 0
