20. Ch6_1D_wave_equation_linear_displacement

20. Ch6_1D_wave_equation_linear_displacement -...

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6.1.2 1D wave equation: arbitrary initial displacement of the string (i) Orthogonality of the eigenfunctions ׬ ݏ݅݊ గ௫ ቁ ݀ݔ ്0 ׬ ݏ݅݊ ቀ గ௫ ቁ ݏ݅݊ ቀ ଶగ௫ ቁ ݀ݔ ൌ0 ׬ ݏ݅݊ ቀ గ௫ ቁ ݏ݅݊ ቀ ଷగ௫ ቁ ݀ݔ ׬ ݏ݅݊ ቀ గ௫ ቁ ݏ݅݊ ቀ ସగ௫ ቁ ݀ݔ 0 0.5 1 term 1: sin 2 ( π x/L) 1 0.5 0 0.5 1 term 2: sin ( π x/L) sin (2 π x/L) 1.5 1 0.5 0 0.5 1 term 3: sin ( π x/L) sin (3 π x/L) 1.5 1 0.5 0 0.5 1 term 4: sin ( π x/L) sin (4 π x/L) x L x L x L x L
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(ii) Comparison of the solution at t =0 and IC term 1: 2/( π ) sin ( π x/100) term 2: 2/(2 π ) sin (2 π x/100) term 3: 2/(3 π ) sin (3 π x/100) term 4: 2/(4 π ) sin (4 π x/100) term 5: 2/(5 π ) sin (5 π x/100) term 10: 2/(10 π ) sin (10 π x/100) Solution at t = 0, keeping 10 terms in the infinite series s ( x ,0) term1 – term2 + term3 term10 Solution at t = 0 with infinite terms gives the exact IC ࢙ሺ࢞, ૙ሻ ൌ ૛ࡴ ሺെ૚ሻ ࢔ା૚ sin
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

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20. Ch6_1D_wave_equation_linear_displacement -...

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