A string stretched between the points 0and pi on the x axis and initially at res

Question

A string stretched between the points 0and pi on the x axis and initially at rest, is released from the position y = f(x). It's motion is opposed by air resistance, which is proportional to the velocity at each point. Let the unit of time be chosen so that the equation of motion becomes

y sub tt (x,t) = y sub xx(x,t)- 2 beta y sub t(x,t) (0< x < pi, t>0)

where beta is a positive constant. Assuming that 0< beta <1, derive the expression

y(x,t) = e^(-beta*t) sum(n =1 to infinity)* B sub n (cos alfa sub n *t + (beta/alfa sub n)* sin alfa sub n*t) sin nx

where alfa sub n = square root(n^2-beta^2), B sub n = Int (0 to pi)f(x) sin nx dx (n = 1,2,.....)

Explanation / Answer

https://docs.google.com/viewer?a=v&q=cache:lEtdqeTASQ8J:www.wou.edu/~schoenfw/Old Courses/PH202 Winter 2012/Solutions/PH202 Chapter 14 solutions.pdf+A+string+stretched+between+the+points+0and+pi+on+the+x+axis+and+initially+at+rest,+is+released+from+the+position+y+=+f(x).+It's+motion+is+opposed+by+air+resistance,+which+is+proportional+to+the+velocity+at+each+point.+Let+the+unit+of+time+be+chosen+so+that+the+equation+of+motion+becomes+y+sub+tt+(x,t)+=+y+sub+xx(x,t)-+2+beta+y+sub+t(x,t)+(0<+x+<+pi,+t>0)+where+beta+is+a+positive+constant.+Assuming+that+0<+beta+<1,+derive+the+expression+y(x,t)+=+e^(-beta*t)+sum(n+=1+to+infinity)*+B+sub+n+(cos+alfa+sub+n+*t+++(beta/alfa+sub+n)*+sin+alfa+sub+n*t)+sin+nx+where+alfa+sub+n+=+square+root(n^2-beta^2),+B+sub+n+=+Int+(0+to+pi)f(x)+sin+nx+dx+(n+=+1,2,.....)&hl=en&gl=in&pid=bl&srcid=ADGEESiufpgII5WEiPtHAIFH61WuutL53X_EaupzVtoK2BCSKyXH3XStuvnO-c9bX6Zt4gnONFuBoPsVwbLdotqTi19Mkt8ZT8BSAtBZl7Im4mg9YiV_rQnVUE2Z7jyhunIkV9JqOb9W&sig=AHIEtbTlcvvUbqDsQK6FvIeDBe4IlkUinw................ ur answer

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