D'Alembert's Method
(Section 3.4) Example 1 (See Example 1, Section 3.4 for details.) To be able to use d'Alembert's method, we need to define the odd extension of a given function NiMtJSJmRzYjJSJ4Rw== , where NiMtJSJmRzYjJSJ4Rw== denotes the initial shape of the string.
Here is the odd extension on the interval [-1,1]. f:= x->piecewise(0<x and x<1/3, 3*x/10, 1/3<=x and x<=1,(3*(1-x))/20) ;
NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lKnBpZWNld2lzZUc2JjMyIiIhOSQyRjIjIiIiIiIkLCRGMiNGNiIjNTMxRjRGMjFGMkY1LCYjRjYiIz9GNSomRj5GNUYyRjUhIiJGKEYoRig= fodd:=x->piecewise(0<=x and x<=1, f(x), -1<=x and x<0,-f(-x)); NiM+JSVmb2RkR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lKnBpZWNld2lzZUc2JjMxIiIhOSQxRjIiIiItJSJmRzYjRjIzMSEiIkYyMkYyRjEsJC1GNjYjLCRGMkY6RjpGKEYoRig= plot(fodd(x),x=-2..2); 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 Now we wish to continue the portionof the graph seen between NiMvJSJ4RywkIiIiISIi and NiMvJSJ4RyIiIg== periodically to the whole real line.
We will use the formula shown in exercise 20 on p.22. For Maple, floor is the same as the greatest integer function mentioned in the text.
We chose the function name extend to remind us that we are extending part of a graph. extend:=(x,p)->x-2*p*floor( (x+p)/(2*p)); NiM+JSdleHRlbmRHZio2JCUieEclInBHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKSwmOSQiIiIqKCIiI0YvOSVGLy0lJmZsb29yRzYjLCQqJiwmRi5GL0YyRi9GL0YyISIiI0YvRjFGL0Y5RilGKUYp plot( extend(x,1),x=-4..4); 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 This gives a periodic extension of the basic graph NiMvJSJ5RyUieEc= on the interval [-1,1]. To get extensions of other functions, compose them with extend .
For example, we will periodically extend our odd extension fodd . In fact, we could make that our main function fstar . fstar:=x->fodd(extend(x,1)); NiM+JSZmc3RhckdmKjYjJSJ4RzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHRigtJSVmb2RkRzYjLSUnZXh0ZW5kRzYkOSQiIiJGKEYoRig= plot(fstar(x) , x=-3..3); 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 We can now use D'Alembert's method to get the shape of the string at various value of NiMlInRH by simply translating then averaging the graphs of the
odd periodic extension NiMtJSZmc3Rhckc2IyUieEc= . For example, since NiMvJSJjRyomIiIiRiYlI1BpRyEiIg== , we have NiMvLSUidUc2JCUieEclInRHKigiIiJGKiIiIyEiIiwmLSUmZnN0YXJHNiMsJkYnRioqKEYqRiolI1BpR0YsRihGKkYsRiotRi82IywmRidGKkYyRipGKkYq .
We define this and plot on the interval [0,1] for NiMvJSJ0RyomJSNQaUciIiIiIiQhIiI= . u:=(x,t)-> 1/2*(fstar(x-1/Pi*t)+fstar(x+1/Pi*t)); NiM+JSJ1R2YqNiQlInhHJSJ0RzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHRiksJi0lJmZzdGFyRzYjLCY5JCIiIiomJSNQaUchIiI5JUYzRjYjRjMiIiMqJkY4RjMtRi82IywmRjJGM0Y0RjNGM0YzRilGKUYp plot( u(x,Pi/3),x=0..1); 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 We can compare this shape to what we would get by using the series solution that we derived in the previous section. Recall that a good approximation of this function was denoted S(10,x,t) in the worksheet for Section 3.3 entitled string.mws. We will recall from this notebook the necessary data for the analytical solution. So, b:=n->2*int(f(x)*sin(n*Pi*x),x=0..1); NiM+JSJiR2YqNiMlIm5HNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwkLSUkaW50RzYkKiYtJSJmRzYjJSJ4RyIiIi0lJHNpbkc2IyooOSRGNSUjUGlHRjVGNEY1RjUvRjQ7IiIhRjUiIiNGKEYoRig= assume('n',integer); b(n); NiMsJCooLSUkc2luRzYjLCQqJiUjbnxpckciIiIlI1BpR0YrI0YrIiIkRitGKiEiI0YsRi8jIiIqIiM1 S:=(N,x,t)->sum( b(n)*sin(n*Pi*x)*cos(n*t),n=1..N); NiM+JSJTR2YqNiUlIk5HJSJ4RyUidEc2IjYkJSlvcGVyYXRvckclJmFycm93R0YqLSUkc3VtRzYkKigtJSJiRzYjJSJuRyIiIi0lJHNpbkc2IyooRjVGNiUjUGlHRjY5JUY2RjYtJSRjb3NHNiMqJkY1RjY5JkY2RjYvRjU7RjY5JEYqRipGKg== plot( [S(10,x,Pi/3),u(x,Pi/3)],x=0..1); 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 It is important to note that the plot that we obtained from d'Alembert's solution is the exact shape of the string.
Let us check the shape of the string when NiMvJSJ0RyomIiIjIiIiJSNQaUdGJw== . This shows that the string has returned ot it's original shape. plot( u(x,2*Pi),x=0..1); LSUlUExPVEc2Ki0lJ0NVUlZFU0c2JDdVNyQkIiIhRitGKjckJCIzZW1tbTthcnpAISM+JCIzKikqKioqKipcaTlSbCEjPzckJCIzW0xMJGU5dWkyJUYvJCIzLCsrdlZBKUdBIkYvNyQkIjNubW1tInpfIjRpRi8kIjMlKioqKipcUGV1aT1GLzckJCIzW21tbVQmcGhOKUYvJCIzJSoqKioqXGkzJm9dI0YvNyQkIjNDTExlKj0pSFw1ISM9JCIzZSoqKlwob1gqeTkkRi83JCQiM2dtbSJ6LzN1QyJGRSQiM2wqKipcUDlDQXUkRi83JCQiMyUpKioqXDdMUkRYIkZFJCIzUyoqKlxQKnpoZFZGLzckJCIzXW1tInpSJ29rO0ZFJCIzUSoqKlxQPmZTKlxGLzckJCIzdyoqKlxpNWBoKD1GRSQiMycpKSoqXCg9JGYlR2NGLzckJCIzV0xMTDNFbiQ0I0ZFJCIzNysrK0R5LCJHJ0YvNyQkIjNxbW07L1JFJkcjRkUkIjNFKioqKlw3PHpib0YvNyQkIjMiKSoqKioqXEtdNF0jRkUkIjNmKSoqKipcKDQmR10oRi83JCQiMyQqKioqKipcUEF2ciNGRSQiMzUrKytEcmNfIilGLzckJCIzKSoqKioqKlxuSGkjSEZFJCIzJSoqKioqKlwtKm95KClGLzckJCIzam1tInoqZXY6SkZFJCIzISoqKipcUHBuc00qRi83JCQiMz4rXTcuJVElR0tGRSQiM2MrXVA0X0ombypGLzckJCIzP0xMTDM0N1RMRkUkIjMjKSkqKipcUCc9JCkpKipGLzckJCIzUUxMJDN4eGxWJEZFJCIzbSoqKlxQTUxeJSkqRi83JCQiMyxMTExMWS5LTkZFJCIzXCsrK10hWz5xKkYvNyQkIjN3KioqXDdvN1R2JEZFJCIzbStdN3k0JClvJCpGLzckJCIzJ0dMTExRKm9dUkZFJCIzcysrK0RmJ1IyKkYvNyQkIjNBKytEIj1sajslRkUkIjNvKipcN0dBWF0oKUYvNyQkIjMxKyt2ViZSPFAlRkUkIjNOKipcUCVvIVJVJSlGLzckJCIzV0xMJGU5RWdlJUZFJCIzNCsrRCJ5ZzQ3KUYvNyQkIjNHTExlUiIzR3klRkUkIjNfKipcaSF6KHlEeUYvNyQkIjNjbW07L1QxJipcRkUkIjNVKyt2VlFTMnZGLzckJCIzJmVtO3pSUWJAJkZFJCIzQyxdNy5DcHdyRi83JCQiM1wqKipcKD0+WTJhRkUkIjMlKioqXCg9NzIpKSlvRi83JCQiMzltbTt6WHU5Y0ZFJCIzITMrXTc4JCl5ZCdGLzckJCIzbCoqKioqKlx5KSlHZUZFJCIzYSsrK0Qjb21EJ0YvNyQkIjMnKikqKipcaV9RUWdGRSQiMyU9K11pNUFDJWZGLzckJCIzQCoqKlw3eSUzVGlGRSQiM04rXTdHR1BRY0YvNyQkIjM1KioqKlxQIVtoWSdGRSQiM0wsK3ZWengrYEYvNyQkIjNrS0xMJFF4JG9tRkUkIjNLLCsrRFJWKCpcRi83JCQiMyEpKioqKipcUCtWKW9GRSQiM2UrK11QJVxObiVGLzckJCIzP21tInpwZSp6cUZFJCIzOStdN2A+MSFRJUYvNyQkIjMlKSoqKioqXCNcJ1FIKEZFJCIzKSoqKioqXDdFP2ZTRi83JCQiM0dLTGU5UzgmXChGRSQiMy0sXTd5KilIZFBGLzckJCIzUioqKlxpPz1icShGRSQiM1osXWkhcEE8VyRGLzckJCIzIkhMTCQzcz82ekZFJCIzTysrXSg9Kj1MSkYvNyQkIjNhKioqXDdgV2w3KUZFJCIzKDQrREo/JD01R0YvNyQkIjMjcG1tbScqUlJMKUZFJCIzaSoqKioqKlwrNCpcI0YvNyQkIjNRbW07YTwuWSYpRkUkIjMpKSoqKlwob0ImND0jRi83JCQiMz1MTGU5dE9jKClGRSQiMzkqKlw3RyFcYSc9Ri83JCQiM3UqKioqKipcUWtcKilGRSQiMycpKioqKipcQU1iZCJGLzckJCIzQ0xMJDNkZzY8KkZFJCIzOCsrdlYiZktDIkYvNyQkIjNJbW1tbXhHcCQqRkUkIjM5NisrK05vZyUqRjI3JCQiM0ErK0Qib0swZSpGRSQiM1cnKipcN3k0P0gnRjI3JCQiM0ErK3Y9NXMjeSpGRSQiM0ctK3Y9Wj1mS0YyNyQkIiIiRitGKi0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiRGK0ZnW2xGaFtsLSUrQVhFU0xBQkVMU0c2JFEieDYiUSFGXVxsLSUqR1JJRFNUWUxFRzYjJSxSRUNUQU5HVUxBUkctJSVWSUVXRzYkO0ZoW2xGZVtsOyQhMicqKioqXEZQbSgqPkYvJCIyKSoqXC0sJjMpPTUhIzwtRmJbbDYjJSVOT05FRy0lK1BST0pFQ1RJT05HNiNGZVtsLSUqTElORVNUWUxFRzYjRistJSxPUklFTlRBVElPTkc2JCQiI1hGK0ZpXWw= At this point, it is easy to animate several graphs as time increases. with(plots): animate( u(x,t),x=0..1,t=0..2*Pi,frames=12); 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What you see in the above pictures are the exact shapes of the string at the stated times. There is no approximation here. D'Alembert's method gives you the precise shape of the string. We can arrange the plots in an array as follows withthe shortcut %
assuming that these commands are executed immediately after the animate command.. display(%,axes=none); 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