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D'Alembert's solution for the one-dimensional wave equation for a semi-infinite string

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Find the solution to:

PDE: u*xx-c(to the power of negative k)u*tt=0 , 0
ICs: u(x,0)=f(x) and u*t(x,0)=g(x), x>=0

BC: u*x(0,t)=0 , t>=0

This BC corresponds to a string with its end point free to move in a vertical direction.

(Please remember to include to boundary condition in your solution. Thanks very much!)

Hello! Thank you for taking the time to try and solve this problem. Please note that I am unable to use math symbols. Thus, I will use * to symbolize a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. Here is the problem: (also note that the PDE is the one-dimensional wave equation)

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Solution Summary

D'Alembert's solution for the one-dimensional wave equation for a semi-infinite string is investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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