Can the Wave Equation Solution Use Only Positive Values of n?

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SUMMARY

The discussion centers on the validity of using only positive values of n in the wave equation solution. Participants assert that while textbooks often simplify the solution by excluding negative n values, this approach is incorrect. The mathematical identities cos(-nkx) = cos(nkx) and sin(-nkx) = -sin(nkx) allow for the transformation of negative n terms into positive ones, but this does not justify omitting negative values entirely. A comprehensive solution requires considering all values of n, including negative ones, particularly when applying Fourier or Laplace transforms.

PREREQUISITES
  • Understanding of wave equations and their general solutions
  • Familiarity with Fourier series and transforms
  • Knowledge of Laplace transforms
  • Basic trigonometric identities and their applications
NEXT STEPS
  • Study the application of Fourier series in solving wave equations
  • Learn about the implications of using Laplace transforms in wave analysis
  • Explore the mathematical properties of trigonometric functions in relation to wave solutions
  • Investigate the concept of superposition in wave mechanics
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Students of physics and mathematics, particularly those studying wave mechanics, as well as educators and researchers looking to deepen their understanding of Fourier and Laplace methods in wave equation solutions.

sommerfugl
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Hello.

If I have this equation:

2jb4b43.png


And this general solution:

290zo5k.png


Would it then be wrong to write the above solution with only positive values of n? In my textbook they often write the result from a superposition with only positive values of n, becasue the negative values of n already are included in the terms obtained for positive values of n.
 
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cos(-nkx)= cos(nkx) and sin(-nkx)= -sin(nkx) so all of the terms with n negative can be changed to n positive.
 
HallsofIvy said:
cos(-nkx)= cos(nkx) and sin(-nkx)= -sin(nkx) so all of the terms with n negative can be changed to n positive.

This is wrong.. either use finite fourier, fourier, laplace. and resolve.. you should not get n=-infiinity... this problem should be so tough...
 

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