User's Wave Equation

Wave Equation

2 U = Utt / ν2

Wikipedia comments: "The wave equation is an important second-order linear partial differential equation for the description of waves - as they occur in physics - such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange."

Do you have a Wave equation to solve? Or any other math equations? For the next few months of 2014, we are willing to help you solve them using Calculus-level programming. To start, copy and modify the source code below in a file we'll call it {abc} 123}.fc where {abc} = your initials and {123} = any number or id; 8 characters max. for filename. Edit your {abc}{123}.fc file, especially lines starting with a "!" character. E-mail us your {abc}{123}.fc file. We will compile & execute it and send you the output file. Try compiling you own code with our FC-Compiler program. Download and install FC-Compiler with more [~60] example problems to choose from. Learn how easy it works.

All "!" characters in columns 1 or 2 must be deleted before compiling. These "!" were added in order to point out areas needing work.

Next, you may modify a copy of this web page and send it to us for viewing. If accepted, we will post your webpage showing your problem with solution. If you want people to be able to contact you, please include your e-mail address on this web page.

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User's Wave Equation Source Code:

For 1-Dimensional Wave Equation use following:

      global all
      problem WavePDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Wave Equation; a PDE Initial
C --- Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
        dynamic U, Ut, Utt
C User parameters ...
!        v = ???	             ! velocity?
!        ipoints = 21          ! grid pts. over x-axis
!        tFinal =  1           ! final time
C x-parameter initial settings: x ==> i
!        xFinal =  1:     ip = ipoints:   dx= xFinal/(ipoints-1)
C t-parameter initial settings: t ==> m
        pi = 4*atan(1):    tPrint=tFinal/ip
        allot U(ip), Ut(ip), Utt(ip)
        call tAxis      !
      end               ! Stmt.s not necessary in IVP, but used in BVP versions
      model tAxis       !
C ... Integrate over t-axis
        t= 0:    tPrt = tPrint:      dt = tPrt / 10
        Initiate janus;  for PDE;
     ~       equations Utt/Ut, Ut/U;  of t;  step dt;  to tPrt
        do while (t .lt. tFinal)
          Integrate PDE;  by janus
          if( t .ge. tPrt) then
            print 79, t, (U( ij), ij = 1, ip)
            tPrt = tPrt + tPrint
          end if
        end do
 79     format( 1h , f8.4, 3x, 20(g14.5, 1x))
      model PDE                         ! Partial Differential Equation
C                                       ! Method of Lines
!        U(1) = U0:      Ut(1)=0:     Utt(1)=0    ! Initial Conditions
        do 20 ij = 2, ipoints-1         ! System of ODEs
          Uxx = (U(ij+1)-2*U(ij)+U(ij-1))/(dx*dx)  !4 2nd order in 'x'
          Utt(ij)= v**2 * Uxx		    ! Wave equation
 20     continue
!        Ut(ip)= ???:        Utt(ip)= ???   ! Final conditions
      Fmodel U0(xx)       ! Initial starting values @ t = 0
!        if( xx .le. 0) then
!          U0 = 0
!        elseif( xx .lt. .5 ) then
!          U0 = ... f(xx)
!          U0 = 0

User's Wave Equation Output:

selected output goes here ... 

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<a href=""><img style="float:left; width:100px" src=""/> <strong>Wave Equation</strong> </a>; Simulation to Optimization, Tweak Parameters for Optimal Solution.

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