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# boundary for waves in a cantilever with endmass

Posted 9 years ago
 Hello, I am currently trying to solve the 1D wave equation for longitudinal waves in a cantilever (BOUN2) with an endmass (BOUN1). The entire system is at rest (INIT1) and is excited with an initial velocity at t=0 at the free end (INIT2). I am unable to generate a solution. With simpler boundary conditions the system is solvable here i am getting the error that INIT2 is not specified on the boundary of the domain. This should yield a fourier series. Could someone help me with the proper definition of the boundary?  c=1; k=1; L=1; v=1; BOUN1= k*Derivative[0, 2][u][L, t] + Derivative[1, 0][u][L, t] == 0; BOUN2= u[0, t] == 0; INIT1= u[x, 0] == 0; INIT2 = Derivative[0, 1][u][L, 0] == -v ; DEQN1 = D[u[x, t], {x, 2}] - c*D[u[x, t], {t, 2}] == 0; solution1= NDSolveValue[{DEQN1, BOUN1, BOUN2, INIT1, INIT2 }, u, {x, 0, L}, {t, 0, 0.00019}] Plot[solution1[L, t], {t, 0, 0.00019}, PlotRange -> All] 
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Posted 9 years ago
 No problem, INIT2 also has a problem as you define a initial condition only at a point in space but it needs a definition for all x. BTW, you probably can find the analytic solution in MM as well using Solve
Posted 9 years ago
 Thank you! Sometimes the easiest solution to a problem is bluntly overlooked. It did not even occur to me to split it up because I was able to solve the system on a piece of paper. I'll implement that now and hopefully resolve this issue.
Posted 9 years ago
 Stephan, Without deriving equations, it seems your boundary conditions are not consistent. I would suggest to set up two differential equations - one for the beam and one for the mass separately. Of course a force would couple the equations. BOUN1 - you can't have a second derivative in a boundary condition if the differential equation is second order