physics - Trying to simulate a 1-dimensional wave -

i'm trying make simplified simulation of 1-dimensional wave chain of harmonic oscillators. differential equation position x[i] of i-th oscillator (assuming equilibrium @ x[i]=0) turns out - according textbooks - one:

m*x[i]''=-k(2*x[i]-x[i-1]-x[i+1])

(the derivative wrt time) tried numerically compute dynamics following algorithm. here osc[i] i-th oscillator object attributes loc (absolute location), vel (velocity), acc (acceleration) , bloc (equilibrium location), dt time increment:

for every i:     osc[i].vel+=dt*osc[i].acc;     osc[i].loc+=dt*osc[i].vel;     osc[i].vel*=0.99;         osc[i].acc=-k*2*(osc[i].loc-osc[i].bloc);      if(i!=0){       osc[i].acc+=+k*(osc[i-1].loc-osc[i-1].bloc);     }     if(i!=n-1){       osc[i].acc+=+k*(osc[i+1].loc-osc[i+1].bloc);     } 

you can see algorithm in action here, click give impulse 6-th oscillator. can see not doesn't generate wave @ generate motion growing total energy (even if added dampening!). doing wrong?

all given answers provide (great) deal of useful information. must is:

• use ja72´s observation performing updates - need coherent , compute acceleration @ node based on positions same iteration batch (i.e. not mix $x⁽k⁾$ , $x⁽k+1⁾$ in same expression of acceleration these amounts belonging different iteration steps)
• forget original positions tom10 said - need consider relative positions - behaviour of wave equation laplacian smoothing filter applied polygonal line - point relaxed using 2 directly connected neighbours, being pulled towards middle of segment determined neighbours
• finally, if want energy conserved (i.e. not have water surface stop vibrating), it's worth using symplectic integrator. symplectic/semi-implicit euler do. don't need higher order integrators kind of simulation, if have time, consider using verlet or ruth-forest they're both symplectic , of order 2 or 3 in terms of accuracy. runge-kutta will, implicit euler, draw energy (much slower) system, you'll inherent damping. explicit euler spell doom - that's bad choice - (esp stiff or undamped systems). there tons of other integrators out there, go ahead , experiment.

p.s. did not implement true implicit euler since 1 requires simultaneously affecting particles. that, must use projected conjugate gradient method, derive jacobians , solve sparse linear systems string of particles. explicit or semi-implicit methods "follow leader" behaviour expect animation.

now, if want more, implemented , tested answer in scilab (too lazy program in c++):

n=50; i=1:n     x(i) = 1; end  dt = 0.02;  k = 0.05; x(20) = 1.1; xold = x; v = 0 * x; plot(x, 'r'); plot(x*0,'r'); iteration=1:10000     = 1:n         if (i>1 & < n)             acc = k*(0.5*(xold(i-1) + xold(i+1)) - xold(i));             v(i) = v(i) + dt * acc;             x(i) = xold(i) + v(i) *dt;         end     end     if (iteration/500  == round(iteration / 500) )         plot(x - iteration/10000,'g');      end     xold = x; end plot(x,'b'); 

the wave evolution seen in stacked graphs below: