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copied!<p>All given answers provide a (great) deal of useful information. What you must do is:</p> <ul> <li>use ja72´s observation for performing the updates - you need to be coherent and compute the acceleration at a node based on the positions from the same iteration batch (i.e. do not mix $x⁽k⁾$ and $x⁽k+1⁾$ in the same expression of the acceleration as these are amounts belonging to different iteration steps)</li> <li>forget about your original positions as tom10 said - you need to consider only relative positions - the behaviour of this wave equation is like a laplacian smoothing filter applied to a polygonal line - a point is relaxed using its two directly connected neighbours, being pulled towards the middle of the segment determined by those neighbours</li> <li>finally, if you want your energy to be conserved (i.e. not have the water surface stop from vibrating), it's worth using a symplectic integrator. Symplectic/semi-implicit Euler will do. You don't need higher order integrators for this kind of simulation, but if you have the time, consider using Verlet or Ruth-Forest as they're both symplectic and of order 2 or 3 in terms of accuracy. Runge-Kutta will, like implicit Euler, draw energy (much slower) from the system, thus you'll get inherent damping. Explicit Euler will spell doom for you eventually - that's a bad choice - always (esp for stiff or undamped systems). There are tons of other integrators out there, so go ahead and experiment.</li> </ul> <p><strong>P.S.</strong> you did not implement true implicit Euler since that one requires simultaneously affecting all particles. For that, you must use a projected conjugate gradient method, derive Jacobians and solve sparse linear systems for your string of particles. Explicit or semi-implicit methods will get you the "follow the leader" behaviour you expect for an animation.</p> <p>Now, if you want more, I implemented and tested this answer in SciLab (too lazy to program it in C++):</p> <pre><code>n=50; for 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'); for iteration=1:10000 for i = 1:n if (i>1 & i < n) then 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) ) then plot(x - iteration/10000,'g'); end xold = x; end plot(x,'b'); </code></pre> <p>The wave evolution is seen in the stacked graphs below: <img src="https://i.stack.imgur.com/U184i.png" alt="enter image description here"></p>

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