You should change these

Add this acceleration value (between -1 and 1) to the previous velocity v_new = v_old + a.

Finally, add this new v value to the physical wave form u_new = u_old + v.

to

Add this acceleration value (between -1 and 1) to the previous velocity v_new = v_old + a x dt

and

Finally, add this new v value to the physical wave form u_new = u_old + v x dt

As the change in velocity is acceleration multiplied by time, and the change in distance is velocity multiplied by time...the dt is a small change in time. The smaller you make the dt, the more accurate the results will be.

You should change these

Add this acceleration value (between -1 and 1) to the previous velocity v_new = v_old + a.

Finally, add this new v value to the physical wave form u_new = u_old + v.

to

Add this acceleration value (between -1 and 1) to the previous velocity v_new = v_old + a x dt

and

Finally, add this new v value to the physical wave form u_new = u_old + v x dt

As the change in velocity is acceleration multiplied by time, and the change in distance is velocity multiplied by time...the dt is a small change in time. The smaller you make the dt, the more accurate the results will be. It's recommended that you check that each part of the velocity graph is going positive and negative (equally) before going onto the position graph. All the best.