#### Energy-Preserving Parameter Changes (Mass-Spring Oscillator)

If the change in or is deemed to be internal'', that is, involving no external interactions, the appropriate accompanying change in the internal state variables is that which conserves energy. For the mass and its velocity, for example, we must have

where denote the mass values before and after the change, respectively, and denote the corresponding velocities. The velocity must therefore be scaled according to

since this holds the kinetic energy of the mass constant. Note that the momentum of the mass is changed, however, since

If the spring constant is to change from to , the instantaneous spring displacement must satisfy

In a velocity-wave simulation, displacement is the integral of velocity. Therefore, the energy-conserving velocity correction is impulsive in this case.

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