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Jack Goodman
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Discrete Wave Equation WITH DAMPENING
I am currently working on how to express the wave-equation in a discrete setting WHEN DAMPENING IS BEING USED. As Chris points out on his page, the continuous form of the wave-equation, without any dampening, is: \[ \frac{ \partial^2 \mathbf{F}}{\partial t^2} = \mathbf{c^2} \nabla^2 \mathbf{F} \]
However, if one were to include dampening, the form of this would change, to:
\[ \frac{ \partial^2 \mathbf{F}}{\partial t^2} = \mathbf{c^2} \nabla^2 \mathbf{F} - \mathbf{k} \frac{ \partial \mathbf{F}}{\partial t} \]
Here, we refer to k as simply the "dampening constant."
This equation arises due to the fact that any dampening will be proportional to the momentum/velocity of the motion taking place within the material the wave is propagating through, or the derivative of the displacement from equilibrium if one views things from the viewpoint of an oscillating spring (see: http://en.wikipedia.org/wiki/Damping).
So, the discretetized form of the DAMPENED equation becomes the following:
\[ \mathbf{F}_{(x, y, t+1)} = 2\mathbf{F}_{(x, y, t)} - \mathbf{F}_{(x, y, t-1)} + \mathbf{c}^2 *( \mathbf{F}_{(x+1, y, t)} + \mathbf{F}_{(x-1, y, t)} + \mathbf{F}_{(x, y+1, t)} + \mathbf{F}_{(x, y-1, t)} - 4*\mathbf{F}_{(x, y, t)} ) - \mathbf{k} *(\mathbf{F}_{(x,y,t)} - \mathbf{F}_{(x,y,t-1)}) \]
Why did we use F(x,y,t)-F(x,y,t-1) as the discrete time-derivative of F here, rather than F(x,y,t+1)-F(x,y,t) like what was used earlier? This is simply a matter of choice, and it remains to be tested whether taking the derivative at time (t+1) might give a more accurate physical model.
If the time-derivative is taken at time t+1 instead of time t, the equation becomes:
\[ \mathbf{F}_{(x, y, t+1)} = 2\mathbf{F}_{(x, y, t)} - \mathbf{F}_{(x, y, t-1)} + \mathbf{c}^2 *( \mathbf{F}_{(x+1, y, t)} + \mathbf{F}_{(x-1, y, t)} + \mathbf{F}_{(x, y+1, t)} + \mathbf{F}_{(x, y-1, t)} - 4*\mathbf{F}_{(x, y, t)} ) - \mathbf{k} *(\mathbf{F}_{(x,y,t+1)} - \mathbf{F}_{(x,y,t)}) \]
So, the equation for the wave-height at time t+1 in terms of the height at previous times, when taking velocity at time t, becomes:
The Discretized Wave Equation with Dampening (velocity at time t used)
\[ \mathbf{F}_{(x, y, t+1)} = (2-\mathbf{k})*\mathbf{F}_{(x, y, t)} + (\mathbf{k} - 1)* \mathbf{F}_{(x, y, t-1)} + \mathbf{c}^2 *( \mathbf{F}_{(x+1, y, t)} + \mathbf{F}_{(x-1, y, t)} + \mathbf{F}_{(x, y+1, t)} + \mathbf{F}_{(x, y-1, t)} - 4*\mathbf{F}_{(x, y, t)} ) \]
While the equation when using velocity at time t+1 is
The Discretized Wave Equation with Dampening (velocity at time t+1 used)
\[ \mathbf{F}_{(x, y, t+1)} = \frac{\mathbf{(2+k)}\mathbf{F}_{(x, y, t)} - \mathbf{F}_{(x, y, t-1)} + \mathbf{c}^2 *( \mathbf{F}_{(x+1, y, t)} + \mathbf{F}_{(x-1, y, t)} + \mathbf{F}_{(x, y+1, t)} + \mathbf{F}_{(x, y-1, t)} - 4*\mathbf{F}_{(x, y, t)} )}{\mathbf{k+1}} \]
However, we did not derive this formula using our usual "laws of physics-based" approach, and so while I believe this formula to be the correct one for the discrete form of dampened harmonic motion, I would like to find a way to derive it purely using physical principles, rather than by discretizing a PDE. The question here is: why would allowing for dampening due to "drag" within the fluid cause one to subtract a term equal to a constant multiplied by the discrete time-derivative of F at (x,y)?
Note: The above equation is, however, certainly far from perfect by any means. This is due to the fact that the dampening of our system is caused by the DRAG within the fluid transmitting the waves. . .however, this drag is only APPROXIMATELY proportional to the momentum/velocity, AND ALSO HAS A TERM PROPORTIONAL TO THE KINETIC ENERGY (velocity-squared), as noted in the above cited Wikipedia article on dampening. While the drag proportional to the kinetic energy is typically ignored in most situations, it would be nice to be able to write it into our formula in order to make things even more precise. However, if we do this, it should be noted that our recurrence-relation for determining the wave-height will become NONLINEAR as a result. I shall continue investigating this problem, and whether it might be worthwhile to include a nonlinear term proportional to the kinetic energy at time t and position (x,y).
Moreover, in the above formula, only the drag proportional to velocity AT POSITION (x,y) has been considered. However, it is valid to ask whether the drag in ADJACENT cells (such as (x-1,y)) at time t-1, could subsequently affect the wave propagation through cell (x,y) at time t. It seems quite possible that this could be the case, and thus, it might be desirable to also include terms involving the drag, proportional to velocity, in cells adjacent to (x,y). I would very much LIKE to show, if possible, that it is completely valid to ignore the drag in adjacent cells when working with the discretized wave equation, as I suspect this is, in fact, the case.
Using Arbitrary Time-Steps and Lattice-Square-Width
In the case with arbitrarily chosen time-steps and "space-steps" (lattice size), while still keeping to a SQUARE lattice, we obtain: $$F_{(x, y, t+\Delta t)} = 2*F_{(x, y, t)} - F_{(x, y, t-\Delta t)} + $$ $$c^2 * ((\Delta t)/(\Delta u))^2 * ( F_{(x+\Delta u, y, t)} + F_{(x- \Delta u, y, t)} + F_{(x, y+\Delta u, t)} + F_{(x, y-\Delta u, t)} - 4 * F_{(x, y, t)} ) - $$ $$k * \Delta t *(F_{(x,y,t)} - F_{(x,y,t-\Delta t)}).$$
Christopher and I are currently working on verifying that in the undamped case (k=0), if one runs the simulation multiple times, each time allowing the time-steps and lattice-size to become smaller (and having them tend, in the limit, TO ZERO), then the solution will converge to the precise analytic solution in the case with CONTINUOUS TIME AND SPACE. There is a major difficulty here, namely, that of "translating," as it were, the Initial-Value-Problem for the Discrete case, into the properly "corresponding" Initial-Value-Problem for the Continuous case. This is still under investigation. It would seem that perhaps it would be best to start with an IVP for the continuous case which has a known, simple solution, and then translate THIS into a corresponding IVP with Discrete time and space, and then run the simulation with those initial conditions, with smaller and smaller time and space steps, seeing if the solutions converge to the continuous case.
Note: I have proposed that we always allow our time-step size to be equal to THE RECIPROCAL OF A POSITIVE INTEGER, thus allowing for ease of computation (in this case, if the time-steps are 1/n, then we can simply program the simulation to only render the new image every n steps, and thus only at integer times).
The addition of a NONLINEAR term into the discrete wave equation
In my discussion of the discrete wave equation, I took note of the fact that the dampening caused by drag while a wave propagates in a fluid medium will not ONLY be proportional to the velocity, but will also have a (usually quite small) term proportional to the velocity-SQUARED. When one includes this term, it becomes necessary to define two different dampening constants, and the equation, in continuous form, becomes:
\[ \frac{ \partial^2 \mathbf{F}}{\partial t^2} = \mathbf{c^2} \nabla^2 \mathbf{F} - \mathbf{k_1} \frac{ \partial \mathbf{F}}{\partial t} - \mathbf{k_2} (\frac{ \partial \mathbf{F}}{\partial t}) * \|\frac{ \partial \mathbf{F}}{\partial t}\| \]
In discrete form, this becomes: \[ \mathbf{F}_{(x, y, t+1)} = (2-\mathbf{k_1})*\mathbf{F}_{(x, y, t)} + (\mathbf{k_1} - 1)* \mathbf{F}_{(x, y, t-1)} + \mathbf{c}^2 *( \mathbf{F}_{(x+1, y, t)} + \mathbf{F}_{(x-1, y, t)} + \mathbf{F}_{(x, y+1, t)} + \mathbf{F}_{(x, y-1, t)} - 4*\mathbf{F}_{(x, y, t)} )-\] \[ \mathbf{k_2}*((\mathbf{F}_{(x,y,t)}-\mathbf{F}_{(x,y,t-1)}))*\|(\mathbf{F}_{(x,y,t)}-\mathbf{F}_{(x,y,t-1)})\| \]
where we clearly now have two different dampening constants. Note that we have included a term equal to the ABSOLUTE VALUE of the velocity (time-derivative of F) in order to ensure that the DRAG FORCE IS ALWAYS OPPOSED IN DIRECTION TO THE CHANGE (velocity) of F.
While the second dampening constant will, in most cases, be quite small, it should still be interesting to incorporate this nonlinear adjustment into the program for the discrete wave-equation and see what phenomena result from it. Conveniently, if we specify \[k_2=0\], then we simply get the earlier linear recurrence with dampening proportional only to the velocity and not its square.
The Heat/Diffusion Equation with Arbitrary Time-Steps and Lattice-Width
If we continue using a square lattice, at least for now, but allow for arbitrary time-steps and lattice-width in the Heat/Diffusion Equation (with a source s),
$$\frac{∂F}{∂t} = k ∇^2 F + s$$
then, by discretizing, we obtain
$$F{(x, y, t+\Delta t)} = s*\Delta t + F{(x, y, t)} +$$
$$k*\frac{\Delta t}{(\Delta u)^2}*(F_{(x+\Delta u, y, t)} + F_{(x- \Delta u, y, t)} + F_{(x, y+\Delta u, t)} + F_{(x, y-\Delta u, t)} - 4 * F_{(x, y, t)})$$
