If we're given a string with mass density $\rho$ in units $\frac{M}{L^3}$ with constant cross-section $A$, tension $T$ in units $\frac{F}{L^2}$, and whose length is $L$; and then we assume that the vertical displacement is small, the slope of the string during displacement is also small (which probably means that we can approximate tension to be constant); then I think it would be correct to say that the governing equation in this case is $$u_{tt}=c^2u_{xx}$$

where $c^2 = \frac{T}{\rho}$. (Please correct me if I'm wrong).

Now a dissipation term must be added, which should be directly proportional to the mass and velocity of the string.

Would it be correct to specify the dissipation term as follows: $-b\underbrace{\rho A L}_\text{mass} (u_t)^2\frac{1}{u}$ (where $b$ is some dimensionless proportionality constant)?

If this is correct, then the equation would become $$u_{tt}=c^2u_{xx}-bm(u_t)^2\frac{1}{u}$$

where $m$ is mass.

  • $\begingroup$ You say "directly proportional to the mass and velocity", but what you write is proportional to the mass and to the square of the velocity (and inversely to the displacement). Whether you should use the first or second power of the velocity largely depends on the Reynolds number of the system, approximately a cylinder moving perpendicular to its axis through a fluid. This isn't really a math question, but fits better in physics. (And they'll tell you what I said about Reynolds numbers, so you should probably think about that before posting there.) $\endgroup$ – Eric Towers May 10 '16 at 1:36
  • $\begingroup$ Here the question concerns a string, not a cylinder with a fluid. $\endgroup$ – sequence May 10 '16 at 12:37
  • $\begingroup$ What do you believe the geometry of a string moving in a background gas or liquid is? $\endgroup$ – Eric Towers May 10 '16 at 20:40
  • $\begingroup$ True, thanks for clarifying. I've asked this questions on Physics.SE, but they put my question on hold as off-topic. $\endgroup$ – sequence May 11 '16 at 0:08

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