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I study a paper on propagation of plane wave, in which equations are made dimensionless.

Equation of motion is

\begin{equation*} c_{ijmn}u_{m,nj} = \ddot{u_i} \end{equation*}

where $c_{ijmn}$ are elastic constant and $u_i$ is component of displacement.

Then equation is made dimensionless by using $x'_i=C_0\eta x_i$ and $u'_i=C_0\eta u_i$ where $\eta$ is entropy, $C_0$ is the longitudinal wave velocity . Similarly heat conduction equation is made dimensionless. Then this system of equations is solved for propagation of plane wave.

Why did we make equation dimensionless?

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    $\begingroup$ So you don't need to deal with dimensions? $\endgroup$
    – paparazzo
    Commented Oct 24, 2015 at 6:50
  • $\begingroup$ Related: physics.stackexchange.com/a/212320/70242. Specifically the second paragraph after the equations is useful; by making dimensionless we find the one or two parameters which characterize the equation. $\endgroup$
    – nluigi
    Commented Oct 24, 2015 at 8:22
  • $\begingroup$ What was the paper? $\endgroup$
    – HDE 226868
    Commented Oct 25, 2015 at 18:06
  • $\begingroup$ It is 'Plane wave propagation and domain of influence in fractional order thermoelastic materials with three phase lag heat transfer'. $\endgroup$
    – niti
    Commented Oct 26, 2015 at 0:39

1 Answer 1

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  1. It makes things easier to understand, basically it removes the clutter. Like others have commented, by non-dimensionalizing a system, we can narrow it down to the crucial parameters that affects the system.

  2. A lot of computational work is reduced by nondimensionalizing the system. This makes sure that we wont use memory uncessasarily. For example, the Boltzman constant is $ \propto 10^{-23} $ There is a limit to how much the computer has in terms of memory. There are things like floating point errors etc.. So what I'd do in practice when solving such a system, is to set $k_B=1$. It is justified because it is only a constant, and the things we want to see are the trends. And the numbers you obtain can be easily rescaled again if you want the actual value.

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    $\begingroup$ I fully agree with your first point. However, I take issue with your second one. Computer memory usage really has very little to do with why it is done. Even the issue of floating point round-off is not very significant, since you wouldn't have any reason to add the (eg) the Boltzman constant to something of order 1. These outlandishly large/small physical constants (eg: Gravitational constant, Boltzman constant, Avogadro's number, ...) are invariably multiplied by something, when only the number of significant digits (not the exponent) is relevant. (contined in next comment) $\endgroup$ Commented Oct 28, 2015 at 21:32
  • $\begingroup$ The reason why problems are non-dimensionalized is that by doing so, the total number of parameters is often reduced. This allows you to do a parametric study of, for example, the Navier-Stokes equations in some geometry by sweeping the values of the Reynolds number rather than sweeping values for both viscosity and density. These computations are costly, so the ability to save a lot of work is the primary driver for non-dimensionalized equations. $\endgroup$ Commented Oct 28, 2015 at 21:35

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