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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?

niti
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1 Answers1

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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.

Candy Man
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