# Why did we make equations dimensionless? [closed]

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?

• So you don't need to deal with dimensions? Commented Oct 24, 2015 at 6:50
• 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. Commented Oct 24, 2015 at 8:22
• What was the paper? Commented Oct 25, 2015 at 18:06
• It is 'Plane wave propagation and domain of influence in fractional order thermoelastic materials with three phase lag heat transfer'.
– niti
Commented Oct 26, 2015 at 0:39

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.