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Take a look, for example, at the telegrapher's equations (let's look at the voltage one). They have the wrong units.

Equation

$u_{x} = Li_{t} + Ri$

*where $u$ is potential in volts $V$, $L$ is inductance in henries $H$, $i$ is current in amperes $A$, $R$ is resistance in ohms $\Omega$.

Unit Analysis

$[\frac{V}{m}] = [V]$

This appears to be consistent across many PDEs. What am I missing?

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I don't understand your “unit analysis”. What are $V$ and $M$? –  Harald Hanche-Olsen May 10 '13 at 19:32
    
Post modified... –  nick_name May 10 '13 at 19:35
    
$L, R$ should be 'per unit length' quantities. $u$ should be in volts, and $i$ in amps. –  copper.hat May 10 '13 at 19:36
    
Oh, I get it. $V$ is supposed to mean Volts, and $M$ is meters? But meters are written with a lowercase m. –  Harald Hanche-Olsen May 10 '13 at 19:36
    
@copper.hat So are you implying that in all PDEs where the units appear off, that the multiplicative constants must be dimensionally normalized? –  nick_name May 10 '13 at 19:38
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2 Answers

up vote 4 down vote accepted

The units are fine. $L$ is inductance per length unit and $R$ is resistance per length unit.

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Right, I guess the variable names are often not reflecting the units in these equations (implicit). –  nick_name May 10 '13 at 19:43
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Changes of units, in any equation, modify the coefficients. If different coefficients are subject to different scaling exponents (which is what it means to be "dimensionally wrong"), it means that for the equation to not depend on choice of units, the coefficients must be dimensionful, and in a specific way. There is a unique set of units for the coefficients making the equation dimensionally consistent, up to multiplying the entire equation by some unit, like going from A=B to A m^2 = B m^2.

Equations that accurately describe physical processes usually are not dependent on choice of units, so it is likeliest in any particular example that the coefficients have the necessary units to cancel apparent discrepancies in the dimensionality of the different terms. As the first answer states, this is true for the telegraph equation, and the principle is more general.

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I guess it depends on which two of harmed/harmful/harmless you consider it analogous to. Or formed/formful/formless, and the same suffices on shape. –  zyx May 11 '13 at 5:16
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