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Dimensions of arguments of trigs and exponentials Watch

    • Thread Starter

    Quick question, do the arguments of trig functions and exponential functions have to always be dimensionless? I've been given four functions for y

    y = A sin (x - ut)
    y = A sin (x + ut)^0.5
    y = A sin (u - (x/t))
    y = A exp [-(x-vt)²]

    and I have to evaluate whether they're acceptable solutions based upon putting them back into the wave equation

    to see whether they work (which I've nearly done, the last one's a bit horrible), then the second step is evaluating this through 'another way', which I presume is looking at the argument. Problem is none of these have dimensionless arguments - they're [m], [m]^0.5, [m] and [m]² respectively, so does this mean none work? Because the maths with the first one implied it worked (unless my calculus was wrong, which is a serious possibility, but I doubt all four would be wrong!)

    Oops, that wasn't such a quick question after all. I hope someone could help me though! Thanks if so!
    • PS Helper

    Say y has dimension Q, then the PDE gives QL^{-2} = \dfrac{1}{(L/T)^2}QT^{-2}, but the RHS is just equal to QL^{-2}, so whatever the dimensions of y are, it's fine. That's assuming I haven't missed anything of course... but I don't think I have.

    It seems clear enough that length has been non-dimensionalised in this question. Generally, there is supposed to be a wavelength L in these equations, e.g. \displaystyle y = A \sin \frac{2\pi (x - u t)}{L}.
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