What function could I use to approximate a curve with this plot?
The function ideally will involve sine and have its envelope defined by different exponentials of the form $y = ae^{bx}$ and $y = ce^{dx}$.
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If $f$ and $g$ are any two functions with $f<g$ you can get a sinewave-ish thing oscillating between $f$ and $g$ by considering $$\frac{f(x)+g(x)}2+\frac{f(x)-g(x)}2\sin(x).$$
Or $$\frac{f(x)+g(x)}2+\frac{f(x)-g(x)}2\phi(x),$$if $\phi$ is any function that oscillates between $-1$ and $1$.
David C. Ullrich
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Just so the answer is clear to anyone else, here $f(x) = ce^{dx}$ and $g(x) = ae^{bx}$. – Jon Aug 16 '18 at 18:44
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$$c\exp\left(dx+(bx-dx+\ln a-\ln c)\sin^2 x\right)$$ Edit $$ce^{dx}+(ae^{bx}-ce^{dx})\sin^2(kx)$$
Empy2
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It looks as good an option as any. In theory we could replace $\sin^2 x$ with, say, $\sin^{2k}(x+\phi)$. – J.G. Aug 16 '18 at 18:16
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1Take logs, get straight lines; put the sinewave between them; and exponentiate to put them back – Empy2 Aug 16 '18 at 18:52