- #1

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If there is a theoretical string (no thickness, etc.) that is non-stretchable tied to two endpoints and is long enough to be able to form a (taut) sin wave (say, y= sin x from x=0 to pi), then the same string makes a new sin wave y= A sin (nx) for an integer n, is there any relatively simple closed-form way to calculate A (as a function of n)? For example, a brute force attack to compare n=1, A=1 to n=2, gives the ratio A=

(2/5)

^{½}E(½)/E(4/5), where E is the elliptical integral of the second kind with parameter m=k

^{2}, making it rather more complicated than desired. If there is no simpler alternative, OK, but it would be nice if there were. Thanks.