Wavelet Discovery

RANT #4: The simplest wave you can imagine would have one crest followed by a single trough with both ends extending thru all of space, say from plus infinity to minus infinity. Its wavelength can't be defined like an ordinary sine wave, say from crest to crest because there's only one crest.

Now a very interesting property of this wave, like the one shown in blue, is that if you place wavelets on either side at a distance of 3.14...(PI) times the unit length of X and add them together they form an almost perfect sine-like wavelength of 2X, that's half the unit length X is measured in. All elementary functions like sine and exponentials can be written as a sum of terms called a power series. As it turns out the series for sine(2X) and the exponential have the same terms (powers of X) in the variable X but different coefficients to those terms. It hard in this format to write out the series but we'll give it a shot by using the carat (^) to indicate an exponent: using Y so x means "times" then
Y e^(-Y^2) = Y - Y^3 + Y^5/2 - Y^7/3x2 + Y^9/4x3x2 - Y^11/5x4x3x2 + ...
and sin(2Y) = 2Y - (2^3/3x2)Y^3 + (2^5/5x4x3x2)Y^5 - 2^7/7x6x5x4x3x2)Y^7 + ...
Now if you use a calculator to (let Y=1 to keep it simple) calculate the left and right side (those demoninators of the coefficients are factorials (!) on the calculator, 5!=5x4x3x2x1), you'll find that 0.36788 = 0.3667 and 0.90929 = 0.90793 and the right sides will get closer to the left sides as more terms are added. Clearly the amplitude of the exponential is about 40% of that of the sine wave. I have found no mention of this "trick" anywhere. You'd think that if you told somebody who's interested in these things about this they'd be interested - guess again! There's more details and graphical illustration at (Ebony Dungeon) Webaddress. We'll chew on this in future blogs but enough for now, it's something to think about, I hope!