Fig. V-iii. (a) shows a series of aligned curves, y = cos(kx), differing in the value of the angular wavenumber (k), as shown. (b) shows superpositions of such curves, as described. From the bottom upwards, the lower three are sums of three, five and nine waveforms respectively, spanning the same range of k but with increasing numbers of intermediate values. Each superposition has been scaled by dividing by the number of contributing waveforms so, in effect, the value of the superposition wave function, Y, at any value of x, will be the mean of the values in the individual component waveforms.
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(5-6) Consider the individual components of the superpositions in figure V-iii. Each one is a function of both x and k (y = f(x,k) ). Now, let’s fix on one specific value of x and look at what happens as we vary k.
① Let's start by fixing x at a value quite close to zero, where we expect – based on what we have seen in figure V-iii – to see some constructive interference: taking x = 0.2π, sketch the graph of y = cos(kx) as a function of k, over a range of, say, k = -2.5 to k = +2.5.
② Now, imagine a series of closely spaced points spanning the range k=0 to k=0.8. Looking at the variation in the value of y over this range of k, mark on your graph a line corresponding approximately to y = <y>, where <y> is the mean value of cos(0.2πk) for this set of points.
③ Mark on your graph the area that is given by:
④ See if you can deduce the relationship between <y> and this area.
⑤ Now see if you can generalise: If we say the range of k is from k1 to k2 , and we generalise the function to f(k,x), what is the corresponding general relationship (for this specific value of x)?
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