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Figure 2.20. The key to finding how the amplitudes of different modes are initially set is understanding the cosine function. In these examples, L = 10 and k = 1 or k = 6. (a) The functions y = cos(nx/L) or y = cos(6nx/L). (b) The product cos(nx/L)cos(6nx/L). (c) The square cos2(nx/L). (d) The square cos2(6nx/L).

These expressions tell us how to find the initial value of the amplitude of each mode. That is, recall that we know n(x, 0). But we have also represented n(x, 0) by ^ k=1Bk(0)cos(kpx/L). If we multiply both n(x, 0) and its series representation by cos(kpx/L), integrate between 0 and L, and take advantage of the relationships between the integrals of cosine, we will obtain

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