po}f =

Note that our problem has now been reduced to solving an ordinary (rather than partial) differential equation. (This is the usual pattern in solving partial differential equations no matter what specific technique is used.) In Eq. (10-14) p and E are known constants whereas w (the frequency of vibration) is a constant as yet unknown. Applying our usual methods to (10-14) we get

[D2 + p;2Jf = 0 D ~ ~ (10-15) Roots = ±iw~ (I 0-16)

f = C sin ( ~ w x + cf>) (10-17) The constants of integration C and cf> must be found using the boundary conditions and in this process the value of frequency w will also come out. We may write for u

:: = (C coswt) cos ( ~wx + cf>) ~w (10-19) 0 = (C cos wt) ~wcos cf> (10-20)

If we choose C = 0 to satisfy (10-20) we get the trivial solution u(x, t) = 0; thus it must be that cos cf> = 0, which occurs for cf> = ±rr /2, ±3rr /2, ±5rr /2, etc.