FOURIER'S THEOREM 97 



the plucked string ( 26). The differential equation, and the 

 terminal conditions, are satisfied by the finite series 



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y=A l sm -j- cos -j- 4- A z sin j- cos -. \- 



. mirct 



+ A m sm j- cos j , ...(1) 



each term of which represents a normal mode of vibration. 

 This makes the initial velocity zero, whilst the initial form is 



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y = A 1 sm -j- 4- A 9 mn ; + ... + A m smj . ...(2) 



II V 



The question we now have to consider is, how to determine the 

 coefficients A lt A z , ... A m so that (2) may represent, as closely 

 as may be, a prescribed initial form 



y-/() ......................... (3) 



There are many reasons why, from the physical point of 

 view, we may be content with an approximate solution of the 

 problem. Leaving aside such questions as the resistance of the 

 air and the yielding of the supports at the ends of the string, we 

 have still to remember that in substituting a mathematical line 

 of matter, capable only of exerting tension, we have considerably 

 over-idealized the circumstances. In the higher normal modes, 

 at all events, the imperfect flexibility, and the uncertainty as to 

 the true nature of the terminal conditions, render this representa- 

 tion somewhat inadequate, so that a solution which professes to 

 determine these modes accurately is open to the criticism that 

 it attempts too much. Again, the assumed initial form in 

 which two straight pieces meet at a point, is one which can 

 only be approximately realized ; if we go too far in this direction 

 we should produce a permanent bend, or kink, in an actual 

 wire. 



The determination of the coefficients in the finite series 

 (2) will depend on the kind of approximation aimed at. For 

 example, we might divide the length of the string into ra + 1 

 equal parts, and choose the coefficients 'so that the functions (2) 

 and (3) should be equal at the m dividing points. The curves 

 represented by these equations will then intersect in m points in 

 addition to the ends. Another method is to make the sum of 



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