Mathematical Physics Vol 1

7.8 Examples

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– the initial velocity (second initial condition, pertaining to the derivative of the variable) ∂ u ∂ t t = 0 = g ( r ) . (7.375) Let us now apply the method of separation of variables, starting from the assumption that u ( r , t )= R ( r ) T ( t ) . (7.376) Under this condition, the initial equation (7.372) is decomposed to two ordinary differential equations (7.378) where k is a constant, undetermined for now. In previous relations, a point above a variable denotes its derivative by the time t , and () ′ the derivative by variable r . We have also introduced a new constant λ 2 = c 2 k 2 . Observe now the first equation. Let us introduce the substitution s = kr ( s is the new variable), where R ′′ + 1 r R ′ + k 2 R = 0 , (7.377) ¨ T + c 2 k 2 G = 0 , This equation represents the Bessel equation (5.45) (for ν = 0), whose general solution (5.98) is R = C 1 J 0 ( s )+ C 2 Y 0 ( s ) . (7.382) Here J 0 is a Bessel function is of first type zero order, and Y 0 a Bessel function of second type zero order. Given that the displacement of the points of the membrane u is always finite, Y 0 → ∞ when s → 0 ⇒ C 2 = 0. It is obvious that C 1̸ = 0, as otherwise R ≡ 0, which would be a trivial solution. Without losing generality, we shall assume that C 1 = 1, from where we obtain R = J 0 ( s )= J 0 ( kr ) . (7.383) On the border r = r 0 we have u ( r 0 , t )= R ( r 0 ) · T ( t )= 0 ⇒ R ( r 0 )= 0 ⇒ J 0 ( kr 0 )= 0 . (7.384) The Bessel function J 0 has an infinite number of real zeros ( α 1 , α 2 ,..., α n ,... ), and thus, from equation J 0 ( kr 0 )= 0 it follows that α m = kr 0 ⇒ k = α m r 0 = k m , m = 1 , 2 ,... (7.385) R ′ = d R d s d s d r = k d R d s , (7.379) R ′′ = d d r k d R d s = d d s k d R d s d s d r = k 2 d 2 R d s 2 . (7.380) By substituting these relations into equation (7.377) we obtain d 2 R d s 2 + 1 s d R d s + R = 0 . (7.381)