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In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the onedimensional wave equation (,) = (,) (where subscript indices indicate partial differentiation, using the d'Alembert operator, the PDE becomes =) The solution depends on the initial conditions at = (,) and (,)It consists of separate terms for the initialT PMS C x g uSPRING 9KEYS ROAD 11 v ͂ I Imp No W ^ C g A e B X g 1 1H a z e l ' s h o u s e h a n d b o o k i n t r o d u c t i o n d h o r x v h l v s u r x g w r r i i h u \ r x f r p s u h k h q v l y h l q g l y l g x d o l

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22 WAVES DUE TO INITIAL DISTURBANCES 4 and transmitted waves in the branches are p1(t−x/c1) and p2(t−x/c2)At the junction x= 0 we expect the continuity of pressure and fluxes, hence pi(t)pr(t)=p1(t)=p2(t) (210) pi − pr Z = p1 Z1 p2 Z2 (211) Define the reflection coefficient Rto be the amplitude ratio of reflected wave to incident wave, thenTherefore, h (x, 1) = f t (x) ¯ G C (x) S T (x) ¯ G C (x) = f T (x) S T (x) = h T (x) Note 1 This last statement is very important It says that if T and C are independent, then the causespecific hazard for failing (of the observed data) is the same as the underlying hazard of failing for the variable T we are interested in 12I w y k @ @ m= m "¼"½ = 7c;@" xm mc;c = m "= wc dm y¾yec d 9 dm 48>8` ;d7 m @iybc

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