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chapter6.page11

# chapter6.page11 - Definition 4.1 The Dirac delta function...

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4. DELTA FUNCTION 11 The graph contains solutions for the following three cases, (1) No external force is applied(blue curve). (2) Force of sin(2 t ) is applied at ﬁrst π seconds and turnoﬀ(red curve). (3) Force of sin(2 t ) is constantly applied (green curve). The initial conﬁguration is that the spring is at equilibrium position x (0) = 0 and is compressed at 1 unit speed (feet/s) x 0 (0) = - 1 . From the graph we can clearly see that when the force is turnoﬀ at t = π its eﬀect is immediately gone(as shown below)! 4. Delta Function Delta function is one of so-called generalized functions, which are not functions in ordinary sense but as an operators that sometimes can be represented by ordinary functions.
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Unformatted text preview: Definition 4.1 . The Dirac delta function at a, δ a ( t ) , is an op-erator that satisﬁes Z ∞ g ( t ) δ a ( t ) dt = g ( a ) for any continuous function g ( t ) . In Physics, if f ( t ) , a ≤ t ≤ b is a force that acts only during a short period of time interval [a, b], the impulse p of force f ( t ) is computed as Z b a f ( t ) dt. δ a ( t ) can be viewed as an instantaneous unit impulse that occurs pre-cisely at the instant t = a. And pδ a ( t ) is an instantaneous p units impulse that occurs precisely at the time t = a. δ a ( t ) is an important function used in modelling real phenomena, for example,...
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