The concept of a generalized function

Introduction

Let y(t) be the displacement at time t of a mass m that is attached to a spring having the force constant k as shown in Fig. 7.1. We assume that the mass is at rest in its equilibrium position [i.e., y(t) = 0] for all t ≤ 0. At time t = 0 we begin to subject the mass to an impulsive driving force

When the duration ∈ < 0 is “small,” this force simulates the tap of a hammer that transfers the momentum

to the mass and “rapidly” changes its velocity from y′(0) = 0 to y′(∈) ≈ p/m.

You should have no trouble verifying that

is a twice continuously differentiable function that satisfies the forced differential equation

my″(t) + ky(t) = f∈(t)

for the motion (except at the points t = 0, t = ∈ where y″∈ is not defined), see Ex. 7.1. Here

so that sin(ωt), cos(ωt) are solutions of the unforced differential equation

my″(t) + ky(t) = 0.

Now as ∈ → 0+, the response function (2) has the pointwise limit

and it is natural to think of y0 as the response of the system to an impulse

of strength

that acts only at time t = 0 as illustrated in Fig. 7.2. The physical intuition is certainly valid, and such arguments have been used by physicists and engineers (e.g., Euler, Fourier, Maxwell, Heaviside, Dirac) for more than two centuries. And for most of …