Point Force Applied to the System

A unit force localized at a point is a unit force distributed over an $\varepsilon$ -interval surrounding the given point. The density of this distributed force is inversely proportional to $\varepsilon$ . More precisely, one defines

$$\delta_\varepsilon (x-\xi) \equiv\left\{ \begin{array}{lcl} ... ...ght. \left[ \frac{\textrm{(Force)}}{\textrm{(length)}}\right],$$

the finite approximation to the Dirac distribution, whose integral, the total force,

$$\int^b_a \delta _\varepsilon (x-\xi)~dx=\int^{\xi+\varepsilon /2} _{\xi-\varepsilon /2} \delta _\varepsilon (x-\xi)~dx=1~~,$$

is unity. Let us apply such a force density,

$$F(x)=\delta _\varepsilon (x-\xi)~~,$$

to a string with constant horizontal tension $T$ . The response of this string is governed by the Poisson equation (4.13), namely

$$T\frac{d^2 G}{dx^2}=-\delta _\varepsilon (x-\xi)~~.$$

Figure 4.2: A distributed unit force applied to a string with tension $T$ . The force is concentrated in an interval of size $\varepsilon$ . As $\varepsilon \to 0$ , the response of the string tends towards the Green's function $G(x;\xi )$ .

Note that the sum of all the vertical forces is necessarily zero. This equilibrium condition is expressed by the statement that (see Fig. 4.2)

$$TG'(\xi^+)-TG'(\xi^-)+1=0$$

or by

$$\boxed{ G'(\xi^+)-G'(\xi^-)=-\frac{1}{T} } ~~,$$

which is known as the jump condition. Here we are using the notation $\xi ^\pm =\xi \pm \varepsilon /2$ with $\varepsilon$ neglegibly small. The other condition that the response $G$ must satisfy is that it be continuous at $x=\xi$ , i.e.

$$\boxed{ G(\xi^+)-G(\xi^-)=0}~~.$$

This continuity condition, the jump condition, together with the boundary conditions lead to a unique response, the Green's function $G(x;\xi )$ of the string.(Why?)