Infinite String as the Limit of a Finite String

It seems that the properties and behavior of an infinite string are irreconcilably different from those of a finite string. However, it is possible to consider the former as a limiting form of the latter. This fact, together with the the associated role of the square root function, is brought out by comparing the solutions of two simple strings, both of length $\vert\ell\vert$ . One extends into the positive, the other into the negative $x$ -direction:

Problem:

Consider the Green's functions $g_+$ and $g_-$ of the two linear systems with symmetrically located domains.

(i)

The first one is governed by the differential

$$\frac{d^2g_+}{dx^2}+\lambda g_+ =-\delta(x-\xi)\quad 0<x,\xi<\ell$$ (462)

$$x=0$$

$$g_+(x=0) =0~,$$

$$x=\ell$$

$$b_1 g_+(\ell ) + b_2 g_+'(\ell ) =0~.$$ (463)

(ii)

The second one is governed by the same differential equation, but on a domain which extends symmetrically to the left:

$$\frac{d^2g_-}{dx^2}+\lambda g_- =-\delta(x-\xi)\quad \ell<x,\xi<0$$ (464)

$$x=0$$

$$g_-(x=0) =0~,$$

$$x=\ell=-\vert\ell\vert$$

$$b_1 g_-(\ell ) + b_2 g_-'(\ell ) =0~.$$ (465)

Thus the domain of $g_+$ is $[0,\ell]$ , while that of $g_-$ is $[-\vert\ell\vert,0]$ .

Compare

the respective Green's functions $g_+$ and $g_-$ in the limit as $\vert\ell\vert\rightarrow\infty$ .

Remark: We shall find three noteworthy results: First of all, each Green's function has two asymptotic limits: one is square integrable, the other is not. Second, these limits are entirely independent of the mixed Dirichlet-Neumann conditions, Eq.(4.64), (4.67). Finally, each of these limits accomodates two radiation conditions: outgoing and incoming.

Solution:

(i) The first step consists of constructing the Green's function in the usual way. This task is based on

$$u_1= \sin \lambda^{1/2} x~~~,$$

which satisfies $u_1(0)=0$ , and on

$$u_2=A\cos \lambda^{1/2} x +B \sin \lambda^{1/2} x$$

whose coefficients $A$ and $B$ are related so as to satisfy the given boundary conditions, Eqs.(4.64) and (4.67) at $x=\ell$ . These boundary conditions demand that

$$A(\underbrace{ b_1 \cos \lambda^{1/2} \ell -b_2 \lambda^{1/2} \si... ...b_1\sin \lambda^{1/2} \ell + b_2 \lambda^{1/2} \cos \lambda^{1/2} \ell}_D ) =0$$

or

$$A=-B\frac{N}{D}~~.$$ (466)

To construct the Green's functions

$$g_{\pm} (x;\xi)=\left\{ \begin{array}{l} \frac{-1}{c}~ u_1(x)~u_2... ..._1(\xi)~u_2(x)\quad\textrm{when }\vert\xi\vert<\vert x\vert \end{array}\right.$$

we need the Wronskian determinant

$$c = \left\vert \begin{array}{cc} u_1&u_2\\ u'_1 & u'_2 \end{array} \right\vert$$

$$= \left\vert \begin{array}{cc} \sin \lambda^{1/2} x & A\cos \lambda... ...in \lambda^{1/2} x +\lambda^{1/2} B\cos \lambda^{1/2} x \end{array} \right\vert$$

$$= -\lambda^{1/2}~A$$

$$= \lambda^{1/2} ~B~\frac{N}{D}$$

Using Eq.(4.68), write down the Green's functions. For $\vert\xi\vert<\vert x\vert$ one has

$$g_{\pm} (x;\xi) = \frac{-1}{c}~ u_1(\xi)~u_2(x)$$

$$= \frac{-1}{ \lambda^{1/2} ~B~\frac{N}{D} }~\sin \lambda^{1/2} \xi \left[ (-)B\frac{N}{D}\cos \lambda^{1/2} x +B\sin \lambda^{1/2} x \right]$$

$$= \underbrace{ \frac{\sin \lambda^{1/2} \xi }{\lambda^{1/2} } \cos ... ...a^{1/2} x }{\frac{N}{D} } }_ {\textrm{\lq\lq solution~to~the~homogeneous~problem''}}$$ (467)

(ii) The second step consists of taking the limit of this Green's function as the boundary $\vert\ell\vert \to \infty$ . Note that there is no $\ell$ -dependence whatsoever in the ``particular solution'' part of $g_{\pm}(x;\xi)$ . In fact, it is totally independent of the specific boundary condition that has been imposed at $x=\ell$ .

This is different for the second part, the ``solution to the homogeneous equation''. It depends on the boundary condition by virtue of the ratio

$$\frac{N}{D} = \frac{ b_1\sin \lambda^{1/2} \ell + b_2 \lambda^{1/... ...ll } {b_1 \cos \lambda^{1/2} \ell -b_2 \lambda^{1/2} \sin \lambda^{1/2} \ell }$$

and hence also on the length $\ell$ . However, in the limit as $\ell \to \infty$ something remarkable happens: the ratio $N/D$ , and hence the Green's function 4.69, becomes independent of the mixed Dirichlet-Neumann boundary condition at $x=\ell$ . In order to determine the value of the limiting ratio

$$\lim_{\ell \to \infty} \frac{N}{D}~~,$$

set

$$\lambda^{1/2}=\alpha +i\beta~~,$$

so that

$$\sin \lambda^{1/2} \ell = \frac{1}{2i} (e^{i\alpha \ell} e^{-\beta\ell}- e^{-i\alpha \ell} e^{\beta\ell})$$

$$\cos \lambda^{1/2} \ell = \frac{1}{2} (e^{i\alpha \ell} e^{-\beta\ell}+ e^{-i\alpha \ell} e^{\beta\ell})~~.$$

Introduce these expressions into the ratio $N/D$ . It is evident that this ratio has no limit when $\beta=0$ . However, for $\beta \ne 0$ one finds that

$$\lim_{\beta\ell \to \infty} \frac{N}{D} =\lim_{\beta\ell \to \infty} \frac{ \frac{b_1}{2i}(-)e^{-i\alpha\... ...a\ell}e^{\beta\ell} +\frac{b_2\lambda^{1/2}}{2i}e^{-i\alpha\ell}e^{\beta\ell} }$$

$$=i$$
and

$$\lim_{\beta\ell \to -\infty} \frac{N}{D} =-i~.$$

Applying these two limits to the Green's function, Eq.(4.69), one obtains

$$\lim_{\beta\ell \to \infty}g_{\pm} (x;\xi) = \frac{\sin \lambda^{1/2} \xi }{\lambda^{1/2} } e^{i\lambda^{1/2... ...or }\ell\to\infty\\ \textrm{lower sign for }\ell\to -\infty \end{array} \right.$$ (468)
and

$$\lim_{\beta\ell \to -\infty}g_{\pm} (x;\xi) = \frac{\sin \lambda^{1/2} \xi }{\lambda^{1/2} } e^{-i\lambda^{1/... ...or }\ell\to\infty\\ \textrm{lower sign for }\ell\to -\infty \end{array} \right.$$ (469)

The allowed values of $\lambda^{1/2}=\alpha+i\beta$ are no longer detrmined by the Dirichelet-Neumann conditions. Instead, for an (in the limit) infinite system the two parts of $\lambda ^{1/2}$ are determined by the two boundary conditions

1. square integrability of $g_{\pm}$ , and
2. outgoing (or incoming) signal propagation condition.

The first condition is met by fulfilling the reqirement that

$$\textrm{for }g_+:~\beta>0 \textrm{i.e. }\lambda \in \textrm{1st Riemann sheet of } \lambda^{1/2}$$

$$\textrm{for }g_-:~\beta<0 \textrm{i.e. }\lambda \in \textrm{2nd Riemann sheet of } \lambda^{1/2}$$

The second condition for outgoing (resp. incoming) signal propagation is met by fulfilling the reqirement that

$$\textrm{for }g_+: \lambda^{1/2}=\left\{ \begin{array}{rl} \vert a\vert+i\epsilo... ...\textrm{incoming (from the \lq\lq right'') wave} \end{array} \right.$$

$$\textrm{for }g_-: \lambda^{1/2}=\left\{ \begin{array}{rl} -\vert a\vert-i\epsil... ...&\textrm{incoming (from the \lq\lq left'') wave} \end{array} \right.$$

These are the mathematical conditions on $\lambda^{1/2}=\alpha+i\beta$ for an asymptotically infinite string. Inserting them into Eqs.(4.70) and (4.71), one finds that the Green's functions satisfying these conditions are

$$\left. \begin{array}{ll} g_{\pm}^{outgoing}&=\displaystyle \f... ...es e^{-i\omega\tau}~~~\textrm{for }\vert\xi\vert<\vert x\vert~.$$

Figure 4.13: Instantaneous amplitude profiles of waves with outgoing and incoming phase velocities. In the first case the source emits energy; in the second case the source absorbs energy.

Figure 4.13 depicts the real part of the graph of these functions.