Solution to Wave Equation via Dirichlet Kernel

In a subsequent chapter we shall study the inhomogeneous Helmholtz wave equation

$$(\nabla ^2+k^2)\psi =f(x,y,z) \quad .$$

It is amusing that the solution to this equation exhibits a property which is readily expressed in terms of a Dirichlet kernel and more generally in terms of a Fourier series. This property is so useful, physically fundamental, and deducible with so little effort that it is worthwhile to give a quick derivation. The property pertains to the field amplitude $\psi (x,y,z)$ when the inhomogeneity (``source'') of the wave equation is concentrated at, say, $2N+1$ sources

$$(x_n,y_n,z_n) \quad n=0,\pm1, \dots , \pm N \quad .$$

In that case, the governing Helmholtz equation is

$$\left[ \frac{\partial ^2}{\partial x ^2} +\frac{\partial ^2}{\par... ...t] \psi (x,y,z)= -\sum^N_{n=-N} A_n~\delta (x-x_n) \delta (y-y_n)\delta (z-z_n)$$ (23)

One can readily show that the solution to this inhomogeneous wave equation is

$$\psi (x,y,z)= \frac{1}{4\pi}\sum^N_{n=-N} A_n \frac{e^{ikR_n}}{R_n}~~.$$

Each term in this solution is proportional to the strength of each corresponding localized source of the wave equation. The quantity

$$R_n =\sqrt{ (x-x_n)^2 +(y-y_n)^2 +(z-z_n)^2}$$

is the distance between $(x,y,z)$ , the point where the field is observed, and $(x_n,y_n,z_n)$ , the location of the $n$ th source point.

We now consider the circumstance where this distance is large. More precisely, we assume that if the sources are distributed along, say, the $x$ -axis,

$$(x_n,y_n,z_n)=(x_n,0,0) \quad n=0, \pm 1,\cdots ,\pm N \quad$$

and the amplitude is observed at, say,

$$(x,y,z)=(x,0,z)$$

so that the distance is

$$R_n =\sqrt{r^2 -2xx_n +x_n^2} \quad \hbox{where} ~~r^2=x^2 +z^2 \quad,$$

then ``large distance'' means that $r$ is so large that

$$\frac{x_n}{r}~\ll ~ 1\quad n=0, \pm 1,\cdots ,\pm N \quad .$$

For such distances the solution has the form

$$\psi (x,y,z)= \frac{1}{4\pi}\sum^N_{n=-N} \frac{A_n}{R_n}\exp \{ikr-ikx \frac{x_n}{r} +ikx_n \frac{x_n}{2r} +(\hbox{negl.~terms}) \}$$ (24)

The long distance assumption can be strengthened by demanding that both

$$\frac{x_n}{r}~\ll ~ 1 \quad \hbox{and}~ kx_n \frac{x_n}{2r}~\ll ~1$$ (25)

be satisfied. This strengthened assumption is called the ``Fraunhofer approximation''. Under this approximation the third contribution to the phase in the exponential of the solution, Eq.(2.4), is so small that this contribution can also be neglected. As a consequence the solution assumes the perspicuous form

$$\psi (x,0,z)= \frac{e^{ikr}}{4\pi r} \sum^N_{n=-N} A_n \exp{ \{ -ikx_n \frac{x}{r} \} } \quad .$$ (26)

Suppose the $2N+1$ sources are equally spaced and hence are located at

$$x_n =\triangle x~n\quad n=0, \pm 1,\cdots ,\pm N \quad .$$

In that case the solution is a $(2N+1)$ -term Fourier series whose coefficients are the source strengths $A_n$ in Eq.():

$$\psi (x,0,z)= \frac{e^{ikr}}{4\pi r}\sum^N_{n=-N} A_n e^{-in\theta }; \quad \theta \equiv \frac{x}{r} (k\triangle x) \quad .$$ (27)

We thus have proved the following fundamental

Theorem 21.1 (Fraunhofer-Kirchhoff)
At sufficiently large distances expressed by Eq.(2.5), the solution to the inhomogeneous Helmholtz wave equation, Eq.(2.3), has the Fourier form Eq.(2.6) whose spectral coefficients are the strengths of the inhomogeneities in that wave equation. If these inhomogeneities are equally spaced, then the solution is a Fourier series, Eq.(2.7).

When all the sources have equal strength, say $A$ , then the solution is proportional to the Dirichlet kernel,

$$\psi (x,0,z)=A \frac{e^{ikr}}{2 r}\delta _N (\theta); \quad \theta \equiv \frac{x}{r} (k\triangle x) \quad ,$$ (28)

which varies with $\theta$ in a way given in Fig. 2.1.

For the sake of completeness it is necessary to point out that the Fraunhofer approximation can always be satisfied by having the separation between the ``observation'' point and the finite source region be large enough. If it is not satisfied, i.e., if

$$\frac{x_n}{r}~\ll ~ 1 \quad \hbox{but}~ kx_n \frac{x_n}{2r}\approx 1 ~\hbox{or}~ kx_n \frac{x_n}{2r} >1 \quad ,$$ (29)

then the third contribution to the phase of the solution, Eq.(2.4), cannot be neglected. This less stringent assumption is called the ``Fresnel approximation''.

Exercise 21.1 (SHIFTED INTEGRATION LIMITS)
Suppose that $f(x+2\pi)=f(x)$ is an integrable function of period $2\pi$ . Show that

$$\int\limits^{2\pi+a}_a f(x)dx=\int\limits^{2\pi}_0f(x)dx$$

where $a$ is any real number.