# Did Gauss anticipate Kirchhoff's diffraction formulation in 1836?

Clemens Schaefer says, in his treatise "Über Gauss' physikalische Arbeiten", that for a very short time in the years 1835/1836 Gauss showed interest in the new diffraction experiments of Fresnel and Fraunhoffer, and he wrote one note. After making Google Translate on the relevant passage from Schaefer's treatise, it reads like the following:

In 1835 the work of the Speyerer Professor F.M.Schwerd appeared: The diffraction defects were analytically developed and represented in pictures from the fundamental laws of the theory of undulation, which had a very great significance for the fixation of the undulation theory. In the meantime Schwerd had put together a "diffraction apparatus" which would enable his relatives to repeat his experiments. Regarding the importance of the problem, it is not strange that Gauss was also busy on it; He wrote to Schumacher on June 7, 1836: "In the last week I have been busy with optical, especially with the very essentive diffraction experiments. Schwerd has done quite a great deal in this field to deduce from a principle these highly varied phenomena, to which he first opened the way. In the meantime, however, there is still a great deal left until the theory can be regarded as complete and exhausting". On July 23, 1836, Gauss also informs Olbers that he has acquired a SHERWER apparatus in order to try diffractive experiments. "These objects become almost as interesting to me as magnetism and galvanism. But now I've had to break it all by force, since I do not have the time". Unfortunately, the split was final: Gauss never had the leisure to return to the diffraction problem. From the estate we have received some notes on General formulas for the effect of a luminous point P on a point p, which apparently come from the same time (1835/36).

After this he says: "The first of these notes is interesting in several respects", and examines the Gaussian attempt to give exact mathematical formulation to the Huygens-Fresnel principle. Gauss's integral formula describes a similar scenario like the one in Kirchhoff's description - wave disturbance of a spherical monochromatic wave passes through an opening in an opaque screen. His formula reflects the basic intuition that waves propagate in a certain direction, while the basic Huygens's principle describe points on a wavefront as sources of spherical waves (without the preferred direction). According to Schaefer, Gauss's formula is the following:

$$\int_S\frac {{ds}}{{Rr}}\frac {{e^{ik(R + r)}(\cos (R,v) - \cos(r,v))}}{{\sin (\omega)}}$$

where $S$ in the opening region, $ds$ an element of area, $R$ is the distance of point in the opening to the source of the wave, $r$ the distance from the point in the opening to the point $p$, $k$ the wavenumber, and $v$ is the normal vector to the opening ($(R,v)$ and $(r,v)$ are the angles between $R$ and $r$ to $v$), and $\omega$ the angle between $R$ and $r$.

Kirchhoff's formula is the following:

$$-\frac {{iA}}{{2\lambda}}\int_S\frac {{ds}}{{Rr}} e^{ik(R + r)}(\cos (R,v) - \cos(r,v))$$

Where $A$ is the amplitude of the emanating wave at point $P_0$. The factor $\cos (R,v) - \cos(r,v)$ in both formulas is called the obliquity factor $K(\chi)$ and is responsible for the fact the waves don't travel backwards. The factor $i = \sqrt {{-1}}$ in Kirchhoff's formula means that the secondary waves are $\pi/2$ out of phase with the primary waves. The factor $\frac {{1}}{{\sin\omega}}$ is very strange and I don't know what Gauss meant in this factor. It seems to me that since there is much similarity in the two formulas, there is some stuff to discuss here. Schaefer also says that the whole Gaussian approach isn't complete since he eventually left these problems

This question is just about getting explanation to Gauss's note. There isn't very much information in the internet about Gauss's less known contributions to optics. If anyone wants to read what Schaefer says in his treatise, the relevant pages are 209-211 on his treatise (that is in Gauss's werke, band XI1).

• Does it matter to you that Kirchhoff’s formula is widely considered incorrect, anyway? E.g. Baker-Copson (1939, §II.1.2) or Born-Wolf (1959, §8.3.2) explain how it’s derived by solving the Helmholtz equation with boundary conditions that it ends up not satisfying. – Francois Ziegler Sep 14 at 6:04
• a) This question is really not very important to me - sometimes i'm trying my luck and asking questions that maybe someone will be able to answer quickly. I'm simply trying to cover the entire body of Gauss's lesser known contributions. – user2554 Sep 14 at 8:18
• b) evev if kirchoff diffraction theory not correct, it's still an advancement relative to the earlier approaches. – user2554 Sep 14 at 8:19
• By the way, the question i regard most urgent to answer is about the question of the volume of tetrahedron in hyperbolic geometry. I made tremendous efforts to bring it to the forefront of stackexchange discussions by asking it on HSM stackexchange, math stachexchange end even mathoverflow, but still with no success. I saw some of my questions attracted your attention, so if you really want to help me please help me with that. – user2554 Sep 14 at 8:25