Fritz Gassmann (1899–1990) was a Swiss mathematician and geophysicist. Gassmann is the eponym for the Gassmann triple and Gassmann's equation. His PhD advisors at ETH Zurich were George Pólya and Hermann Weyl. He was a geophysics professor at the ETH Zurich. Gassmann was honored with an SEG Special Commendation in 1990.
Fritz Gassmann published his classic paper “Elastic waves through a packing of spheres” in GEOPHYSICS in 1951. At that time I was a member of a motley group of Gassmann’s graduate students—one mathematician, one theoretical physicist, one experimental physicist, one civil engineer, and two geologists (there was no undergraduate geophysics program in those days)—who were blissfully unaware of its fundamental significance. This is hardly a surprise considering our inexperience.
The broad background of our graduate student group proved helpful in many ways. Just two incidents may illustrate the point. The other geologist had to deal with a well rounded clastic sediment, and he figured that he should be able to deduce the grain size distribution from just a single thin section or polished surface. He enlisted the help of the mathematician. The latter kept scratching his head for quite a while (remember this was 1950) but did come up with the solution eventually. My own field was gravity measurements, and I was dissatisfied with the very-near station terrain corrections (the middle- and far-field sections were well covered by graticules in the literature). I put the theoretical physicist to work and he came up with slanting “cake pieces” which produced satisfactory results. Gassmann himself, broadly based in geophysics, developed graticules for the evaluation of the gravity effects of two- and three-dimensional bodies. It was a laborious undertaking in those precomputer days. Yet, it enabled us to get away from such artificial bodies as spheres, cylinders, and half-plates.
The first fallout of Gassmann’s work was a minor correction by J. E. White (GEOPHYSICS, 1953). Gassmann, a stickler for precision and accuracy, much like the late King Hubbert, was mortified that a √2 had escaped the proofreading graduate student, and the culprit got moved to the backbench of the coffee room.
Stickler as he was, such terms as anisotropy and heterogeneity figured prominently in his lectures, and he made sure that all students had a full understanding of their significance. In those days, homogeneity and isotropy were the order of the day to reduce the mathematics to manageable proportions. Gassmann made it clear that such computations delivered guidelines rather than precise values. Only in 1959, when carrying out refraction work in North Africa, were the French forced to consider anisotropy (Dunoyer et al., Geophysical Prospecting) for the highly slanted raypath (reflection was essential vertical incidence only). When I started to teach geophysics in Calgary in 1966 and used these concepts, my colleagues shook their heads and commented that only a “European square” could be interested in such minor details. This was a general view that prevailed well into the 1970s as is best documented by the comment on anisotropy found in the 1976 text Applied Geophysics by Telford et al. We have come a long way, and today these terms are household words in geophysics, reservoir engineering, and hydrology. Later reactions were more favorable and the “Gassmann equation” found a permanent home in Bob Sheriff’s Encyclopedic Dictionary of Exploration Geophysics. One may well ask the question as to how it came about that Gassmann, with his training as a mathematician, chose to write a paper regarding the rack as a porous medium. The answer to this question is both interesting and relevant in our times.
Gassmann, contrary to many colleagues, never sat next to God. He was a shy man and played his cards close to his chest. As a result, some of the following comments are somewhat speculative. Gassmann’s interest in geophysics was triggered by a two-year stint, after graduation, at the Swiss Federal Earthquake Service. Later he taught mathematics at the Aarau high school (most high-school teachers in Switzerland had PhDs), and Gassmann continued his activities in geophysics as a hobby. In 1928 he started giving lectures on geophysics at the Swiss Federal Institute of Technology (ETH). In 1942 he was elected associate professor and head of the Institut für Geophysik at ETH. For a country almost devoid of natural resources this is an amazing early emergence of geophysics (Heiland came to the Colorado School of Mines in 1926). It seems that the staunch promoter of geophysics in the earth science curriculum at this time was my undergraduate professor of petrography, Paul Niggli, obviously a man of remarkable foresight.
Let us back up to Gassmann’s 1951 paper that deals with packing of spheres. This idea is no doubt borrowed from Niggli. In his 1948 textbook Gesteine und Minerallagerstätten, Niggli uses the packing of spheres as an approximation for a well sorted and well rounded clastic sediment. This procedure has the advantage that it can be handled in a mathematical manner. It also demonstrates that such a material in its loosest configuration has a porosity of 47.6% and in its densest packing the porosity is 25.9%. Unrelated to the problem at hand this immediately demonstrates that such a material cannot be compacted to less than 26% porosity unless it is crushed or contaminated (thus becoming poorly sorted). This purely theoretical consideration has now found a practical application in the new truck runaway lanes.
Gassmann used Niggli’s model of Ping-Pong balls as Figure 2 in his 1951 paper. For his purpose it permitted the calculation of seismic wave velocities because the grain contacts are clearly defined and ordered. The packing of spheres is both a fairly realistic model of a well sorted and rounded clastic sediment and also amenable to rigorous analysis. It is deplorable that it is not given more exposure in current textbooks. Given the above facts there can be little doubt that communication between Niggli and Gassmann was instrumental in producing this paper that is basic to modern geophysics. As the Americans would say: “Niggli got the ball and passed it to Gassmann, who ran with it.” This is an early success story for the interdisciplinary approach.
In times when we no longer have geophysicists or seismologists, but migrators, inverters, AVO analysts, aquisitioners, anisotropists, interpreters, etc., we need the intradisciplinary in addition to the interdisciplinary dialog. The Gassmann paper shows that it pays to talk to each other.