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You probably learned that the Earth has a crust, a mantle, a liquid outer core and a solid inner core. The NCERT Class 11 chapter on the interior of the Earth notes that the deepest drilling has reached only about 12 km, so how do we know? Much of the answer comes from earthquake waves, and the inner core was found by a Danish seismologist who looked closely at signals other people treated as noise. This post explains who Inge Lehmann was, the evidence she used, a travel-time calculation in her style, and what her way of working can teach you.
Who was Inge Lehmann?
Lehmann was born in Copenhagen on 13 May 1888 and died there on 21 February 1993, aged 104, according to the Dansk Kvindebiografisk Leksikon. She studied mathematics at the University of Copenhagen, with a stay at Newnham College, Cambridge, and worked for years as a calculator for an insurance company.
In 1925 she became an assistant setting up seismographs for the Danish geodetic survey, learning the subject almost from scratch. In 1928 she became state geodesist and head of the seismic department of the new Geodetic Institute, the Dansk Biografisk Leksikon records, responsible for stations in Copenhagen and Greenland.
How do earthquake waves reveal the inside of the Earth?
The same NCERT chapter explains that P-waves are faster and can travel through solids, liquids and gases, while S-waves travel only through solids. Waves also bend when they pass into material where their speed changes.
That creates shadow zones. NCERT states that seismographs within 105° of the epicentre record both P and S waves, those beyond 145° record P-waves but not S-waves, and the band between 105° and 145° is a shadow zone for both. The missing S-waves beyond 105° show that part of the core is liquid. On the surface, 105° is a long way: with NCERT's radius of 6,378 km, the arc is 2π × 6,378 × 105/360, about 11,700 km from the epicentre.
What did Lehmann notice?
Bruce Bolt's Royal Society memoir of Lehmann, whose text is reproduced by UCLA, explains the puzzle. Beyond about 105° from an earthquake, P-waves went missing, as a liquid core predicts. Yet improved seismographs of the 1920s and 1930s picked up extra P-type waves arriving in the gap between about 105° and 142°, which a single uniform core could not explain. Some seismologists suggested the energy was diffracted around the core.
Lehmann's Danish and European stations were far from the big earthquakes of the South Pacific, so they recorded these waves well. Bolt notes that a damaging earthquake in New Zealand on 16 June 1929 was well recorded across Europe, and comparing those records she could see the unexplained onsets clearly. In her 1936 paper, titled simply P′, she wrote: "A hypothesis will be here suggested which seems to hold some probability, although it cannot be proved from the data at hand." The hypothesis was a small inner core in which waves travel faster.
How did a simple model make her case?
Bolt describes how she stripped the problem down. She assumed constant P-wave speeds, about 10 km/s in the mantle and 8 km/s in the core, so the rays became straight lines and travel times could be found with basic geometry. Then she added a small central core with its own speed and adjusted it until the predicted times matched the records. A speed of about 10 km/s and a radius of about 1,400 km worked.
You can try the same style of reasoning for the ray that goes straight through the centre. Use NCERT's radius of 6,378 km and core boundary at 2,900 km depth, so the core radius is 6,378 − 2,900 = 3,478 km.
Mantle: the ray crosses it twice, 2 × 2,900 = 5,800 km. At 10 km/s that takes 580 s.
Core with no inner core: 2 × 3,478 = 6,956 km at 8 km/s takes 869.5 s. Total: 580 + 869.5 = 1,449.5 s, about 24.2 minutes.
Core with a 1,400 km inner core: the outer core part is 2 × (3,478 − 1,400) = 4,156 km at 8 km/s, which is 519.5 s. The inner core part is 2,800 km at 10 km/s, which is 280 s. Total: 580 + 519.5 + 280 = 1,379.5 s, about 23.0 minutes.
The faster inner core makes this wave arrive about 70 s earlier. Real Earth speeds change with depth, so these are model numbers, but they show how a travel-time difference becomes evidence for a hidden layer. A common error is to forget that the ray crosses each shell twice. Wave speed and refraction at a boundary are covered in our post on the travelling wave equation.
The New Carlsberg Foundation summarises the result: she argued that the inner core is made of a different material with a higher wave speed, and numerous later studies showed it is solid. NCERT now teaches that the outer core is liquid, the inner core solid, and the core mostly nickel and iron.
What did she do after 1936?
She retired from the Geodetic Institute in 1953 but kept publishing; more than half of her 58 publications came after retirement, the Kvindebiografisk Leksikon says. She used signals from underground nuclear explosions and early electronic computers in the United States and Canada to model the upper mantle. Her last article appeared in 1987, the year she turned 99. Honours included election as a Foreign Member of the Royal Society in 1969 and an honorary doctorate from Columbia University in 1964; the American Geophysical Union founded an Inge Lehmann Medal in 1996.
What can a student take from how Lehmann worked?
Do not throw away the odd data point. The weak arrivals in the shadow zone were the discovery.
Propose the simplest model that explains everything. One extra layer accounted for the signals.
Learn a new field properly. She came to seismology as a mathematician and studied it station by station.
Keep working at your own pace. Much of her output came after the age of 65.
Waves bending at a boundary are hard to picture from a flat diagram. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read. For another scientist who found structure hidden in data, see our post on Marie Tharp and the ocean floor.
Next step: work out the surface distance to 145°, the far edge of the P-wave shadow, then sketch both shadow zones on a circle.
