For my urban patterns research, I have developed the dataset with housing units by census tract for 59 large urban areas delineated for each census from 1950 to 2010. (More information is in the posts describing the data and urban area definition.) For the first, most basic inquiry, I examined the overall densities of those areas and how these have changed over the period.
Given the tremendous growth of suburban areas, my expectation was that densities would be declining significantly over time. The maximum density did decrease greatly over the period, from nearly 3,000 housing units per square mile to just over 1,900 units. The mean density across the 59 urban areas likewise declined steadily, though far less dramatically.
Looking at the changes in densities over time for the individual areas, however, yielded a very different, more complex picture. I divided the areas into quartiles based on their change in density from 1950 to 2010. The first two quartiles did indeed see significant declines in density. For areas in the third quartile, however, housing unit densities were more or less stable over the period. These areas had relative small changes, ranging from a decline of 200 units per square mile to an increase of just over 100 units per square mile. But most surprising were the 15 urban areas in the top change quartile, which experienced increases in housing unit densities over the period up to a maximum jump of over a thousand units per square mile.
As a group, the areas in the top quartile with increasing densities had among the lowest densities in 1950, half the mean density of areas in the first quartile. By 2010, the density increases for this group gave them the highest mean density for all of the quartiles.
The locations of the urban areas with declining, stable, and increasing densities is very striking, as can be seen on this map:
Most of the areas that saw declines in densities were located in the Northeast and Midwest (and densities dropped for all of the areas in those regions). The areas where densities increased were primarily in the West and in Florida. And almost all of the areas with relatively stable densities were in the South.
More detail on this research is presented in the paper Density of Large Urban Areas in the U.S., 1950-2010.