ADVANCES IN APPLIED MECHANICS VOLUME 8 This Page Intentionally Left Blank ADVANCES IN APPLIED MECHANICS Ed itors TH. H. L. DRYDEN VON KLRMLN Managing Editor G. KUERTI Case Institute of Technology, Cleveland, Ohio Associate Editors F. H. VAN DEN DUNGEN L. HOWARTH VOLUME 8 1964 ACADEMIC PRESS NEW YORK AND LONDON COPYRIGHTQ1964, BY ACADEMICPRESSINC. ALL RIGHTSRESERVED N O PART OF T H I S BOOK MAY B E REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM T H E PUBLISHERS. ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published b y ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l LIBRARY OF CONCRliSS CATALOG CARD NUMBER:48-8503 P R I N T E D I N T H E U N I T E D STATES OF AMERICA CONTRIBUTORS TO VOLUME 8 BERNARD D. COLEMAN, Mellon Institute, Pittsburgh, Pennsylvania HERSHELMARKOVITZ, Mellon Institute, Pittsburgh, Pennsylvania N. N. MOISEEV,Computing Centre of the U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. E. L. RESLER, JR., Cornell University Ithaca, N e w York H. S. RIBNER,Institute of Aerophysics, University of Toronto, Canada V. V. RUMYANTSEV, Institute of Mechanics of the U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. W. R. SEARS, Cornell Universit31, Ithaca, N e w Yurk V This Page Intentionally Left Blank Theodore yon Khrmhn: A Tribute The death of von KhrmAn in May 1963 has robbed this series of one of its editors. Even if he had not been so closely associated with the “Advances” it would nevertheless have been fitting that a tribute to one so distinguished should appear as a frontispiece to this volume. KBrmAn was one of the giants of our subject whose influence over more than 50 years has had incalculable effects on its growth. Thus he can be said to have been one of a select band present at the birth of real continuum mechanics in the early years of this century and who lived to act as guide, counselor, and source of inspiration to it throughout its graduate and postgraduate career. KirmAn went to Gottingen from Budapest in 1906 a t a time when, it is scarcely necessary to remark, Prandtl and his associates were developing boundary-layer theory and with it were injecting realism into fluid mechanics. During his 6-year stay there Kirmin developed the theory of the KarmLn street, contributed to the theory of the strength of slender columns, and collaborated with Max Born on vibrations in lattices and their relation to specific heat. From Gottingen Karmin moved in 1912 to Aachen where he remained, apart from service with the Austro-Hungarian Aviation Corps in World War I , until 1930. In 1930 Kirmin was appointed Director of the Guggenheim Aeronautical Laboratory at the California Institute of Technology which then became the Mecca of many a would-be scientist and engineer. If I discuss this period more than his time in Aachen it is because I know it better. No one could minimize the importance of the work he did in Aachen in building up the Aeronautical Institute or of the work he did there on the turbulent boundary layer (including the logarithmic law), on the momentum integral concepts, and on the rotating-disc solution of the Navier-Stokes equations all of which are still producing repercussions. For many of us, working under K i r m h ’ s direction in the 1930’s was an unforgettable experience - inspiring and in every way enjoyable. He was the most accessible of men and had built up a staff and school around him which was the envy of all. At that time Pasadena was home for Kirmin.. He lived with his mother and sister in S. Marengo, and together they did al