In 1981 I was approached by Klaus Peters to assist in publishing a selection of the papers of Kurt Otto Friedrichs. With some reluctance Friedrichs has agreed to help in making the selection although in the initial stages only one paper [47-2] was in his opinion worthy of being included. With some coaxing on my part the selection was made mainly by Friedrichs despite his frequent modest grumble that it did not make sense to include "that" paper as "X" had subsequently improved either the result or the proof. Later P.D. Lax and L. Nirenberg identified appropriate papers and, perhaps not so remarkably, there was almost complete overlap. The vast majority of the papers are in partial differential equations and spectral theory where Friedrichs had the greatest impact. For these Peter Lax and Tosio Kato have written the commentaries. Friedrichs made fundamental contributions in the theory of asymptotics; two papers, including Friedrichs' Gibbs Lecture, have been reviewed by W. Wasow. The very profound contributions Friedrichs made to applied mathematics are represented by some papers in elasticity, mainly written with J.J. Stoker and now reviewed by F. John and in magneto-hydrodynamics where Friedrichs' original notes have been reproduced and his contributions to the subject reviewed by Harold Weitzner. In both these areas Friedrichs recognized and clarified approximations made in engineering and physics.

The work presented in this report originated in connection with a research project in magneto-hydrodynamics conducted by Professor H. Grad at the Institute of Mathematical Sciences at New York Univer­ sity; the report was completed while the author was a consultant at Los Alamos Scientific Laboratory. The report contains an expository analysis of the mathematical structure of magneto-hydrodynamic theory; specifically, of the theory of the interaction of magnetic fields with conducting compressible fluids. One of the major aims is to emphasize the remarkably close parallelism between this theory and ordinary gas dynamics. While hydromagnetic shocks and small disturbance waves have been treated before, it seems that hydromagnetic 'simple waves" are here treated for the first time. In a typical example it is shown that - as in gas dynamics - simple one-dimensional flow problems can be solved with the aid of shocks and simple waves. Throughout the exposition a familiarity with the basic notions of gas dynamics - covered e.g. by the first three chapters of Super­ sonic ~low and Shock Waves - is presupposed. The bibliography refers only to those papers which are concerned with the subjects treated. No reference is made to a number of sig­ nificant contributions to hydromagnetics concerned with subjects - such as astro- and geophysical applications, effect of turbulence, stabil­ ity questions - which are not covered in this report.

In 1981 I was approached by Klaus Peters to assist in publishing a selection of the papers of Kurt Otto Friedrichs. With some reluctance Friedrichs has agreed to help in making the selection although in the initial stages only one paper [47-2] was in his opinion worthy of being included. With some coaxing on my part the selection was made mainly by Friedrichs despite his frequent modest grumble that it did not make sense to include "that" paper as "X" had subsequently improved either the result or the proof. Later P.O. Lax and L. Nirenberg identified appropriate papers and, perhaps not so remarkably, there was almost complete overlap. The vast majority of the papers are in partial differential equations and spectral theory where Friedrichs had the greatest impact. For these Peter Lax and Tosio Kato have written the commentaries. Friedrichs made fundamental contributions in the theory of asymptotics; two papers, including Friedrichs' Gibbs Lecture, have been reviewed by W. Wasow. The very profound contributions Friedrichs made to applied mathematics are represented by some papers in elasticity, mainly written with J.J. Stoker and now reviewed by F. John and in magneto-hydrodynamics where Friedrichs' original notes have been reproduced and his contributions to the subject reviewed by Harold Weitzner. In both these areas Friedrichs recognized and clarified approximations made in engineering and physics.

In 1981 I was approached by Klaus Peters to assist in publishing a selection of the papers of Kurt Otto Friedrichs. With some reluctance Friedrichs has agreed to help in making the selection although in the initial stages only one paper [47-2] was in his opinion worthy of being included. With some coaxing on my part the selection was made mainly by Friedrichs despite his frequent modest grumble that it did not make sense to include "that" paper as "X" had subsequently improved either the result or the proof. Later P.D. Lax and L. Nirenberg identified appropriate papers and, perhaps not so remarkably, there was almost complete overlap. The vast majority of the papers are in partial differential equations and spectral theory where Friedrichs had the greatest impact. For these Peter Lax and Tosio Kato have written the commentaries. Friedrichs made fundamental contributions in the theory of asymptotics; two papers, including Friedrichs' Gibbs Lecture, have been reviewed by W. Wasow. The very profound contributions Friedrichs made to applied mathematics are represented by some papers in elasticity, mainly written with J.J. Stoker and now reviewed by F. John and in magneto-hydrodynamics where Friedrichs' original notes have been reproduced and his contributions to the subject reviewed by Harold Weitzner. In both these areas Friedrichs recognized and clarified approximations made in engineering and physics.

The work presented in this report originated in connection with a research project in magneto-hydrodynamics conducted by Professor H. Grad at the Institute of Mathematical Sciences at New York Univer­ sity; the report was completed while the author was a consultant at Los Alamos Scientific Laboratory. The report contains an expository analysis of the mathematical structure of magneto-hydrodynamic theory; specifically, of the theory of the interaction of magnetic fields with conducting compressible fluids. One of the major aims is to emphasize the remarkably close parallelism between this theory and ordinary gas dynamics. While hydromagnetic shocks and small disturbance waves have been treated before, it seems that hydromagnetic 'simple waves" are here treated for the first time. In a typical example it is shown that - as in gas dynamics - simple one-dimensional flow problems can be solved with the aid of shocks and simple waves. Throughout the exposition a familiarity with the basic notions of gas dynamics - covered e.g. by the first three chapters of Super­ sonic ~low and Shock Waves - is presupposed. The bibliography refers only to those papers which are concerned with the subjects treated. No reference is made to a number of sig­ nificant contributions to hydromagnetics concerned with subjects - such as astro- and geophysical applications, effect of turbulence, stabil­ ity questions - which are not covered in this report.

In 1981 I was approached by Klaus Peters to assist in publishing a selection of the papers of Kurt Otto Friedrichs. With some reluctance Friedrichs has agreed to help in making the selection although in the initial stages only one paper [47-2] was in his opinion worthy of being included. With some coaxing on my part the selection was made mainly by Friedrichs despite his frequent modest grumble that it did not make sense to include "that" paper as "X" had subsequently improved either the result or the proof. Later P.D. Lax and L. Nirenberg identified appropriate papers and, perhaps not so remarkably, there was almost complete overlap. The vast majority of the papers are in partial differential equations and spectral theory where Friedrichs had the greatest impact. For these Peter Lax and Tosio Kato have written the commentaries. Friedrichs made fundamental contributions in the theory of asymptotics; two papers, including Friedrichs' Gibbs Lecture, have been reviewed by W. Wasow. The very profound contributions Friedrichs made to applied mathematics are represented by some papers in elasticity, mainly written with J.J. Stoker and now reviewed by F. John and in magneto-hydrodynamics where Friedrichs' original notes have been reproduced and his contributions to the subject reviewed by Harold Weitzner. In both these areas Friedrichs recognized and clarified approximations made in engineering and physics.

The work presented in this report originated in connection with a research project in magneto-hydrodynamics conducted by Professor H. Grad at the Institute of Mathematical Sciences at New York Univer­ sity; the report was completed while the author was a consultant at Los Alamos Scientific Laboratory. The report contains an expository analysis of the mathematical structure of magneto-hydrodynamic theory; specifically, of the theory of the interaction of magnetic fields with conducting compressible fluids. One of the major aims is to emphasize the remarkably close parallelism between this theory and ordinary gas dynamics. While hydromagnetic shocks and small disturbance waves have been treated before, it seems that hydromagnetic 'simple waves" are here treated for the first time. In a typical example it is shown that - as in gas dynamics - simple one-dimensional flow problems can be solved with the aid of shocks and simple waves. Throughout the exposition a familiarity with the basic notions of gas dynamics - covered e.g. by the first three chapters of Super­ sonic ~low and Shock Waves - is presupposed. The bibliography refers only to those papers which are concerned with the subjects treated. No reference is made to a number of sig­ nificant contributions to hydromagnetics concerned with subjects - such as astro- and geophysical applications, effect of turbulence, stabil­ ity questions - which are not covered in this report.

In 1981 I was approached by Klaus Peters to assist in publishing a selection of the papers of Kurt Otto Friedrichs. With some reluctance Friedrichs has agreed to help in making the selection although in the initial stages only one paper [47-2] was in his opinion worthy of being included. With some coaxing on my part the selection was made mainly by Friedrichs despite his frequent modest grumble that it did not make sense to include "that" paper as "X" had subsequently improved either the result or the proof. Later P.O. Lax and L. Nirenberg identified appropriate papers and, perhaps not so remarkably, there was almost complete overlap. The vast majority of the papers are in partial differential equations and spectral theory where Friedrichs had the greatest impact. For these Peter Lax and Tosio Kato have written the commentaries. Friedrichs made fundamental contributions in the theory of asymptotics; two papers, including Friedrichs' Gibbs Lecture, have been reviewed by W. Wasow. The very profound contributions Friedrichs made to applied mathematics are represented by some papers in elasticity, mainly written with J.J. Stoker and now reviewed by F. John and in magneto-hydrodynamics where Friedrichs' original notes have been reproduced and his contributions to the subject reviewed by Harold Weitzner. In both these areas Friedrichs recognized and clarified approximations made in engineering and physics.

In 1981 I was approached by Klaus Peters to assist in publishing a selection of the papers of Kurt Otto Friedrichs. With some reluctance Friedrichs has agreed to help in making the selection although in the initial stages only one paper [47-2] was in his opinion worthy of being included. With some coaxing on my part the selection was made mainly by Friedrichs despite his frequent modest grumble that it did not make sense to include "that" paper as "X" had subsequently improved either the result or the proof. Later P.D. Lax and L. Nirenberg identified appropriate papers and, perhaps not so remarkably, there was almost complete overlap. The vast majority of the papers are in partial differential equations and spectral theory where Friedrichs had the greatest impact. For these Peter Lax and Tosio Kato have written the commentaries. Friedrichs made fundamental contributions in the theory of asymptotics; two papers, including Friedrichs' Gibbs Lecture, have been reviewed by W. Wasow. The very profound contributions Friedrichs made to applied mathematics are represented by some papers in elasticity, mainly written with J.J. Stoker and now reviewed by F. John and in magneto-hydrodynamics where Friedrichs' original notes have been reproduced and his contributions to the subject reviewed by Harold Weitzner. In both these areas Friedrichs recognized and clarified approximations made in engineering and physics.

The work presented in this report originated in connection with a research project in magneto-hydrodynamics conducted by Professor H. Grad at the Institute of Mathematical Sciences at New York Univer­ sity; the report was completed while the author was a consultant at Los Alamos Scientific Laboratory. The report contains an expository analysis of the mathematical structure of magneto-hydrodynamic theory; specifically, of the theory of the interaction of magnetic fields with conducting compressible fluids. One of the major aims is to emphasize the remarkably close parallelism between this theory and ordinary gas dynamics. While hydromagnetic shocks and small disturbance waves have been treated before, it seems that hydromagnetic 'simple waves" are here treated for the first time. In a typical example it is shown that - as in gas dynamics - simple one-dimensional flow problems can be solved with the aid of shocks and simple waves. Throughout the exposition a familiarity with the basic notions of gas dynamics - covered e.g. by the first three chapters of Super­ sonic ~low and Shock Waves - is presupposed. The bibliography refers only to those papers which are concerned with the subjects treated. No reference is made to a number of sig­ nificant contributions to hydromagnetics concerned with subjects - such as astro- and geophysical applications, effect of turbulence, stabil­ ity questions - which are not covered in this report.