# George Carrier's early papers

Even before the award of his Ph.D., Carrier was submitting papers. The first was

*Provided stress analysis for the case of anisotropic rotating disks of uniform thickness*(1943) and the second was*The thermal-stress and body-force problems of the infinite orthotropic solid*(1944). He published two further papers in 1945, both related to work he had undertaken for his Ph.D. thesis. These are*On the vibrations of the rotating ring*, and*On the non-linear vibration problem of the elastic string*. We give below the Introduction and a review of each of three 1944-45 papers. We also give the Introduction and a review of two papers Carrier published in 1946, the second being a continuation of the first.**1. The thermal-stress and body-force problems of the infinite orthotropic solid (1944), by G F Carrier.**

**1.1. Introduction.**

Elastic problems dealing with orthotropic materials have had considerable investigation in recent years, but up to the present time, such investigation has been largely limited to a consideration of the problems involving thin plates of this material. In the present paper, two problems dealing with the stresses and displacements in an infinite elastic orthotropic solid are solved, and in each case the results are obtained in terms of three independent displacement potentials. The two solutions are: 1) the displacement potentials arising from an arbitrary distribution of temperature within a finite region of the solid (the temperature being measured from an arbitrary datum) and 2) the potentials arising from an arbitrary distribution of body force within a finite region. Each of these problems reduces to the solution of three simultaneous partial differential equations, which are transformed, through the use of Fourier integrals, into individual solutions for each potential. The expressions for these potentials are reduced to the form of Newtonian potential integrals for those cases where sufficient symmetry of the material properties exists to allow such a reduction. In the more complicated cases, the results are still expressed in closed form in terms of definite integrals.

**1.2. Review by Herman William March in**

*Mathematical Reviews*MR0010496 (6,26i).The states of elastic deformation associated with an arbitrary distribution either of temperature or of body forces in a finite region of an infinite orthotropic elastic solid are expressed in each case in terms of three independent displacement potentials. Each of these potentials is represented by an integral containing a Green's function. This function can be readily determined for two dimensional problems and for certain three dimensional problems.

**2. On the vibrations of the rotating ring (1945), by G F Carrier.**

**2.1. Introduction.**

An interesting addition to the group of problems dealing with thin elastic rings is the analysis of the vibration of a circular ring which is rotating with constant speed about its geometric axis. In this paper, the small bending vibrations of the unconstrained ring are analysed and the frequencies at which such vibrations can occur are determined. For various problems of the partially constrained ring, it is shown that the "free vibrations" differ essentially in character from those of the free ring, exhibiting a group of natural modes characterised by linear combinations of trigonometric functions. The forced vibrations of both the free and supported rings are also treated.

**2.2. Review by Dio Lewis Holl in**

*Mathematical Reviews*MR0013370 (7,144b).Thin rings rotating with constant speed about their geometric axes are analysed for characteristic frequencies at which bending vibrations occur. The ring may be free or it may be supported at n evenly spaced intervals such that there is no relative tangential displacement of the points of support. The rotating points of support may admit either radial or no radial movement. In the supported ring the physical problem is that of determining what periodic forces applied at the supports are capable of sustaining a motion wherein the support points have no relative tangential displacement at any time. Solutions are found satisfying this condition for particular values of the frequencies or an associated eigenvalue. The author asserts that the set of solutions has the property of completeness and that the motion may be fully described for any initial conditions which have the appropriate periodicity in the angular variable. The treatment also includes forced vibrations and elastically supported rings.

**3. On the non-linear vibration problem of the elastic string (1945), by G F Carrier.**

**3.1. Introduction.**

It is well known that the classical linearised analysis of the vibrating string can lead to results which are reasonably accurate only when the minimum (rest position) tension and the displacements are of such magnitude that the relative change in tension during the motion is small. The following analysis of the free vibrations of the string with fixed ends leads to a solution of the problem which adequately describes those motions for which the changes in tension are not small. The perturbation method is adopted, using as a parameter a quantity which is essentially the amplitude of the motion. The periodic motions arising from initial sinusoidal deformations are closely approximated in closed form. The method is applied to motions not restricted to a single plane and finally the exact solution for the transmission of a localised deformation is indicated.

**3.2. Review by Norman Levinson in**

*Mathematical Reviews*MR0012351 (7,13h).The motion of an elastic string is considered for the case where changes in tension during the motion cannot be neglected. This nonlinear problem is handled by the perturbation method using as the parameter a quantity closely associated with the "amplitude'' of the motion. An approximation in explicit form is given for motions arising from initial sinusoidal deformations. Motions not confined to a plane as well as the transmission of a localised deformation are considered.

**4. On the propagation of small disturbances in a moving compressible fluid (1946), by G F Carrier and F D Carlson.**

**4.1. Introduction.**

Although the propagation of sound waves in moving media has received considerable attention, little information is available concerning the propagation of such disturbances in rotational streams or concerning the propagation of transient rotational phenomena. It is shown in the present paper that the wave fronts associated with those parts of a disturbance which are derivable from a potential propagate in a rotational stream according to those laws which they are already known to obey in an irrotational stream. It is further shown that the rotational disturbances drift with the stream rather than propagate relative to the moving fluid. The analysis consists of an application of conventional perturbation procedures to the Navier-Stokes and continuity equations. The equations so derived are treated according to the theory of characteristics. The results obtained lead to a general expression for the Mach lines of an arbitrary supersonic flow and also suggest a new method of wind tunnel calibration which eliminates the necessity of placing an obstacle in that portion of the stream being calibrated. Finally, predictions are carried out as to the nature of pulses which are formed at a surface and then propagate through a boundary layer into a uniform stream.

**4.2. Review by David Gilbarg in**

*Mathematical Review*s MR0015998 (7,499c).A general stream is considered. The propagation of small disturbances in it is analysed by applying standard perturbation techniques to the Navier-Stokes and continuity equations and then treating the derived equations by the theory of characteristics. A simplification is achieved by replacing the energy equation by the condition that the changes in state from undisturbed to disturbed stream are isentropic; this assumption is justified by an order of magnitude argument. However, no restrictions are placed on the undisturbed stream, so that the analysis is otherwise quite general. It is shown that rotational disturbances move with the stream (except for diffusion), whereas the irrotational portions of a disturbance propagate relative to this general stream as they would in an irrotational flow. Specific application is made to the propagation of pulses through a simplified boundary layer into a uniform stream. The wave fronts thus predicted are found to be in qualitatively good agreement with those observed in Schlieren photographs.

**5. The propagation of waves in orthotropic media (1946), by G F Carrier.**

**5.1. Introduction.**

The present article is an extension of a previous paper dealing with the elasticity problems of orthotropic media.1 Here, the displacement potentials which define the dynamic phenomena in such media are discussed.

**5.2. Review by H W March in**

*Mathematical Reviews*MR0017190 (8,120g).The methods of a previous paper [4] are extended to discuss the displacement potentials of dynamic phenomena in orthotropic media. Consideration is limited to media which are isotropic in planes parallel to one of the planes of elastic symmetry and for which a certain relation among the elastic constants is satisfied. The number of independent elastic constants of such media is thereby reduced to four.

Last Updated January 2021