Carl Eckart

The approximation of one matrix by another of lower rank

The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution.A hypothetical interpretation of the canonic components of a score matrix is discussed.

The Scattering of Sound from the Sea Surface

It is assumed that the equation of the sea surface is z = ζ(xyt), and that the time average Φ = 〈ζ(x′y′t)ζ(x″y″t)〉 is a function only of ξ = x″ − x′, η = y″ − y′. The scattering coefficient of long‐wave sound is calculated and shown to be σ = (c2k2/4π)2∫∫Φ(ξη) exp[−ik(aξ+bη)]dξdη, where 2π/k is the wavelength of the sound, and a, b, c are, respectively, the sum of the x, y, z direction cosines of the incident and scattered rays. For short‐wave sound, the formula is more complicated, but independent of the wavelength of the sound, as it should be. However, it is shown that experiments with short waves yield much less information about the sea surface than do those with long waves. This is unfortunate, since the latter experiments are much more difficult than the former.

Variation Principles of Hydrodynamics

It is shown that the Lagrangian equations for the motion of both incompressible and compressible fluids can be derived from variation principles. As has been pointed out by C. C. Lin, an important feature of these principles is the boundary condition: The coordinates of each particle (and not merely the normal component of its velocity) must be specified. A systematic application of known results from the calculus of variations reveals new interrelations between such hydrodynamic results as Bernoullis principle and the circulation theorem. Their derivation is both simplified and systematized. Clebschs transformation is found to have an important relation to the problem of integrating the vorticity equation. The general solution of this problem for nonbarotropic flow is obtained. This reduces the second‐order Lagrange equations to a set of first‐order equations, in which the potential of the irrotational component replaces the pressure. The entropy gradients and a remarkable quantity, defined as the time...

A principal axis transformation for non-hermitian matrices

each two nonparallel elements of G cross each other. Obviously the conclusions of the theorem do not hold. The following example will show that the condition that no two elements of the collection G shall have a complementary domain in common is also necessary. In the cartesian plane let M be a circle of radius 1 and center at the origin, and iVa circle of radius 1 and center at the point (5, 5). Let d be a collection which contains each continuum which is the sum of M and a horizontal straight line interval of length 10 whose left-hand end point is on the circle M and which contains no point within M. Let G2 be a collection which contains each continuum which is the sum of N and a vertical straight line interval of length 10 whose upper end point is on the circle N and which contains no point within N. Let G = Gi+G2 . No element of G crosses any other element of G, but uncountably many have a complementary domain in common with some other element of the collection. However, it is evident that no countable subcollection of G covers the set of points each of which is common to two continua of the collection G. I t is not known whether or not the condition that each element of G shall separate some complementary domain of every other one can be omitted.

Internal Waves in the Ocean

Salinity, temperature, and pressure gradients all cause the density of sea water to vary with depth in the ocean, and the density gradient affects the motion of the waters. A quantity N, having the units radians per second, can be defined using the density gradient, the velocity of sound, and the acceleration of gravity.The simplest motions have the form of horizontally progressive waves of frequency ω, wave number κ, and velocity V. If h is the amplitude of the vertical displacement of the water and z the vertical coordinate, then V2(d2h/dz2) + [N2(z) − ω2]h = 0; this equation is formally identical with Schrodingers wave equation. The stream function of these waves is φ = Vh(z) sin (κx − ωt), and the variable part of the pressure is − ρ dφ/dz, while the vorticity is R = −N2φ/V2. The wave may be described as a lattice of vortices moving with velocity V. In the ocean N(z) ordinarily has one or two maxima called thermoclines. The analogy with the quantum‐mechanical problems of one and two potential minima ...

The Theory of Noise in Continuous Media

It is shown that the mathematical solution of problems involving the propagation of noise is materially aided by the introduction of the space correlation function ψ(x1, x2, τ), defined as the average, over t, of p(x1t)p(x2t−τ), p being the acoustic pressure in the noise field. The differential equations satisfied by ψ are derived. Its relation to Ψ(α, β, γ) is discussed, α, β, γ being the propagation vector of a sinusoidal wave, and Ψ/2ρc2 being the density of potential energy in the α, β, γ space.The theory of uniform noise fields, both isotropic and anisotropic, is developed in detail. The anisotropy caused by a reflecting plane is discussed. The radiation of noise by a vibrating plane is discussed, neglecting the reaction of the radiation on the motion of the surface. In particular, it is shown that to this approximation the sea surface cannot radiate subsonic energy into the high atmosphere because the velocity of the surface gravity waves is less than the velocity of sound in air.The theory of noise...

Some Transformations of the Hydrodynamic Equations

The Lagrangian equations of hydrodynamics are transformed to general coordinates. Since they involve two sets of variables (dependent and independent) two transformations are involved. The independent coordinates are usually taken to be the time and the initial positions of the particles of the fluid. Other kinds of independent variables are often useful. If they are chosen so that they describe the motion of a second fluid, the Lagrangian equations become a tool for comparing the motion of the two fluids. If the second fluid obeys the same laws as the first, the comparison is between two modes of motion of the same fluid. The study of this problem leads to a new derivation of the Eulerian equations from the Lagrangian. If the two modes of motion differ only slightly, the perturbation equations are obtained in generalized coordinates. They satisfy a variation principle. This can be transformed so as to exhibit the general form of a pseudo potential energy function that is useful in the study of stability ...

Extension of Howard's Circle Theorem to Adiabatic Jets

L. N. Howard proved an inequality concerning the complex velocity of the propagation of waves disturbing the laminar, unidirectional flow of an incompressible stratified fluid. It is shown that this theorem is closely related to the virial theorem of hydrodynamics. It can be generalized for the steady motion of a compressible fluid, provided the streamlines are either parallel straight lines or coaxial circles. The connection of the circle and virial theorems provides an interpretation of the former in terms of the energy of the perturbation. A positive potential energy always exerts a stabilizing influence, as does the angular velocity in the case of circular streamlines. This latter is diminished for short waves; it is also complicated by the fact that radial variations of the angular velocity may contribute negative terms to the potential energy. Zonal flow, such as the jet stream of the Earths atmosphere, is generally baroclinic. In the barotropic case, the potential energy of the perturbation is sim...

The Rotation and Vibration of Linear Triatomic Molecules

The rotation and vibration of a linear molecule cannot be treated by the method of rotating axes if it contains more than three atoms. The definition of the axes for the triatomic case may be based on the fact that the three atoms always determine a plane. The relation of this definition to that for the nonlinear molecules is discussed, and the appropriate normal coordinates are introduced. The wave equation of the problem is derived and its approximate solution is discussed. The fact that the Eulerian angles do not enter into the zero‐ and first‐order terms of the Hamiltonian makes it possible to calculate some of the second order terms of the expression for the energy values with complete generality. These are the terms that arise from the rotation and from the coupling between the rotation and vibration. They are found to be ℏ2[J(J+1)−l2]/2A+C, where A is the value of the moment of inertia at equilibrium, and C=[ℏ2(V2+1)/2A]{s2(V1+12)[(ω1/ω2)+(ω2/ω1)] +c2(V3+12)[(ω3/ω2)+(ω2/ω3)]}−ℏ2/2A. In this express...

Surface Wake of a Submerged Sphere

The problem of a sphere moving in an infinite homogeneous incompressible liquid has been discussed by many writers. The corresponding problem for a semi‐infinite liquid with a free surface has not been treated earlier.It is shown that the surface wake of such a submerged sphere is approximately the same as that which would be caused by a traveling pressure disturbance in the atmosphere above the free surface. This is, essentially, a consequence of the Bernoulli theorem.If the motion is sufficiently slow, the surface reacts to this equivalent pressure as a barometer (equilibrium theory). For more rapid motions, dynamic effects reduce the response of the surface, but leave a wake in the region already traversed by the sphere. The calculation of this wake involves the usual distinction between incoming and outgoing waves, which is introduced in the Fourier transform of the solution. The resulting integrals are evaluated by Kelvins approximate method of stationary phase.