X1 Direction Research Articles

Electromagnetic radiation from an AT-cut quartz plate under lateral-field excitation

Equations of linear piezoelectricity (with the quasistatic approximation) for the quartz plate and Maxwell’s equations for the electromagnetic field in the surrounding vacuum are solved for the thickness-shear vibrations of rotated Y cut of quartz plate. For an AT-cut plate vibrating near the thickness-shear resonance excited by a uniform, lateral electric field of magnitude 1 V/m, the electromagnetic energy radiated from each face is about 0.13 μW/cm2. The radiated power is about 0.1 μW/cm2 if the plate is excited by a shearing face traction which produces a strain of 10−5. Present solution is compared in detail with Mindlin’s [Int. J. Solids Struct. 9, 697 (1972)] solution of equations of piezoelectromagnetism (without the quasistatic approximation) for the thickness-shear vibrations excited by shearing face traction. It is found that the percent difference in radiated powers computed from these two solutions, due to the quasistatic approximation, is in the order of β2(=v̂2/ĉ2), where v̂ is the velocity of the x1-thickness-shear wave, ĉ is the velocity of electromagnetic wave of E3 propagating in the x2 direction in quartz, and β≊10−5.

On the Schrödinger equation with time-dependent electric fields

SynopsisWe prove existence and uniqueness of solutions of i(∂ψ/∂t) = (−Δ+x1g(t)+q(x))ψ, ψ(x, s) = ψs (x) in ℝ3 for potentials q(x) including the Coulomb case. Existence and completeness of the wave operators is established for g(t) periodic with zero mean and q(x) short-range, smooth in the x1 direction. We characterize scattering and bound states in terms of the period operator.

High‐frequency vibrations of rectangular crystal plates coated with multiple strip electrodes

The two‐dimensional equations of motion for the vibrations of crystal plates developed by Lee and Nikodem are extended to incorporate the mass effect of electrode platings on a crystal plate. These, together with the unplated equations of motion, are combined to govern the motion of an elastically coupled piezoelectric resonator system with multiple strip electrodes. A rectangular AT‐cut quarts plate with the strip electrodes which are parallel to the X1 axis and cover the full length of the plate in X1 direction are considered, X1 being the digonal axis of the crystal. Closed form solution is obtained for the coupled thickness‐twist and face‐shear vibrations, which satisfies the traction‐free conditions at edges of the plate and the continuity conditions for traction and displacement at the junctures of the plated and unplated regions of the plate. The calculated results are compared with experimental values with close agreement. Then the dependence of the inter‐resonator coupling, resonance frequencies, and vibrational modes on the electrode mass, on the dimensions of the electrodes' configuration, and on the plate geometry are studied and analyzed.

Theory of regular electrostatic beams of charged particles

In some papers concerned with the exact solution of the equations of a nonrelativistic single-energy beam of charged particles (e.g., [1, 2]), the opinion has been expressed that, while the method of separation of the variables has possibilities, serious difficulties can arise in obtaining the actual systems with separated variables. In particular, it has become popular, when investigating regular electrostatic flows, to transform to a coordinate system connected with the trajectory. In this system the velocity vector only has one component, say v={ix t, 0}, so that flow only occurs in the x1 direction (x1 flow). We shall also refer to a single-component flow, as in [3], This method is thought (e.g., [4]) to be effective for a wide class of flows. The question of the coordinate systems that allow flows in the1 direction is more specialized than the general problem of separation of variables. The concept of an x1 flow is discussed in §1 from the point of view of its utility for obtaining solutions of the regular electrostatic beam equations. Transformation to a coordinate system connected with the trajectory is found to be only justifiable for four orthogonal systems: cartesian, cylindrical, helical-cylindrical, and spherical. It is shown that, in the case of two-dimensional systems on a plane with conformal metric, the condition required for an x1 flow and the conditions for the space to be euclidean can be used effectively to establish the existence in the given class of coordinate systems of an x1 flow starting from a fictitious emitter (§2). The usual tensor notation is employed.