Nontrivial Wave Research Articles

Wave solutions for the deterministic host-vector epidemic

Recent papers by Atkinson and Reuter(1), Brown and Carr(5), and Barbour(2) have proved several important results concerning wave solutions of the usual deterministic model for the spatial spread of an epidemic, such as measles or influenza. In particular Atkinson and Reuter showed that non-trivial wave solutions, in a population along a line, only exist provided the contact distribution function has an exponentially bounded tail, and that the speed of propagation c must be at least some critical value c0. They constructed a solution for c > c0, which Barbour later showed to be the unique solution, modulo translation, at that speed. Brown and Carr showed that a solution was also possible at speed c0, though it has not been possible to show uniqueness at this speed.

Mathematical theory of non-linear waves on the surface of a magnetic fluid

Consider a ferrofluid occupying the region <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z \leq 0</tex> . If the fluid is subjected to a sufficiently strong magnetic field, surface waves appear. Under certain assumptions we shall prove that when the field strength becomes greater than a critical strength, a nontrivial wave solution on the surface of the ferrofluid bifurcates from the trivial solution. We then study the stability of these nontrivial solutions.

Exact Error Bounds for the Phase Velocity in an Acoustic Wave Guide

Waves propagating in an infinite elastic cylinder with finite cross section are considered. Let $f(x,y)e^{i(\omega t - \beta z)} $ be such a wave which does not produce zero stress at the surface of the cylinder. Exact bounds are found on how much $\omega $ must be changed (say to $\tilde \omega $) so that for the same $\beta $ there is a nontrivial wave $\tilde f(x,y)e^{i(\tilde \omega t - \beta z)} $ which has zero stress at the surface. The bound is a function of the stress produced by the original wave and the cross section of the cylinder.

New, Attenuated Wave Mode in He II Films

It is shown that if normal-fluid motion with a viscous force is allowed in He II films, then a new wave mode is predicted by the equations of motion. The film is described here by four equations: conservation of mass, conservation of energy, the usual superfluid thermohydrodynamic equation, and the normal-fluid equation with a viscous force per unit volume. The viscosity term is $\ensuremath{\omega}\ensuremath{\rho}\mathcal{r}{v}_{\mathrm{nx}}$, in which ${v}_{\mathrm{nx}}$ is the velocity of the normal fluid, $\ensuremath{\rho}$ is the bulk helium density, $\ensuremath{\omega}$ is the angular frequency of the wave, and $\mathcal{r}$ is a dimensionless viscosity parameter. Two nontrivial wave modes are allowed: One corresponds to third sound, with a comparatively small attenuation, and the other is the new mode. Both modes are simultaneously waves in local film height, local film temperature, and superfluid and normal-fluid velocity. The new mode is heavily overdamped with an attenuation that rises from about 400 ${\mathrm{cm}}^{\ensuremath{-}1}$ at $\mathcal{r}={10}^{\ensuremath{-}2}$, to about ${10}^{4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ at $\mathcal{r}={10}^{3}$, for waves of ${10}^{3}$ cps in a 250 \AA{} film. The velocity of the mode decreases with temperature in an interesting way: The maximum velocity at 1.2\ifmmode^\circ\else\textdegree\fi{}K is about 900 cm/sec, at 2.0\ifmmode^\circ\else\textdegree\fi{}K about 160 cm/sec, and at 2.15\ifmmode^\circ\else\textdegree\fi{}K about 60 cm/sec. It is shown that these attenuations and velocities are of the same kind as have recently been observed as anomalies in third sound. It is suggested that this new mode is generally excited along with third-sound waves and that normal fluid can move in He II films.