Page:Philosophical magazine 23 series 4.djvu/111

Rh and the energy

whence

for the axis of $$x$$, with similar expressions for the other axes, V being the volume, and $$r$$ the radius of the vortex.

Prop. XIX. — To determine the conditions of undulatory motion in a medium containing vortices, the vibrations being perpendicular to the direction of propagation.

Let the waves be plane-waves propagated in the direction of $$z$$, and let the axis of $$x$$ and $$y$$ be taken in the directions of greatest and least elasticity in the plane $$xy$$. Let $$x$$ and $$y$$ represent the displacement paralll to these axes, which will be the same throughout the same wave-surface, and therefore we shall have $$x$$ and $$y$$ functions of $$z$$ and $$t$$ only.

Let X be the tangential stress on unit of area parallel to $$xy$$, tending to move the part next the origin in the direction of $$x$$.

Let Y be the corresponding tangential stress in the direction of $$y$$.

Let $$k_1$$ and $$k_2$$ be the coefficients of elasticity with respect to these two kinds of tangential stress; then, if the medium is at rest,

$X=k_{1}\frac{dx}{dz},\ Y=k_{2}\frac{dy}{dz}.$

Now let us suppose vortices in the medium whose velocities are represented as usual by the symbols $$\alpha,\beta,\gamma$$, and let us suppose that the value of $$\alpha$$ is increasing at the rate $$\tfrac{d\alpha}{dt}$$, on account of the action of the tangential stresses alone, there being no electromotive force in the field. The angular momentum in the stratum whose area is unity, and thickness $$dz$$, is therefore increasing at the rate $$\tfrac{1}{4\pi}\mu r\tfrac{d\alpha}{dt}dz$$; and if the part of the force Y which produces this effect is Y', then the moment of Y' is $$-Y'dz$$,so that $$Y'=-\tfrac{1}{4\pi}\mu r\tfrac{d\alpha}{dt}$$.

The complete value of Y when the vortices are in a state of