# UCLA Physics 180E

AbstractUsing an antenna to generate whistler waves in a plasma using a helium source and a magnetic field probe to measure perturbations in the magnetic field, a biased disc was placed behind the wave generator to create ducting. Ducted waves were seen to propagate towards density minimum when in a region of high magnetic field, and then along a density maximum when in a region of lower magnetic field. A simulation was created using theory, which was found not to agree with measured results.

# Introduction

Whistler waves were heard as early as 1886 on long telephone lines. In 1953, Storey showed that whistlers originate from lightning discharges, propogating along field-aligned tubes of density enhancement or depletion. The ﬁrst experiment to investigate whistler waves in detail was by Stenzel in 1975, in a different plasma with a much higher density but otherwise comparable plasma parameters for this experiment [1]. This experiment seeks to find out what occurs to a plasma and whistler waves when there is a density depletion or duct. In the lab a duct is made in the plasma by placing a a copper circular disc between the wave launch antenna and the plasma source. The disc is biased to collect electrons and in turn creates a density depletion [1]. Using Appleton’s equation, simulations of ducted waves were compared to measured ducted waves as well as finding the polarization of these waves due to ducting.

# Theory

The study of whistler waves in ducting will still be determined by the dispersion relation given by the approximation of Appleton’s Equation

$\eta^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{(\omega_{ce}^2\sin^2\theta)^2}{4(1-\frac{\omega_{pe}^2}{\omega^2})}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)}$

One of the remarkable aspects of ducted whistler waves is that the phase velocity travels at a separate angle to the group velocity and direction of propogation of the wave. The group velocity is given by $$\vec{V}_{gr}=\frac{\mathrm\partial\omega}{\mathrm \partial\vec{k}}$$ and the phase velocity is given by $$\vec{V_{ph}}=\frac{\omega}{k}$$. The group velocity can be rewritten as $$V_{group}=\frac{c}{\frac{d}{df}(\eta f)}$$, which for whistler waves becomes approximately $$V_{group}=2 c \frac{\sqrt{f}}{f_{pe}f_{ce}\cos{2\theta}}(f_{ce}\cos{\theta}-f)^{\frac{3}{2}}$$ where $$\theta$$ is the angle between the group velocity and the magnetic field. It can then be shown that this angle is represented as $\theta_{group}=a\tan\left(\frac{\sin\theta\left(\cos\theta-2\frac{f}{f_{ce}}\right)}{1+\cos\theta\left(\cos\theta-2\frac{f}{f_{ce}}\right)}\right)$ Consider the case when the wave propagation angle is small with respect to the magnetic field and there are no collisions, then appletons equation reduces to $\eta=\sqrt{1-\frac{\omega_{pe}^2}{\omega(\omega-\omega_{ce}\cos\theta)}}$ and plotting the index of refreaction surface gives conditions for wave propagation in a duct [3]. It is seen that for waves with $$\omega>\frac{\omega_{ce}}{2}$$ will travel along density maximums, while waves with $$\omega<\frac{\omega_{ce}}{2}$$ will travel in density minimums [3].

Experimental setup for the purpose of ducting as seen from the front (opposite the source). The copper ducting plate is placed about 1.5 cm in the negative radial direction with the whistler wave emitter taken as centered.