Question 4.2.4: Derive the refraction law and reflection coefficients at the......

The refraction law is determined by the two following requirements: the frequency of the refracted wave (wave 2) is equal to that of the incident wave (wave 1), and the x-components of the wave-vectors, \vec{k}_{1} and \vec{k}_{2}, are also equal to each other. Thus, in the case of a shear Alfven wave, one gets: ω_{1} = k_{1z}V_{A1} = ω_{2} = k_{2z}V_{A2}; k_{1x} = k_{1} \sin θ_{1} = k_{2x} = k_{2} \sin θ_{2}. Since V_{A} ∝ ρ^{−1/2}, it follows from the first of these equations that k_{2} \cos θ_{2} = k_{1} \cos θ_{1}(ρ_{2} ρ_{1})^{1/2}, which together with the second equation yield

(ρ_{2})^{1/2} \tan θ_{2} = (ρ_{1})^{1/2} \tan θ_{1}

For a compressional Alfven wave with the dispersion relation ω = kV_{A}, the above requirements result in

(ρ_{2})^{1/2} \sin θ_{2} = (ρ_{1})^{1/2} \sin θ_{1},

which looks like the standard Snell’s law, with the refractive index of a medium being proportional to a square root of its density. Hence, there is no refracted wave if ρ_{2} < ρ_{1} and the angle of incidence exceeds θ_{r} = \sin^{−1} (ρ_{2}/ρ_{1})^{1/2}, the angle of total reflection.

In order to obtain reflection coefficients of waves one should consider the boundary conditions for the magnetic and electric fields at the interface: the continuity of the tangential components of \vec{B} and \vec{E} . In the case of a shear Alfven wave these are b_{y} and

E_{x} = − \frac{1}{c}Bv_{y}= \frac{k_{z}B^{2}}{4πcρω} b_{y}

(see Problem 4.2.1). In the upper fluid of density ρ_{1} there are two waves, the incident one (wave 1) and the reflected one (wave 3), with the amplitudes of b_{y} equal to b_{1} and b_{3} and k_{3z} = −k_{1z}. In the lower fluid of density ρ_{2} there is only the refracted wave 2. Thus, the above-stated boundary conditions imply:

b_{1} + b_{3} = b_{2}, \frac{k_{1z}}{ωρ_{1}} (b_{1} − b_{3}) = \frac{(b_{1} − b_{3})}{ρ_{1}V_{A1}} = \frac{k_{2z}}{ωρ_{2}} b_{2} = \frac{b_{2}}{ρ_{2}V_{A2}}        (4.26)

Then, by taking into account that V_{A2}/V_{A1} = (ρ_{1}/ρ_{2})^{1/2}, it follows from (4.26) that the reflection coefficient

R = \left( \frac{b_{3}}{b_{1}}\right)^{2} = \left[ \frac{1 − (ρ_{2}/ρ_{1})^{1/2}}{1 + (ρ_{2}/ρ_{1})^{1/2}}\right]^{2}

Note that this reflection coefficient does not depend on the angle of incidence.

Similarly, in the case of a compressional Alfven wave, one should require the continuity of b_{x}, b_{z} and E_{y} = −Bv_{x}/c. Thus, if v_{1,2,3} are velocity amplitudes of each of the three waves shown in Figure 4.4, the above requirements yield (with the help of equations (4.22-4.23)):

v_{z} = 0, v_{y} = − \frac{k_{z}B}{4πρω}b_{y}, v_{x} = \frac{B}{4πρω} (k_{x}b_{z} − k_{z}b_{x})        (4.22)

b_{x} = − \frac{k_{z}B}{ω} v_{x}, b_{y} = − \frac{k_{z}B}{ω} v_{y}, b_{z} = \frac{k_{x}B}{ω} v_{x}      (4.23)

v_{1} + v_{3} = v_{2}, k_{1z}(v_{1} − v_{3}) = k_{2z}v_{2}

(note that the continuity of b_{z} is ensured by the first of the above relations, because all three waves share the same ω and k_{x}). By using the dispersion law (4.25), one gets then the following reflection coefficient:

ω = kV_{A}        (4.25)

R = \left(\frac{v_{3}}{v_{1}} \right)^{2} = \left(\frac{1 − δ}{1 + δ} \right)^{2} , δ = \frac{(ρ_{1}/ρ_{2})^{1/2} \cos θ_{1}}{[1 − (ρ_{1}/ρ_{2}) \sin^{2} θ_{1}]^{1/2}} , θ_{1} < θ_{r}.