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Advanced Physics Homework Help
Visualizing the Paraboloidal Wave
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[QUOTE="TSny, post: 6798246, member: 229090"] From the link, they are considering spherical waves where $$U(\vec r) = \frac{A_0}{r}e^{-jkr}.$$ The phase is ##kr##. The true wavefronts are all spherical. For large distances from ##r = 0##, small patches of the wavefronts can be approximated by paraboloid-shaped wavefronts. (For very large distances, the wavefronts can be approximated by plane wavefronts.) Consider moving out along the ##z## axis to some point ##P## with coordinates ##(x, y, z) = (0, 0, z_0)##. The phase of the wave at that point will be ##kz_0##. We look for points in the vicinity of ##P## for which the wave has the same phase. We assume ##z_0## is large enough so that points in the vicinity of ##P## will have coordinates ##(x, y, z)## satisfying ##x \ll z## and ##y \ll z##. For these points, $$r = \sqrt{x^2 + y^2 + z^2} \approx z + \frac{x^2+y^2}{2z} .$$ The condition that the phase ##kr## at these points be the same as at ##P## is $$k\left[ z + \frac{x^2+y^2}{2z}\right ] = k z_0$$ From this show that points near ##P## on the wavefront passing through ##P## have coordinates ##(x, y, z)## which satisfy $$z-z_0 \approx -\frac{x^2+y^2}{2z_0} .$$ Describe the shape of the locus of points ##(x, y, z)## satisfying $$z-z_0 = -\frac{x^2+y^2}{2z_0} .$$ [/QUOTE]
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Visualizing the Paraboloidal Wave
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