Consider cylindrically symmetric solutions to the wave equation (f does not depend on or :). This...

 

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Consider cylindrically symmetric solutions to the wave equation (f does not depend on or :). This is a wave emitted by a line source, for example. a) Simplify the 3D wave equation to a 1D (in space) partial differential equation (plus a time derivative) by invoking cylindrical symmetry of the solutions (i.e. the wave solutions should only depend on the radial direction ). You should end up with a differential equation with a derivative in space along only one dimension. Hint: Write out the Laplacian in cylindrical coordinates, as given in the notes, and then apply symmetry to simplify it. You should end up with 18 음(음)1(사)글(0) f(r.)= b) Show that monochromatic solutions to this equation are Bessel functions by manipulating the equation until it has the form of Bessel's equation: d² d (²) F(p=0. >)=0, where n is a constant. (Hint: Turn it into a dp Helmholtz equation first, then use a change of variables. If you start from the equation above, your change of variables will be pabr.] c) Find the linear combination of these solutions that gives, for large enough r, exp(ikr) √r This is evidently the analogue of a spherical wave for cylindrical symmetry. Note that this solution deviates from the nice "spherical-like" form for small enough r. [Hint: Look up the asymptotic forms of the Bessel functions for larger] Consider cylindrically symmetric solutions to the wave equation (f does not depend on or :). This is a wave emitted by a line source, for example. a) Simplify the 3D wave equation to a 1D (in space) partial differential equation (plus a time derivative) by invoking cylindrical symmetry of the solutions (i.e. the wave solutions should only depend on the radial direction ). You should end up with a differential equation with a derivative in space along only one dimension. Hint: Write out the Laplacian in cylindrical coordinates, as given in the notes, and then apply symmetry to simplify it. You should end up with 18 음(음)1(사)글(0) f(r.)= b) Show that monochromatic solutions to this equation are Bessel functions by manipulating the equation until it has the form of Bessel's equation: d² d (²) F(p=0. >)=0, where n is a constant. (Hint: Turn it into a dp Helmholtz equation first, then use a change of variables. If you start from the equation above, your change of variables will be pabr.] c) Find the linear combination of these solutions that gives, for large enough r, exp(ikr) √r This is evidently the analogue of a spherical wave for cylindrical symmetry. Note that this solution deviates from the nice "spherical-like" form for small enough r. [Hint: Look up the asymptotic forms of the Bessel functions for larger]

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