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Plane Waves

Module by: Paul Padley. E-mail the author

Summary: A simple expression for a plane wave

Plane Waves

We want to find the expression for a plane that is perpendicular to k k , where k k is a vector in the direction of propagation of the wave.The plane is the set of points that has the same projection onto the vector k k That is any point r r that satisfies k r = c o n s t a n t k r = c o n s t a n t is a point on the planeNow consider the function ψ ( r ) = A e i k r ψ ( r ) = A e i k r we see that the magnitude of ψ ( r ) ψ ( r ) is the same over every plane that is defined by k r = c o n s t a n t k r = c o n s t a n t we want to construct harmonic waves, ie. they should repeat every wavelength along the direction of propagation so they should satisfy ψ ( r ) = ψ ( r + λ k k ) ψ ( r ) = ψ ( r + λ k k ) where λ λ is the wavelengththen we must have A e i k r = A e i k ( r + λ k k ) = A e i k r e i k k λ / k = A e i k r e i k λ A e i k r = A e i k ( r + λ k k ) = A e i k r e i k k λ / k = A e i k r e i k λ This is true if e i λ k = 1 = e i 2 π e i λ k = 1 = e i 2 π or λ k = 2 π λ k = 2 π k = 2 π λ k = 2 π λ This should have a familiar look to it! Finally we want these waves to propagate in time so you should be able to guess the answer from our work on mechanical waves ψ ( r ) = A e i ( k r ω t ) ψ ( r ) = A e i ( k r ω t )

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Lenses

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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