# lect_8_2 - Coordinate Systems and Separation of Variables...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Coordinate Systems and Separation of Variables Revisiting the homogeneous wave equation… 1 2 2 2 2 = ∂ ∂ + ∇ t c ψ ψ where previously in Cartesian coordinates, the Laplacian was given by 2 2 2 2 2 2 2 z y x ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ We are now faced with a spherical polar coordinate system, with the motivation that we might employ the separation of variables technique to solve the wave equation where spherical symmetries are involved. Spherical Polar Coordinates θ φ θ φ θ cos sin sin cos sin r z r y r x = = = [ ] [ ] x y z y x z y x r 1 2 2 1 2 2 2 tan tan − − = + = + + = φ θ ) , , ( θ φ r y φ θ r r 2 2 2 2 2 2 2 2 sin 1 sin sin 1 1 φ θ θ θ θ θ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ = ∇ r r r r r r ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ = ∇ r r r r 2 2 2 1 Assuming variations only in r gives… Thus for a vibrating sphere, we can say immediately that the relevant wave equation is given by the following form of the Helmholtz equation in terms of...
View Full Document

## This note was uploaded on 02/24/2012 for the course MECHANICAL 2.067 taught by Professor Davidbattle during the Spring '04 term at MIT.

### Page1 / 12

lect_8_2 - Coordinate Systems and Separation of Variables...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online