Classical Wave Theory - Physical Chemistry - Lecture Notes, Study notes of Physical Chemistry

Download Classical Wave Theory - Physical Chemistry - Lecture Notes and more Study notes Physical Chemistry in PDF only on Docsity! Classical Wave Theory Introduction to Differential Equations ordinary differential equation 2 2 2 ( ) ( ) 0d y x k y x dx − = generalized solution for linear diffeq 1 2( ) kx kxy x c e c e−= + Notice: k could be real or imaginary. The solution to another equation with + sign would involve ikx Euler’s formula cos sin ie iθ θ θ± = ± One Dimensional Vibrating String position along line position above line v velocity of wave t time x u ≡ ≡ ≡ ≡ 2 2 2 2 2 1 v u u x t ⎛ ⎞ ⎛ ⎞∂ ∂ =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ Partial differential equation Docsity.com SOLUTION: on)quantizati as emerge (willindex string oflength angle phase frequencyangular waveof amplitude sin)cos(),( 1 ≡ ≡ ≡ ≡ ≡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛⋅+= ∑ ∞ = n l A l xntAtxu nn n n φ ω πφω example of frequency and speed: velocity 2l m/sec angular frequency πn2 Hz normal frequency (υ ) n Hz so we see that velocity and frequency are related: l n nn ππυω v2 == Verify the solution to the wave equation in these ways: • test boundary condition at x=0 • test boundary condition at x=l Docsity.com