From solving the 2 dimensional wave equation on a disc
From solving the 2 dimensional wave equation on a disc
Creates an animation of the solution to the wave equation on the real line, u_{tt} = c^2 u_{xx}, with initial conditions either u(x,0) = f(x), u_t (x,0) = 0 or u(x,0) = 0, u_t (x,0) = f(x).
Creates an animation of the solution to the wave equation on the bounded interval [0,l], u_{tt} = c^2 u_{xx}, with initial conditions either
- u(x,0) = f(x), u_t (x,0) = 0, or
- u(x,0) = 0, u_t (x,0) = f(x).
and the two boundry conditions are
i) u(0,t) = 0 or
ii) u_t (0,t) = 0
and
i) u(l,t) = 0 or
ii) u_t (l,t) = 0.
If the initial conditions are 1, then
u(x,0) = \sum bn sin(n π x /l) = f(x) or
u(x,0) = \sum bn sin((n+1/2) π x/l) = f(x) or
u(x,0) = \sum bn cos((n+1/2) π x/l) = f(x) or
u(x,0) = \sum bn cos(n π x /l) = f(x).
If on the other hand initial conditions are 2, then
u_t (x,0) = \sum bn n π c / l * sin(n π x /l) = f(x) or
u_t (x,0) = \sum bn (n+1/2) π c/l * sin((n+1/2) π x/l) = f(x) or
u_t (x,0) = \sum bn (n+1/2) π c/l * cos((n+1/2) π x/l) = f(x) or
u_t (x,0) = \sum bn n π c / l * cos(n π x /l) = f(x).
These bn are the ones you need to calculate to create the animation. This should probably be done by expanding f(x) into a fourier series.
Creates an animation of the solution to the two dimensional wave equation on the bounded interval [0,1]x[0,1], u_{tt} = c^2 (u_{xx} + u_{yy}, with initial conditions either
- u(x,y,0) = f(x,y), u_t (x,0) = 0, or
- u(x,0) = 0, u_t (x,0) = f(x).
and the boundry conditions
i) u(0,y,t) = 0
ii) u(1,y,t) = 0
iii) u(x,0,t) = 0, and
iv) u(x,1,t) = 0.
When solving this you get an expression like
u(x,y,t) = \sum_{n,m} a_{n,m} sin(π t \sqrt{n^2+m^2}) * sin(n π x) * sin(m π y)
these a_{n,m}:s are the ones you need to calculate from the initial condition if you want to create an animation not already calculated by me.
OBS! This creates a gif file as the animation is very slow and without a considerable drop in framerate cannot be played in realtime.
Creates an animation of the solution to the two dimensional wave equation on the disc [0,ρ]x[0,2π]. Δu = u_{tt} u(ρ,φ,t) = 0 u(r,φ,0) = f(r,φ) u_t(r,φ,0) = 0
This one is done numerically as the coefficients that need to be calculated are not nice to do by hand. Simply define an f and you'll be good to go.If the number of frames are about 100, it takes a couple of seconds to run, choose a higher number than that and be prepared to wait for a while.