COMPUTATIONAL MODELING FOR ENGINEERING MECN 6040 Professor: Dr. Omar E. Meza Castillo Department of Mechanical Engineering

FINITE DIFFERENCE METHODS FOR HYPERBOLIC PROBLEMS Wave Equation

PARTIAL DIFFERENTIAL EQUATIONS ▪ Our study of PDE’s is focused on linear, second-order equations ▪ The following general form will be evaluated for B 2 - 4AC (Variables – x and y/t)

B 2 -4AC Category Example < 0 Elliptic Laplace equation (steady state with 2 spatial dimensions) = 0 Parabolic Heat conduction equation (time variable with one spatial dimension) >0 Hyperbolic Wave equation (time-variable with one spatial dimension) PARTIAL DIFFERENTIAL EQUATIONS

WAVE EQUATION Mechanical vibrations of a guitar string, or in the membrane of a drum, or a cantilever beam are governed by a partial differential equation, called the Wave Equation. To derive the wave equation we consider an elastic string vibrating in a plane, as the string on a guitar. Assume u(x,t) is the displacement of the string away from its equilibrium position u=0.

u(x,t) x u T(x+  x,t) T(x,t) x x+  x   WAVE EQUATION

Newton’s Law: F=ma Vertical : where ρ is the mass density function of the string Horizontal : As we assume there is only vertical motion of the string T(x+  x,t) T(x,t)   WAVE EQUATION

LetThen, So, F vert = ma implies, from the previous slide, that Or, WAVE EQUATION

Now, tan β is just the slope of the  string at x+Δx and tan  is the slope at x. Thus, So, T(x+  x,t) T(x,t)   WAVE EQUATION

Now, take the limit as Δx -> 0, to get This the Wave Equation in one dimension (x) WAVE EQUATION

WAVE EQUATION BOUNDARY CONDITIONS Since the wave equation is second order in space and time, we need two boundary conditions for each variable. We will assume the string x parameter varies from 0 to L. Typical set-up is to give boundary conditions at the ends of the string in time: u(0,t) = 0 u(1,t) = 0 u(0,t) = 0 u(1,t) = 0 and space initial conditions for the displacement and velocity: and space initial conditions for the displacement and velocity: u(x,0) = f(x) u t (x,0)=g(x) u(x,0) = f(x) u t (x,0)=g(x)

WAVE EQUATION BOUNDARY CONDITIONS

ONE DIMENSIONAL WAVE EQUATION ▪ We will solve this equation for x and t values on a grid in x-t space: Approximatesolution u ij= u(x i, t j ) at grid points

FINITE DIFFERENCE APPROXIMATION ▪ To approximate the solution we use the finite difference approximation for the two derivatives in the wave equation. We use the centered-difference formula for the space and time derivatives:

FINITE DIFFERENCE APPROXIMATION ▪ Then the wave equation (u tt =c 2 u xx ) can be approximated as Or, Or, Let p = (ck/h) Solving for u i,j+1 we get:

ONE DIMENSIONAL WAVE EQUATION ▪ Note that this formula uses info from the j-th and (j-1)st time steps to approximate the (j+1)-st time step:

ONE DIMENSIONAL WAVE EQUATION ▪ The solution is known for t=0 (j=0) but the formula uses values for j-1, which are unknown. So, we use the initial condition u t (x,0)=g(x) and the centered difference approximation for u t to get ▪ At the first time step, we then get

ONE DIMENSIONAL WAVE EQUATION ▪ Simplifying, we get ▪ So,