Welcome to AE Resources

Converted document

The Wave Equation

Step five, apply initial conditions F and G are not defined at this point and will be defined by initial condition. For a long string that it can be approximated by infinite length, with the following initial condition one can find f and g

(24) u(x, 0) = f(x) (∂u)/(∂t) = g(x)

The first equation is the initial shape and the second one is the initial velocity. Of course since the wave equation is second order in time, once needs two initial conditions. For an infinite string the finial solution can be written as

(25) u(x, t) = (1)/(2)⎛⎝f(x − Ct) + f(x + Ct) + (1)/(C)^x + Ct⌠⌡_x − Cth(s)ds⎞⎠

This solution shows that the initial shape and the average of initial velocity will travel along the two characteristic line of wave equations.

Separation of variable is a much better way of solving wave equation with different boundary conditions.

Boundary Conditions

As a second-order PDE, boundary conditions must be specified at all edges. A common boundary condition is the clamped boundary condition, where the solution is zero along the edges.

(26) u(x = 0, t) = 0 u(x = L, t) = 0

Computational Modeling

The wave equation is well-suited to computational methods. Finite differencing techniques can work quite well, provided a stable scheme is chosen.

← Previous Page