Math Help

1-D Wave Equation (String Vibrations) Click here to go to an example.

In a Nut Shell: The 1-D wave equation (string vibration, see figure below) is governed

by the following partial differential equation:

∂²u/∂t² = a² ∂²u/∂x² ---------------------- (1)

where u = u(x,t) = the displacement of the string

x = the position along the string

t = the time that the displacement occurs at x

and

a is the speed of wave propagation (material constant)

Note: Since the partial differential equation is second order in both its derivatives with

respect to x and t you need two boundary conditions and two initial conditions.

The desired outcome is to predict the displacement of the string, u(x,t), subject to

its boundary and initial conditions (provided in the tables below).

The common boundary conditions at the ends of the string are as follows:

a. Fixed end at x = 0 u(0,t) = 0

b. Fixed end x = L u(L,t) = 0

c. Sliding end condition at x = 0 ∂u(0,t)/ ∂x = 0

d. Sliding end condition at x = L ∂u(L,t)/ ∂x = 0

or any combination of these boundary conditions

The common initial conditions at the ends of the string are as follows:

a. Prescribed displacement at t = 0 u(x,0) = f(x)

b. Prescribed speed of string at t = 0 ∂u(x,0)/∂t = g(x)

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