Welcome to AE Resources

Converted document

The Wave Equation

Physically, this means that a disturbance in the initial domain will not be felt throughout the entire domain. In other words, it takes time for a distrubance to propagate through the domain. These disturbances travel along the characteristic directions of the equation.

Derivation of wave equation on a vibrating string

Figure 3↓ shows a very small piece of a string under tension. This derivation makes following assumptions:

Figure 3 A very small piece of a string under tension

Displacements are small.
The tension (T) in the spring is constant, this is the same as assuming the dominant motion is in y direction.

In the following equations, u is the displacement in y direction. m is the mass per unit of length, T is the (constant) tension in the string. Since the dominant motion is in y direction, one needs to concentrate on equilibrium equations in this direction. Newton’s second law can be applied to this small incremement of string in the y direction.

(4) ⎲⎳F_y = mdx(∂²u(x, t))/(∂t²)

(5) − Tsin(θ) + Tsin(θ + (∂θ)/(∂x)dx) = mdx(∂²u(x, t))/(∂t²)

Since θ is small, by using Taylor series and neglecting higher order terms, one can approximate sin(θ) by θ and sin(θ + (∂θ)/(∂x)dx) by θ + (∂θ)/(∂x)dx so

(6) − Tθ + Tθ + T(∂θ)/(∂x)dx = mdx(∂²u(x, t))/(∂t²)

on the other hand from the geometry of the problem θ = (∂u)/(∂x). Finally one have

(7) T(∂²u(x, t))/(∂x²) = m(∂²u(x, t))/(∂t²)

(8) (∂²u(x, t))/(∂t²) = (T)/(m)(∂²u(x, t))/(∂x²)

Equation 8↑ as the same as eq. 2↑ when (T)/(m) = C². In next section, one can show that wave equation has unique solution only if (T)/(m) is a positive number.

Solution Procedure — D’Alembert’s solution

There are quite a few different ways of solving wave equations. One way is separation of variables. Another way of solving the wave equations is know as D’Alembert’s solution. The main idea is to transform the wave equation (eq. 2↑) from x and t domain to ξ and η in order to simplify the equation.

Step one, change of variables In this method, one use the following change of variables is applied:

(9) ξ = x − Ct η = x + Ct

← Previous Page