Wave Equation

Wave Equation:

The equation is given as;

describe the vertical vibration of an infinite stretched elastic string, where u (x, t) is the vertical displacement of the string from its rest position at position x, time t. Let the initial displacement and velocity be given as

We take Fourier transform of the wave equations and the initial conditions with respect to the x variables and denoted by u^ (ξ, t) the fourier transform F (u(x, t)). The using the derivatives properties of the Fourier transform,

Using the translation property and the convolution theorem,