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General Wave Equation

Let us consider a disturbance propagating along say, X-axis. A disturbance in some sort of a displacement of the medium which occurs at point x, at time t. If this disturbance is moving with a constant velocity v along the X-axis, it would move a distance Δ x = v Δ t, in time Δ t. That is, the disturbance reaches the point x’ = x + Δ x at time t’ = t + Δ t. Assuming that the given disturbance does not wither away as it moves, i.e. it keeps its shape intact, we infer that the displacement produced at point (x’, t’) must be same as displacement at (x, t). Denoting the displacement as the function ƒ(x, t), we find

ƒ(x, t) = ƒ(x’, t’) = ƒ(x + Δ x, t + Δ t)

A function which uniquely satisfies above condition is

ƒ(x, t) = ƒ(x – vt)

where, v = Δ x/Δ t, is the velocity of propagation of the disturbance.

The above equation, in general, represents a wave motion, i.e. a disturbance which moves along positive X-axis. A disturbance moving along negative X-axis is obtained by replacing v by – v; that is, the function

g (x, t) = g(x + vt)

represents, in general, a wave travelling along negative X-axis.

Any partial differential equation whose solutions are the functions of the type given under represents wave equation. For example, if we put z = x – vt, we find

Both the above equations represent possible forms of wave equations in one dimension. However, it is valid only for wave moving in positive X-direction. The above eq. by virtue of having v², expresses the wave equation for waves moving either towards +ve, or –ve X-axis. Hence, we shall refer to the above eq. as the standard equation of a travelling wave in one dimension.