Waves  (Page 5/6)

$$y=y\left(x,t\right)$$

We can further specify the nature of the mathematical function by associating the speed of the wave in our consideration. Let “v” be the constant speed with which wave travels from the left end to the right end. We notice that wave function at a given position of the string is a function of time only as we are considering displacement at a particular value of “x”. Let us consider left hand end of the string as the origin of reference (x=0 and t=0). The displacement in y-direction (disturbance) at x=0 is a function of time, “t” only :

$$y=f\left(t\right)$$

The disturbance travels to the right at a constant speed “v”. Let it reaches a point specified as x=x after time “t”. If we visualize to describe the origin of this disturbance at x=0, then time elapsed for the disturbance to move from the origin (x=0) to the point (x=x) is “x/v”. Therefore, if we want to use the function of displacement at x=0 as given above, then we need to subtract the time elapsed and set the equation as :

$$y=f\left(t-\frac{x}{v}\right)=f\left\{\frac{\left(vt-x\right)}{v}\right\}$$

Division by a constant like speed of wave does not change the nature of argument. Hence :

$$y=f\left(vt-x\right)$$

We can exchange terms of the argument of the wave function as well :

$$y=f\left(x-vt\right)$$

The exchange of terms aound negative sign introduces a phase difference between waves represented by two forms. The coorect order of terms depend on the state of motion of the particle at x=0 and t=0. We shall explore this aspect in detail when cosidering transverse harmonic waves in a separate module.

The wave functions represent displacement of unidirectional wave. This describes the displacement (disturbance) along the string which moves in positive x-direction. Extending the derivation, the function representing a wave moving in negative x-direction is :

$$y=f\left(x+vt\right)$$

It is the sign (minus or plus) separating two terms of the argument that determines the direction of wave with respect to positive x-direction. We should, however, be careful that all function of the forms as indicated above may not represent a wave. For example, the function should evaluate to a finite value as displacement needs to be finite. For this reason, a function given here under is invalid wave function (not defined for x=0 and t =0) :

$$y=\frac{1}{x+vt}$$

On the other hand, following functions are bounded by finite values and hence are valid wave function :

$$y=A_o{e}^{-\left(t-\frac{x}{\lambda /T}\right)}$$

$$y=A_o\sin \left(\omega t-kx\right)$$

Wave equation

The propagation of wave is governed by a differential equation. For wave in one dimension, the equation is given as :

$$\frac{d^{2}u}{d{t}^2}=k\frac{d^{2}u}{d{x}^2}$$

Here “k” is a constant equal to square of the wave speed. The parameter “u” is disturbance or amplitude parameter, which can be displacement, pressure or electric field depending on the wave in question. The parameter “u” is a function of position (x) and time (t). This equation can be written with respect to “y” and “z” directions to represent two or three dimensional waves.

Confining ourselves to one dimensional wave, putting $k=v^2$ and u = y for displacement from the mean position, the differential equation takes the form :

$$\Rightarrow \frac{d^{2}y}{d{t}^2}=v^2\frac{d^{2}y}{d{x}^2}$$