Waves  (Page 6/6)

A general solution of the wave equation in one dimension is given as :

$$y\left(x,t\right)=f\left(ax\pm bt\right)$$

The solution for the wave moving in positive x-direction is :

$$y\left(x,t\right)=f\left(ax-bt\right)$$

A comparison of this function with the equation derived earlier $y=f\left(x-vt\right)$ , we see that speed of the wave is :

$$\Rightarrow v=\frac{dx}{dt}=\frac{b}{a}$$

Interpreting wave function

What does wave function represent? Here, it is helpful to recall that wave, after all, is energy. A wave function should represent the wave (energy form) – not motion of a particle. We shall see, here, that wave function -apart from describing motion of individual particles - also represents the motion of “disturbance” as required. It represents something which has no material existence. How does it do so?

In the earlier section, the mathematical wave function has been developed to determine disturbance at any position “x” and time instant “t”. Note the important point – it gives disturbance of all particles on the string at any given instant – not a single particle. In order to understand the difference, let us consider a sinusoidal function valid for wave on a taught string :

$$y\left(x,t\right)=A\sin k\left(x-vt\right)$$

At the point of the origin, x = 0, the function is :

$$y\left(x=\mathrm{0,}t=t\right)=A\sin \left(-kvt\right)=-A\sin kvt$$

This function is a sine function in time “t”. As we are aware, this is an equation of simple harmonic motion. It describes the motion of the particle executing SHM at x=0. Similarly, the function representing motion of the particle in “y” direction at x = a is :

$$y\left(x=a,t=t\right)=A\sin k\left(a-vt\right)$$

Clearly, the sinusoidal expression at a given position describes motion of a single particle at that point – not that of wave i.e. the motion of disturbance. The speed of the particle in y-direction is given by the time derivative of particle’s displacement from its mean position:

$$v_p=\frac{dy}{dt}$$

Thus, we see that we can use wave function to interpret the motion associated with the particle or small string segment in y-direction.

Now, let us look at the function as a function of “x” along which wave is considered to travel. At time instant t = 0, the shape of the wave is sinusoidal and the equation representing the shape is :

$$y\left(x=x,t=0\right)=A\sin kx$$

The snapshots corresponding to time instants t = 0,T/4, T/2, 3T/4 and T, where “T” is the time period of oscillation in y –direction, are shown in the figure. Looking at the figures, let us ask this question to ourselves: what does change with “x”? We know that string particle is not moving in x-direction. Definitely, an expression involving change in “x” does not represent motion of a particle.

Looking closely at the figure, we note that a disturbance shown by letter “A” has moved in the positive x-direction at the successive time instants. We can make the same observation with respect to the maximum displacement at “B”. These markings at “A” and “B” here show the amount of disturbance i.e. displacement in y- direction. They have actually moved to right in the successive snapshots. Key here is to understand that we are talking about disturbance represented by letters “A” and “B” – not the particles at those points. We should keep in our mind that string particle has no lateral movement in x-direction. Keeping this in our mind, we can say that motion in x-direction, as described by the wave function, represents motion of a disturbance or more sophistically the motion of a wave. As such, the speed of wave is given by :

$$v=\frac{dx}{dt}$$

This expression gives the speed of the “disturbance” or plainly speaking “wave”. In other words, mathematical representation of wave (energy) is equivalent to the motion of "disturbance". This concept of wave motion is further verified by emphasizing that the argument of sinusoidal function is constant for a given “disturbance” as it travels in x-direction (Note that we are observing motion of a particular disturbance). This implies :

$$y\left(x,t\right)=A\sin k\left(x-vt\right)=\text{constant}$$

$$k\left(x-vt\right)=\text{constant}$$

Differentiating with respect to “x”,

$$k\frac{dx}{dt}=kv$$

$$v=\frac{dx}{dt}$$

We can look at wave motion in yet another way. It can be considered to be the motion of the “wave form” or “the shape of the disturbance”. Look at the figure below. A wave form at an instant is displaced by a distance “∆x” in time interval “∆t”. The speed, at which this wave form is moving, is again obtained by the ratio of displacement in x-direction and time interval or as the time derivative of “x”.