2.1 Transverse harmonic waves  (Page 5/6)

In the first case, the slope is positive and hence particle velocity is negative. It means particle is moving from reference origin or mean position to negative extreme position. In the second case, the slope is negative and hence particle velocity is positive. It means particle is moving from positive extreme position to reference origin or mean position. Thus two forms represent waves which differ in direction in which particle is moving at a given position.

Once we select the appropriate wave form, we can write wave equation in other forms as given here :

$$y\left(x,t\right)=A\sin \left(kx-\omega t\right)=A\sin k\left(x-\frac{\omega }{kt}\right)=A\sin \frac{2\pi }{\lambda }\left(x-vt\right)$$

Further, substituting for “k” and “ω” in wave equation, we have :

$$y\left(x,t\right)=A\sin \left(\frac{2\pi }{\lambda }x-\frac{2\pi }{T}t\right)=A\sin 2\pi \left(\frac{x}{\lambda }-\frac{t}{T}\right)$$

If we want to represent waveform moving in negative “x” direction, then we need to replace “t” by “-t”.

Speed of wave along a taut string

The wave speed, in general, depends on the medium of propagation and its state. In the case of string wave, the wave motion along a stretched string approximates harmonic transverse wave under simplified assumptions. We consider no loss of energy as the wave moves along the string.

For a simplified consideration, we assume that inertial property (mass per unit length) is small and uniform. A light string is important as we consider uniform tension in the string for our derivation. It is possible only if the string is light. When the string changes its form, the elastic force tends to restore deformation without any loss of energy. This means that we are considering a perfectly elastic deformation.

The mechanical transverse wave speed on a taut string is a function of the inertial property (mass per unit length) and tension in the string. Here, we shall investigate the motion to drive an expression of speed. For this we consider a single pulse and a small element of length ∆l as shown in the figure. Let mass per unit length be μ, then the mass of the element is :

$$\text{Δ}m=\mu \text{Δ}l$$

The small element is subjected to tensions in the string. The element does not move sideways along the circular path in the case of transverse wave motion. Rather it is the wave form which approximates to a curved path. For our mechanical analysis, however, the motion of wave is equivalent to motion of small string along the curved path having a radius of curvature “R” moving at a speed equal to the wave speed, “v”. Thus, we can analyze wave motion by assuming as if the small element is moving along a circular path for a brief period. Now, the string wave moves with constant speed and as such there is no tangential acceleration associated with the analysis. There is only centripetal acceleration in the radial direction.