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Longitudinal waves are characterized by the directions of vibration (disturbance) and wave motion. They are along the same direction. It is clear that vibration in the same direction needs to be associated with a “restoring” mechanism in the longitudinal direction.
Consider the case of sound wave. The wave comprises alternating compressions and rarifications. The compressed zone is characterized by higher pressure, which tends to expand the air in the zone. Thus, there is alteration of pressure as the zone transitions from compression to rarification and so on.
It is intuitive to note that all medium types (solid, liquid or gas) support longitudinal waves.
We shall attempt here to evolve a mathematical model of a traveling transverse wave. For this, we choose a specific set up of string and associated transverse waves traveling through it. The string is tied to a fixed end, while disturbance is imparted at the free end by up and down motion. For our purpose, we consider that pulse is small in dimension; the string is light, elastic and homogeneous. These assumptions are required as we visualize a small traveling pulse which remains undiminished when it moves through the string. We also assume that the string is long enough so that our observation is not subject to pulse reflected at the fixed end.
For understanding purpose, we first consider a single pulse as shown in the figure (irrespective of whether we can realize such pulse in practice or not). Our objective here is to determine the nature of a mathematical description which will enable us to determine displacement (disturbance) of string as pulse passes through it. We visualize two snap shots of the traveling pulse at two close time instants “t” and “t+∆t”. The single pulse is moving towards right in the positive x-direction.
Three positions along x-axis named “1”,”2” and “3” are marked with three vertical dotted lines. At either of two instants as shown, the positions of string particles have different displacements from the undisturbed position on horizontal x-axis. We can conclude from this observation that displacement in y-direction is a function of positions of particle in x-direction. As such, the displacement of a particle constituting the string is a function of “x”.
Let us now observe the positions of a given particle, say “1”. It has certain positive displacement at time t = t. At the next snapshot at t=t+∆t, the displacement has reduced to zero. The particle at “2” has maximum displacement at t=t, but the same has reduced at t=t+∆t. The third particle at “3” has certain positive displacement at t=t. At t=t+∆t, it acquires additional positive displacement and reaches the position of maximum displacement. From these observations, we conclude that displacement of a particle at any position along the string is a function of “t”.
Combining two observations, we conclude that displacement of a particle is a function of both position of the particle along the string and time.
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