14.3.3. Period, Angular Frequency and Frequency

Fig. 14.7 shows again a sinusoidal plot. It describes not the shape of the wave at a certain instant but the displacement of an element (at any fixed location) of the medium as a function of time. We may for simplicity, with φ = 0 and monitor the motion of the element say at 0x=. We then get (0, ) sin( )y t a t ω = − sina tω = −.

This chunk discusses how we can observe the behavior of a sinusoidal wave at a specific location over time. By fixing a position, we can see how the displacement of the medium oscillates as the wave travels. For example, if we take the position at x = 0, the displacement y as a function of time t is expressed using the sine function, which varies between its maximum and minimum values. This tells us how high or low a point on the wave will move at any given moment.

Imagine watching a single point on a flowing river as waves pass by. While the waves travel downstream, that specific point on the river waters rises and falls, illustrating the oscillation akin to how the wave's displacement moves up and down in a sinusoidal manner.

Now, the period of oscillation of the wave is the time it takes for an element to complete one full oscillation. That is sin sin ( T)a t a tω ω− = − + sin( T)a tω ω = − + Since sine function repeats after every 2π, T 2ω π= or 2 Tπω= (14.7).

The period T is defined as the duration taken for one complete cycle of oscillation at a fixed point in the medium. Because of the oscillatory nature of the sine function, which repeats every 2π, we can derive the period, showing that it is inversely related to the wave's frequency. This means that if the wave oscillates quickly, it will have a short period, while slower oscillations will result in a longer period.

Think of a pendulum swinging. If you measure the time it takes to swing back and forth once, that's its period. A pendulum that swings faster will complete each swing in less time, whereas a slower pendulum will take longer, illustrating the relationship between frequency and period.

ω is called the angular frequency of the wave. Its SI unit is rad s –1. The frequency ν is the number of oscillations per second. Therefore, 1 T 2ωνπ= = (14.8) ν is usually measured in hertz.

In this chunk, we define angular frequency (ω) as a measure of how quickly an object oscillates or completes a cycle in radians per second. The frequency (ν) is then derived from the period, showing how many cycles occur each second and is measured in hertz (Hz). Notably, angular frequency and frequency are related through a constant factor involving 2π, highlighting the cyclical nature of wave motion.

Imagine a Ferris wheel going around. If it completes more rotations in a shorter time, it has a higher frequency. Similarly, if you consider how much angular distance (in radians) it covers per second, that gives you its angular frequency. So, a faster-turning Ferris wheel makes a lot more rotations, just like a quicker oscillation means a higher frequency.

In the discussion above, reference has always been made to a wave travelling along a string or a transverse wave. In a longitudinal wave, the displacement of an element of the medium is parallel to the direction of propagation of the wave. the displacement function for a longitudinal wave is written as, s(x, t) = a sin (kx – ωt + φ)

Unlike transverse waves where displacement is perpendicular to the wave direction, in longitudinal waves, the movement of particles happens in the same direction as the wave is traveling. The equation presented describes how the displacement of particles varies over time and position, much like in transverse waves but aligned with the wave's motion.

Consider a slinky toy when you push and pull it in a straight line. The coils compress and expand along the direction of the push—this represents a longitudinal wave where the movement is in the same direction as the wave travels, similar to sound waves in air which compress and rarefy as they propagate.

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