Displacement Relation in Progressive Wave

Waves come in a variety of shapes and sizes. A wave is an energy transfer disturbance that transports energy from one location to another. When a wave moves, only energy, not matter, is exchanged.

The medium is the substance through which a wave travels. That medium oscillates back and forth, eventually returning to its initial position.

Progressive Wave

A progressive wave is a wave that travels from one point in the medium to another. A transverse or longitudinal progressive wave can exist.

A progressive wave is a wave that progresses from one point A in the medium to another point B in the medium. A travelling wave, also known as a progressive wave, is a wave that travels in the same direction in the same medium without changing. Furthermore, there are two types of progressive waves: transverse and longitudinal waves.

Plane progressive Wave

When the particles of a medium vibrate simply harmonically around their mean positions throughout the propagation of a wave through it, the wave is said to be a plane progressive harmonic wave.

The particles in a medium tend to oscillate harmonically about their mean positions throughout the transmission of a wave; in this case, the wave is referred to as a plane progressive harmonic wave.

Displacement relation for a progressive harmonic wave

Generally the displacement in a sinusoidal wave propagating in the positive x- direction is given by:

yx,t=asin⁡(kx-ωt+ϕ)

Here, a is the amplitude of the wave

K is the angular wave number

is the angular frequency

(kx-ωt+ϕ) is the phase and 

is the phase constant or phase angle.

The oscillation of a string element is represented by the sine function and the time-dependent phase of a wave, and the amplitude of the wave determines the extremes of the element’s displacement. The initial phase angle is the name given to the constant.

Simple Harmonic progressive wave

This waveform continues to travel in the same direction without changing its shape. Furthermore, the medium’s particles tend to move in a harmonic manner around their mean position, with the same amplitude and period.

Wavelength of a progressive Wave

For a progressive wave, the wavelength is the distance measured between two successive points of the same phase at a given time. This is twice the distance measured between two consecutive nodes or antinodes in the case of a stationary wave. The propagation constant is denoted by the letter ‘k.’ the radian per metre, or rad m-1, is the SI unit. 

k=2πλ

Frequency and period of a progressive wave

The time period ‘T’ of a wave oscillation is the amount of time it takes for any component of the medium to complete one full oscillation. The following relationship connects this to ‘’ or angular frequency:

 ω=2πT

Here, wave frequency “v” is mentioned as 1T and it is also related to angular frequency as

v=2π

It can alternatively be described as the number of oscillations prepared per unit time in a string element as the wave passes through it. In most cases, this is calculated in Hertz.

Periodic Travelling Wave

A periodic travelling wave is a one-dimensional periodic function that moves at a constant speed. As a result, it is a unique type of spatiotemporal oscillation in which both space and time are periodic.

Many mathematical equations, including self-oscillatory systems, excitable systems and reaction–diffusion–advection systems, depends on periodic travelling waves. These types of equations are commonly employed as mathematical models in biology, chemistry, and physics, and various empirical examples of periodic travelling waves have been discovered.

Sinusoidal Wave

A sine wave, also known as a sinusoidal wave or simply a sinusoid, is a mathematical curve defined in terms of the sine trigonometric function. It’s a smooth periodic function and a sort of continuous wave. It can be found in many domains, including mathematics, physics, engineering, signal processing, and many more.

Its most basic form in function of time (t) is:

yt=Asin2π ft+ψ=Asin(ωt+ψ)

Here,

A is amplitude,

F is ordinary frequency

ω=2πf is angular frequency and,

ψ=phase .

In physics, the sine wave is significant because it preserves its wave structure when combined with another sine wave of the same frequency, phase, and magnitude. This characteristic is unique to the periodic waveform. This attribute gives it significance in Fourier analysis and identifies it acoustically.

By its ease of representation and some specific favourable qualities, sine waves are one of the most commonly used waveforms. A sine wave, also known as a sinusoidal wave, is a smooth, repeated oscillation described by a curve. The sine wave is defined as “a wave shape whose amplitude is always proportional to the sine of its displacement angle at all points in time.”

By combining sine waves, any wave can be created. The sine wave has a pattern that repeats itself. The wavelength is the length of the sine wave’s repeating portion.

Generation of Sine Waves

There are several methods for generating sine waves. They are as:

• Quartz Crystal Oscillator
• Negative Resistance Oscillator
• Basic Single Coil AC Generator
• Phase Shift Oscillator
• Wein Bridge Oscillator etc.

Conclusion

In this article we have studied waves and progressive waves. At last we discussed Sinusoidal waves and traveling waves. A wave in which the medium’s particles move in a gradation of speeds along the direction of wave propagation, with the faster overtaking the slower and being overtaken by the faster.