Displacement Relation For a Progressive Wave

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.

• Transverse Waves: Waves in which the particle movement in the medium is measured perpendicular to the wave direction or wave travel direction are known as transverse waves.
• Longitudinal Waves: Particle displacement persists in the same direction as the wave travels with Longitudinal Waves.

Here you can learn more about transverse and longitudinal waves.

Plane Progressive Harmonic Wave

The particles in a medium tend to oscillate harmonically around their mean locations throughout the transmission of a wave. In this instance, the wave is referred to as a plane progressive harmonic wave.

Simple Harmonic Progressive Wave

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

Characteristics of SHM Wave

• While the wave travels through a medium, all particles or components of the medium display SHM
• As they vibrate, all particles keep the same amplitude
• The medium is used to conduct energy
• The entire number of particles vibrates at the same time.

Plane Progressive Harmonic Wave: Displacement Relation

The displacement of a sinusoidal wave flowing in the x-direction (positive) is given below in terms of a planar progressive harmonic wave:

                                   y = a sin(kx – t + )

The amplitude of the wave is denoted by ‘a’, the angular wave number is denoted by ‘k’, and the angular frequency is denoted by ‘’. The phase is written as (kx – t + ), where is the phase angle and t is the phase constant.

The sine function, like the time-dependent phase (wave), resembles the oscillation of a string component, however the amplitude of the wave specifies the component’s displacement extremes. It’s vital to remember that the initial phase angle is the constant.

Wavelength of 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=2/

Frequency and Period of Progressive Wave

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

                                          =2/T

The wave frequency ‘v’ is expressed as 1/T and is connected to angular frequency as follows:

                                         v=/2

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

Phase

The phase of the function is defined as the argument (kx – t +) of the oscillatory term sin (kx – t +). It describes the wave’s current stage of motion. Points on a wave that travel in the same direction and rise and fall at the same time are said to be in phase. Points on a wave that go in opposing directions, such as one rising while the other falls, are said to be anti-phase.

Amplitude

The greatest displacement of a particle in a wave from its equilibrium location is measured in amplitude.

Angular Wave Number

In terms of cycles per unit distance, the wavenumber is the spatial frequency of a wave. Similar to the idea of frequency, it may also be described as the number of waves that exist across a certain distance.

Angular Frequency

The angular displacement per unit time of the rate of change of phase of a waveform is known as angular frequency.

We may express it mathematically as,

=2T = 2f                                         

where T is the frequency and f is the time period of the sinusoidal function that represents the wave.

Intensity of Progressive Wave

We get a nice sensation in the ear when we hear the sound of a violin and the instrument flute or harmonium, but we get an unpleasant sensation in the ear when we hear the sound of a pistol, a horn, a motor car, and so on.

The loudness of a sound is determined by the strength of the sound wave and the sensitivity of the ear.

The intensity is commonly defined as the amount of energy traversing per unit area per unit time in a direction perpendicular to the wave’s propagation path.

W m–2 is the unit of measurement for intensity.

Progressive Wave – Important Points

• Each particle in the medium produces a vibration that is centred on its mean location. The disruption spreads from one particle to the next
• The medium’s particles vibrate with equal amplitude around their mean locations
• Each particle, or we may say, each subsequent particle of the medium, moves in the same direction as its predecessor along the wave’s propagation path, albeit at a later period.
• In most cases, no particle stays in the rest position indefinitely. The particles are temporarily at rest at extreme locations twice throughout each vibration. Various particles arrive at different positions at different times.
• Wave crests and troughs define the transverse evolution of the waves. The longitudinal waves are characterised by compressions and rarefactions
• There is also an energy transfer that occurs across the medium in the direction of progressive wave propagation
• When particles travel through the mean location, they all have the same maximum velocity
• The displacement of the particle’s velocity and acceleration are the same when separated by the equation m, where m is an integer.

Conclusion

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 dictates the extremes of the element’s displacement. The initial phase angle is the name given to the constant.