Displacement Relation in a Progressive Wave

A wave that travels continuously in the medium from one point to another in the same direction without any change in its amplitude is termed as progressive. 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. 

Waves help in the movement of energy without requiring the transport of stuff. Waves are disturbances that transmit energy from one location to another without carrying substance through a medium. To transfer energy from one spot in a medium to another, waves require oscillating or vibrating sources.

To understand the displacement relation in a progressive wave, let’s take an example of a wave propagating in the positive X-direction. Its displacement will be expressed by:

y(x,t) = a sin (kx-ωt+ φ)

Here ‘a’ is taken as the amplitude of the wave, k is valued as the angular wave number, ω as the angular frequency, (kx – ωt + φ) is the phase, and φ 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 dictates the extremes of the element’s displacement. The initial phase angle φ is given to the constant.

Amplitude – A particle’s maximum displacement in the wave from a point of equilibrium is represented by the amplitude. The phase of a wave’s motion describes its current condition. In determining the wave’s displacement relationship, the wavelength plays an important role. It is comparable to the concept of frequency in that it is defined as the distance over which the shape of the wave is repeated.

Wavelength – A progressive wave’s wavelength is the distance between two consecutive points of the same phase at the same time. It is twice the distance between two successive nodes or antinodes in a stationary wave. The propagation constant, or angular wave number, is k.

SI unit of propagation constant k is radian per metre, or rad m-1.

Type of progressive waves

A progressive wave can either be longitudinal, orbital  or transverse.

Transverse waves – Transverse waves are those in which the medium’s particles are shifted perpendicular to the wave’s propagation direction.

Longitudinal waves – Longitudinal waves are those in which the medium’s particles are shifted along the wave’s propagation direction. If one of the spring’s ends is moved back and forth in the spring’s direction. Then waves are created in which the spring displacement is parallel to the wave propagation direction, and these waves are known as longitudinal waves.

Orbital waves – Particles move in circular orbits as they move over the interface, which involves components of both longitudinal and transverse waves. As a result, ocean surface waves are orbital waves (also called interface waves).

Evaluating intensity of a progressive wave

We get a nice sensation in the ear when we hear the sound of a violin and the instrument flute or harmonium, whereas 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 intensity 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 direction.

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