Progressive Wave

Progressive waves are types of waves that generally move in the same direction without changing the value of amplitude. In physics, waves are a flow of energy. Waves are of various types, some are periodic, and some are non-periodic. Progressive waves are non-periodic. They move in the perpendicular direction to the motion of particles. The displacement of wave formulas describes the displacement of progressive waves. Displacement relation in a progressive wave equation demonstrates the successful relationship between any progressive wave and its motion. This relation also describes the motion of particles moving with the progressive waves. 

Equation Of Progressive Wave

The progressive wave equation defines the movement of particles flowing with the progressive waves in the same medium. In the motion of particles with progressive waves, the particles show simple harmonic motion due to their small size. The simple harmonic motion of all the particles contains the same amplitude. However, their phase differs under some circumstances. 

Derivation Of Equation Of Progressive Wave 

For understanding the progressive wave equation, assume the progressive wave is moving from origin 0 to the positive direction of the x-axis. The value of displacement at any instant point is taken in respect of the Y-axis and written as, 

y = a sin ωt …… Equation (1)

In the above equation, a is the value of the amplitude of the motion and ω = 2πn

The value of displacement of a particle from origin 0 to any point on the X-axis is written as, 

y = a sin (ωt – φ) …. Equation(2)  

In the above equation, φ = phase difference. 

In the same medium, if the two particles contain the distance λ, then they have a phase difference of 2π. Then the value of phase difference becomes, 

φ = 2π/λ x

Displacement Of Wave Formula

The wave’s phase and amplitude describe a displacement of the wave formula. This formula is a part of the displacement relation in a progressive wave equation. The displacement in progressive waves contains the same frequency and is considered moving on the Y-axis. Here, the displacement will be denoted by the function y. 

Hence, 

y(x,t)= A sin(kx – ωt + φ) 

X = the motion of waves on X-axis

So, the above equation describes the displacement of the wave formula. This formula also describes displacement relation in a progressive wave equation. 

Types Of Progressive Waves

There are two types of progressive waves:

• Longitudinal progressive wave 
• Transverse progressive wave 

Longitudinal Progressive Waves 

The longitudinal progressive waves are those waves that move in the direction of motion of the particles. The motion of particles and longitudinal progressive waves remains in a parallel direction. These waves make a 0°angle with their particles. These waves are defined using phenomenons like compression and refraction. 

Transverse Progressive Waves 

Transverse waves move in the perpendicular direction to the movement of particles. The motion of particles and transverse progressive waves remain in the perpendicular direction. These waves make a 90°angle with their particles. These waves are defined using phenomena like trough and crest. 

Characteristics Of Progressive Waves 

Some of the characteristics of progressive waves are:

• In progressive waves, each particle present in the medium will execute the simple harmonic motion at its mean position. However, other particles in the surrounding area are also affected by this simple harmonic motion. 
• All the particles move at their mean position and the constant amplitude. 
• The phase difference of simple harmonic motion in the particles of progressive waves remains within the phase value of 0 to 2π. 
• All the particles move at the same intensity as their predecessor. Although, due to vibration, these particles also exchange energy. 
• All the particles present in the progressive waves contain constant velocity and move with a constant acceleration.
• Particles of wave motion never remain at rest. They continuously show momentum, due to which they release energy in the surroundings. 

Conclusion 

Progressive waves are also called travelling waves. They are called travelling waves because they continuously move in the direction of motion. The progressive wave equation describes the displacement relation in a progressive wave equation. However, the displacement relation in a progressive wave equation describes how the value of displacement changes with the change in wave nature. If the wave gets compressed or refracted, the value of displacement increases or decreases. All the particles present in the progressive wave contain the exact nature. If the velocity of one particle increases or decreases, then the velocity of successive particles also increases or decreases.