A travelling wave, also known as a progressive wave, is a wave that travels constantly in the same direction without changing its amplitude.

Traveling waves are another name for progressive waves. It continues to unveil itself. A progressive wave is defined by its ability to travel indefinitely without stopping or changing in amplitude or direction. A propagating dynamic disturbance is defined by the presence of one or more quantities.

Equation of a plane progressive wave

The displacement of a vibrating particle in a medium through which a wave flows can be represented by an equation. As a result, each particle of a progressive wave performs a simple harmonic motion with the same period and amplitude but different phases.

Assume that a progressive wave goes from left to right along the positive X axis, starting at the origin. The instant displacement of a particle is given by

amplitude of vibration of the particle

At a given instant, the displacement of the particle P at a distance x from the origin is given by,

If two particles are separated by a distance , they will have a phase difference of . As a result, the particle P’s phase at a distance x is

Since , the equation can be written as,

We know that, .So, we can write equation (3) as

If the wave is travelling in the opposite direction, the equation will be,

• Variation of phase with time:

At a constant distance, the phase changes continuously with time.

At a given distance x from O, let be the phase of the particle at given time respectively.

In the time interval ,the phase change of a particle is . If = T, then = 2. This demonstrates that the phase of a particle becomes the same after a time period T.

• Variation of phase with distance:

At a given time t, the phase changes periodically with distance x. Let be the phase for two particles at a distance respectively from the origin O at a given time t.

Now,

Therefore,

When the wave goes from left to right, the forward points are out of phase, as indicated by the negative sign.

The phase difference between two particles with a path difference is   when and .

Types of Progressive waves 

In general, progressive waves are divided into two types:

• Transverse waves
• Longitudinal waves

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

Production of Transverse waves

To illustrate the motion of the source in generating waves in a medium, consider a loose spring coil (slinky spring). The slinky is a soft spring with a short initial length but a long extended figure.

Consider a horizontal spring with one end that is fixed. A pulse of a wave with a displacement pattern will move along the spring when the free end is shifted from side to side. This demonstrates that particle movement is perpendicular to the propagation path of a wave, resulting in transverse waves.

Longitudinal waves: Longitudinal waves are those in which the medium’s particles are shifted along the wave’s propagation direction.

Production of Longitudinal waves

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.

Characteristics of progressive wave

1. Each particle in the medium vibrates in relation to its mean location. The disruption spreads from one particle to the next.
2. The medium’s particles vibrate with the same amplitude around their mean positions.
3. Along the propagation of the wave, each subsequent particle of the medium follows in the footsteps of its predecessor, although at a later period.
4. Every particle’s phase shifts from 0 to .
5. There is no such thing as a particle that is always at rest. The particles are momentarily at rest at extreme positions twice throughout each vibration, with distinct particles achieving the position at various times.
6. Crests and troughs distinguish transverse progressive waves. Compressions and rarefactions characterise longitudinal waves.
7. Energy is transferred across the medium in the direction of progressive wave propagation.
8. When the particles pass through the mean position, they all have the same maximum velocity.
9. The particle separated by has the same displacement, velocity, and acceleration, where m is an integer.

Intensity of Progressive Wave

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

The intensity of the sound wave and the sensitivity of the ear determine the loudness generated by the sound.

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.

The S.I. unit of intensity is watt per square metre ().

Conclusion

A progressive wave is defined as one in which the wave profile goes forward with the wave’s speed. Waves in both the transverse and longitudinal directions are progressive. In a progressive wave, the amplitude and frequency of particle vibrations are the same, but the phase of the vibration varies from point to point along the wave.

Longitudinal waves are those in which the medium’s particles are shifted along the wave’s propagation direction. Whereas, Transverse waves are those in which the medium’s particles are shifted perpendicular to the wave’s propagation direction.