A wave which generally travels continuously in a medium of the same direction without the change in its amplitude is known as a travelling wave or a progressive wave.
Let us consider the example of a progressive wave on a string. Here, we will describe the relation.
In the subjects of Maths and Physics, we will see that this topic is very much related. A propagating dynamic disturbance is known as the waves of one or even more than one quantity. We will discover further about the waves and its progressiveness in this article.
What is Progressive Wave
An equation generally can be formed to represent generally the displacement of a particle that is vibrating in a medium through which a wave passes. Thus, we can see that each particle of a progressive wave executes simple harmonic motion of the same period and amplitude differing in phase from each other.
Let us now assume that a wave that is the progressive generally travels from the origin O along the positive direction of X-axis from left to right. The displacement of a particle at a given instant is as follows:
y = a sin ωt …... (1)
where we can see that a is the amplitude of the vibration of the particle and then ω = 2πn.
The displacement of the particle denoted by letter P at a distance x from O at a given instant is given by,
y = a sin (ωt - φ) …... (2)
If two particles are said to be separated by a distance by symbol λ they will differ by a phase of 2π. Therefore we can say that the phase denoted by symbol φ of the particle P at a distance
x is φ = 2π/λ x
y = a sin *ωt - 2πx/λ …... (3)
Since we see that the symbol ω = 2πn = 2π (v/λ), the equation is given by
y = a sin [(2πvt/λ) - (2πx/λ)]
y = a sin 2π/λ (vt – x) …... (4)
Since ω = 2π/T we see that the equation (3) can also be written as,
y = a sin 2π *t/T – x/λ) …... (5)
If the wave that generally travels in the direction which is opposite the equation generally becomes
y = a sin 2π (t/T + x/λ) …... (6)
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Progressive Wave - Important Points
Each particle which is present in the medium executes the vibration which is about its mean position. The disturbance that progresses is from one particle to another.
The particles of the medium vibrate with the same amplitude about their mean positions.
Each particle or we can say that the successive particle of the medium performs a motion similar to that of its predecessor along the direction of the propagation of the wave, but later in time.
The phase of every particle that changes from 0 to 2π.
No particle generally remains permanently at the position which is at rest. Twice during each vibration, the particles are momentarily at rest at extreme positions. The particles which are different attain the position at different times.
The transverse progression of the waves is characterised by crests and troughs. The waves which are the longitudinal are characterised by compressions and rarefactions.
There is an energy transfer as well which is across the medium in the direction of propagation of progressive waves.
All the particles generally have the same maximum velocity when they pass through the mean position.
The displacement of the velocity and acceleration as well of the particle separated by the equation that is mλ are the same, where m is an integer.
Intensity of Progressive Wave
If we generally hear the sound which is produced by violin and the instrument flute or harmonium, we get a pleasing sensation in the ear whereas the sound which is produced by a gun, horn a motor car, etc., generally produces unpleasant sensation in the ear.
The loudness that is generated by the sound that depends on the intensity of the sound wave and sensitivity of the ear.
The intensity is generally defined as the amount of energy which is crossing per unit area per unit time that is perpendicular to the direction with respect to the propagation of the wave.
Intensity is measured in the W m–2.