Progressive Wave: Characteristics, Equation and Sample Questions

Progressive waves are described as waves carrying energy from the initial place to the neighboring particles. These waves are expressed as mathematical equations, in the case when these waves are bounded and single-valued. Progressive waves are also described as continuously traveling in the same direction and amplitude. They are also known as traveling waves. The various aspects these waves possess are wavelength, frequency, and velocity. The two broad categories in progressive waves are transverse and longitudinal waves. In this article, we will have a look at the progressive wave and its equation.

Keyterms: Wave, Wavelength, Velocity, Energy, Frequency, Amplitude, Transverse waves, Longitudinal waves, Integer

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What is Progressive wave?

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Progressive waves aid in the transmission of vibratory motion in the onward direction. Progressive waves have the characteristics of moving in a forward direction, which also involves the action of carrying with itself the energy to the surrounding particles also.

Progressive Wave

Characteristics

The other characteristics of progressive wave include:

• These waves move with a definite velocity.
• The pressure and density have the same changes occurring at all medium points.
• Every particle present has a similar motion to its forebear, along with the time propagation. The propagation of time is accompanied later in the time interval.
• During the movement, a transfer of energy occurs across the medium, across the particles of the wave. The transfer occurs in the same direction as that of the wave.
• The displacement, velocity, and acceleration are separated by mλ, here 'm' is an integer.
• The phrase of change of every particle is from 0 to 2π
• No particle in the wave remains in rest. During the body's vibration, each particle comes to rest momentarily twice at extreme positions. The positions obtained by various particles are different at different places.

Progressive Wave characteristics

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Transverse and Longitudinal waves

Progressive waves are further categorized into Transverse and Longitudinal waves.

• Transverse waves are the waves in which particles of the medium are in motion, which is parallel to the proportion of the wave. These waves are two-dimensional, and it is possible to align and polarise them. They can only be produced on solid and liquid surfaces. They depict crests and troughs.
• Whereas, in Longitudinal waves, the direction of the particle is parallel to the motion of the way. Longitudinal waves are one-directional and cannot be aligned and polarised. The waves can be produced in all three states, which include: solid, liquid, and gas. They direct compression and refraction.

Longitudinal and Transverse waves

Also read: Difference between Transverse and Longitudinal waves

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Equation representation of Progressive waves

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A mathematical equation has been used to represent the displacement of the particle vibrating in a medium. The removal is determined by a particle through which waves pass, which aids in concluding that a progressive wave executes simple harmonic motion of the same period and amplitude differing in phase from each other.

Let us assume that a progressive wave is traveling from Origin O and in a positive direction concerning X-axis. The focus will be from right to left. The displacement can be expressed as:

y = a sinωt (1)

Here, it is noticed that 'a' is the amplitude of the vibration of the particle; therefore, ω = 2πn. Further, the displacement of the particle is denoted by P, which is assumed to be at a distance 'x' from origin. Assume the above-stated pointers; the equation can be written as:

y = a sin(ωt - φ) (2)

Further, if two particles are assumed to be separated by a distance, symbolized as λ, they will differ by a phase of 2π.

x is φ = 2π/λ x

y = a sinωt - 2π.x/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 sin2π/T – 2π.x/λ) (5)

If the wave generally travels in the direction which is opposite, the equation becomes typically

y = a sin 2π (t/T + x/λ) (6)

Read More:- Wave theory of Light

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Important Terminology in Progressive wave

Some of the important terminologies for progressive wave includes:

• Phase

The argument (kx – ωt + φ) of the oscillatory term sin(kx – ωt + φ) is known as the phase of the function. It is used to describe the state of motion of the particles. The points on the wave, traveling in the same direction, which occur to fall at the exact moment, are said to be phases to each other. At the same time, it appears to move in the opposite direction, which leads to the falling of one wave and rising of the other. These are known as anti-phase with each other.

Phase

• Angular frequency

Angular displacement per unit time is described as the rate of change of waveform is known as angular frequency. It is represented as

ω = 2π/T= 2πf

• Amplitude

The maximum displacement attained by a particle in a wave from its equilibrium position is amplitude. A wave can attain both positive and negative amplitude, but in the case of the magnitude of displacement, the positive amplitude is only considered.

Amplitude and Wavelength of a wave

• Wavelength

Distance between two identical waves parallel to the wave's propagation direction is called wavelength. The comparable wavelength can be the crest or trough of a wave. It is measured in meters.

• Angular wave number

The spatial frequency of a wave, generally measured in terms of cycle per unit of distance, is known as angular wavenumber. It is also known as the number of tides that prevail over a particular space.

Also Read:

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Things to remember

• Progressive waves are described as waves carrying energy from the initial place to the neighboring particles. They are also described as continuously traveling in the same direction and amplitude.
• None of the particles in the wave remains at rest during the body's vibration; each particle comes to rest momentarily twice at extreme positions. The positions obtained by various particles are different at different places.
• The argument (kx – ωt + φ) of the oscillatory term sin (kx – ωt + φ) is known as the phase of the function, describing the state of motion of particles as a phase.
• The angular rate of change of waveform per unit time is known as angular frequency.
• Transverse waves are classified into waves in which particles of the medium are in motion, which is parallel to the proportion of the wave. These waves are two-dimensional.
• In Longitudinal waves, the direction of the particle is parallel to the motion of the way. Longitudinal waves are one-directional.
• The displacement, velocity, and acceleration are separated by mλ, here 'm' is an integer.

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