Progressive Wave : Displacement Relation

What Is Progressive Wave?

A wave which travels continuously in a medium in the same direction without a change in its amplitude is called a travelling wave or a progressive wave. In this section, we will derive a function that will describe the propagation of a wave in a medium and gives the shape of the progressive wave at any instant of time during its propagation.

Let us consider the example of a progressive wave on a string. Here, we will describe the relation of displacement of any element on the string as a function of time and the vibration of the elements of the string along the length at a given instant of time.

Let y(x,t) be the displacement of an element at a position x and time t about the y-axis. Considering the wave to be periodic and sinusoidal, the displacement of the element at a position x and time t from the y-axis can be given as,

y (x, t ) = a sin (kx – ωt + φ ) …………………………………..(a)

We can write the above equation as a linear combination of sine and cosine functions as,

y (x, t) =A sin (kx – ωt ) + B cos (kx – ωt ), …………(b)          

The equations (a) and (b) represent the transverse wave moving along the X-axis, where y(x,t) gives the displacement of the elements of the string at a position x at any time t. Hence, the shape of the wave can be determined at any given time.

y(x, t) = a sin (kx + ωt + φ ),

The above equation represents a transverse wave moving along the negative direction of the X-axis.

The parameters that describe a harmonic wave entirely are ‘a’, ‘φ’, ‘k’, and ‘ω’, where a is the amplitude, φ is the initial phase change, k is the angular wavenumber, and ω is the angular frequency. Let us now learn in detail what these quantities represent.

Consider the sinusoidal graph shown above. Here, the plot shows a wave travelling in the positive X direction.

The point of maximum positive displacement is called a crest, and that of maximum negative displacement is called a trough.

Amplitude

Amplitude is the magnitude of maximum displacement of a particle in a wave from the equilibrium position.

The image above shows the positive and negative amplitude in the case of a sinusoidal wave. But as we consider the magnitude of the displacement, the amplitude is always a positive quantity.

Phase

The argument (kx – ωt + φ) of the oscillatory term sin (kx – ωt + φ) is defined as the phase of the function. It describes the state of motion of the wave. Points on a wave which travel in the same direction, rising a falling together, are said to be in phase with each other. Points on a wave which travel in opposite directions to each other, such that one is rising while the other is falling, are said to be in anti-phase with each other.

Wavelength

Wavelength (λ) is the distance between two identical points, such as a crest or a trough on a wave parallel to the wave’s propagation direction. It can also be defined as the distance over which the wave shape repeats itself. It is measured in metres (m).

Angular wave number

The wavenumber is the spatial frequency of a wave in terms of cycles per unit distance. It can also be defined as the number of waves that exist over a specified distance, analogous to the concept of frequency.

Angular frequency

Angular frequency is defined as the angular displacement per unit time of the rate of change of phase of a waveform.

Mathematically we can represent it as,

where T is the time period of the sinusoidal function representing the wave and f is the frequency.

Frequently Asked Questions – FAQs

Q1

What are the two types of progressive waves?

There are two types of a progressive wave:

1. Longitudinal Wave
2. Transverse Waves

Q2

Do progressive waves have a constant phase?

The amplitude of each particle is the same but the phase changes continuously

Q3

How do particles move in a progressive wave?

A transverse wave has oscillations that are perpendicular to the direction of energy transfer whereas longitudinal oscillates parallel.

Q4

What are the properties of progressive waves?

Properties of progressive waves are: Amplitude, wavelength, period, double periodicity, frequency and velocity.

Watch the video and learn more about superposition of waves

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