The waves of water or waves on a string travel in a particular direction. But, have you ever imagined how is this direction decided? What is the influence of this movement? In order to understand this concept indepth, it is important to learn about progressive wave and its displacement aspect.
A progressive wave is a term given to a wave that travels from a specific point A in the medium to another point B. In simple terms, a wave that continuously travels in a medium in the same direction minus the changes is known as a traveling wave or progressive wave. Furthermore, a progressive wave is of two types; namely, transverse wave and longitudinal wave.
Learn more about transverse and longitudinal waves here.
Plane Progressive Harmonic Wave
During the transmission of a wave via a medium, the particles present in the medium tend to vibrate harmonically about their mean positions; in this case, the wave is tagged as a plane progressive harmonic wave.
Simple Harmonic Progressive Wave
This waveform continuously travels in a specific direction without any transition in form. Furthermore, the particles of the medium tend to move harmonically about their mean position carrying the same amplitude and period.
Characteristics of SHM Wave
Plane Progressive Harmonic Wave: Displacement Relation
Considering a plane progressive harmonic wave, the displacement of a sinusoidal wave traveling in the xdirection (positive) is mentioned below:
y = a sin(kx – ωt + φ)
Here, ‘a’ denotes the amplitude of the wave, angular wave number is denoted by ‘k’, whereas ‘ω’ is the angular frequency. The phase is mentioned as (kx – ωt + φ), and φ signifies the phase angle or phase constant. Furthermore, the sine function, as well as the timedependent phase (wave), resemble the oscillation of a string component, whereas the amplitude of wave defines the extremes of the component’s displacement. It is important to remember that, the constant ‘ϕ’ is termed as initial phase angle.
Wavelength of Progressive Wave
The wavelength ‘λ’ for a progressive wave is basically the distance measured between two succeeding points of the same phase at a particular time. Considering a stationary wave, this is twice the distance measured between two successive nodes or antinodes. The propagation constant is defined as ‘k’. The SI unit is calculated in radian per meter or rad m^{1}.
k=2π/λ
The time period ‘T’ of wave oscillation is the duration taken by any component of the medium takes to travel over one complete oscillation. This is related to ‘ω’ or angular frequency through the following relation:
ω = 2π/T
In this, wave frequency ‘v’ is mentioned as 1/T and is also related to angular frequency as:
v = ω/2π
It can also be defined as the number of oscillations/unit time prepared in a string element while the wave passes through it. This is usually calculated in Hertz.
In mathematical terms, the superposition principle can be described as given below. Let's say y_{1}(x, t) and y_{2}(x, t) is the displacements produced by two waves in the medium. Let P be the point where these two ways come and meet. Now using the principle of superposition to find the resultant displacement (y).
Y = y_{1}(x, t) + y_{2}(x, t)
If two or more waves are travelling and meeting at one point in a medium and the wave functions for the individual waves are given by,
Y = f_{1}(x  Vt)
Y = f_{2}(x  Vt)
Y = f_{n}(x  Vt)
The resultant wave after displacement is given by,
102 videos411 docs121 tests

1. What is the principle of superposition? 
2. How does interference occur? 
3. What is the difference between constructive and destructive interference? 
4. Can waves interfere with each other even if they have different frequencies? 
5. How can interference be used in practical applications? 

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