Waves are a vital component of physics because they enable us to understand how energy travels across great distances without the actual movement of matter. How particles in the medium move as a progressive wave come through is one of the first things.
Displacement Relation in a Progressive Wave
A progressive wave is one that advances while carrying energy. The medium’s particles do not follow the wave, instead, they merely oscillate around their mean (rest) positions. We now employ what is known as the displacement relation to quantitatively characterise this oscillation.
What is Displacement in a Wave?
Every particle is moved from its rest position when a wave travels through a medium. A particle’s displacement is the amount it deviates from its equilibrium position at any given time.
The displacement is dependent upon two factors:
The particle’s ( x) location within the medium.
The time (t) at which the particle is observed.
As a result, the displacement is expressed as a function of time and position.: y(x, t)
Displacement Relation for a Progressive Wave
The general form of a displacement relation for a progressive (traveling) wave moving along the positive x-axis is:
y(x,t) = A sin(kx – ωt + ϕ)
Where:
y = displacement of the particle at position x and time t,
A = amplitude of the wave (maximum displacement),
k = wave number = 2π / λ (where λ is the wavelength),
ω = angular frequency = 2πf (where f is the frequency),
ϕ = initial phase (depends on how the wave started).
If the wave moves along the negative x-axis, the formula becomes:
y(x,t) = A sin(kx +ωt + ϕ)
The Terms
Amplitude (A): It tells how much the particles are moving. Larger amplitude means bigger oscillations.
Wave Number (k): It tells how tightly packed the waves are. A smaller wavelength (closer crests and troughs) means a larger k.
Angular Frequency (ω): It tells how fast the particles are oscillating.
Phase (ϕ): It adjusts the wave to match initial conditions at
t = 0(kx – ωt): This combination tells us how the wave travels through space and time.
Important Points
For a fixed time, displacement varies sinusoidally with position x.
For a fixed position, displacement varies sinusoidally with time t.
The wave shape moves forward without changing its form (in an ideal medium).
Simple Way to Remember
Imagine a moving “sine wave” that flows ahead like a smooth serpent, with each particle just sliding up and down to follow the wave without going anywhere.
This entire motion is simply described mathematically by the displacement relation.
Example
Suppose we have a wave with:
Amplitude A = 2 cm,
Wavelength λ = 4 m,
Frequency f = 2 Hz,
Initial phase ϕ = 0.
First, calculate:
k = 2π / λ = 2π / 4 = π / 2 rad/m,
ω = 2πf = 4π rad/s.
Thus, the displacement relation is:
y(x,t) = 2sin π / 2 x − 4πt
This equation fully describes how the particles of the medium are vibrating as the wave moves.
Conclusion
The displacement relation provides a clear and comprehensive picture of the behaviour of a progressive wave. It enables us to forecast the wave’s form and speed at any given time and illustrates how displacement varies with position and time.
The distance a medium particle travels from its equilibrium (rest) location during a progressive wave’s passage is referred to as displacement. It varies with time and situation.
The displacement y(x,t) of a particle in a progressive wave moving in the positive x-direction is given by:
y(x,t) = A sin(kx – ωt + ϕ)
Where A is the amplitude, k is the wave number, ω is the angular frequency, and ϕ is the initial phase.
The term (kx − ωt) represents the phase of the wave. It describes how displacement changes with position x and time t as the wave moves forward.
For a wave traveling in the negative x-direction, the displacement relation changes to:
y(x,t) = A sin(kx + ωt + ϕ)
The sign before ωt becomes positive.
The greatest deviation of the particles from their mean position is known as amplitude A. It demonstrates that the wave’s energy is proportional to its amplitude.
Wave number (k) tells how many wave cycles fit into a certain distance. It is related to wavelength by k = 2π / λ.
Angular frequency (ω) tells how fast the particles are oscillating and is related to frequency by ω = 2πf.