Displacement Relation in a Progressive Wave: Meaning, Equation, Class 11 Notes

A progressive wave is a type of wave that continuously propagates with the same amplitude, travelling outward from the source. 

It's also called a travelling wave. 

To learn the displacement relation, let’s find out the characteristics of the progressive wave. 

Characteristics of a Progressive Wave

• All particles that oscillate in a progressive wave have the same amplitude. A progressive wave to continuously propagate requires an idealised uniform medium where the amplitude is constant.
• These waves transfer energy and momentum from one point to another within the same medium. There is only energy transfer without net mass transport.
• Every particle in a progressive wave follows Simple Harmonic Motion at its mean position. Each particle also has phase changes from 0 to 2π as the wave passes through. 
• All progressive waves transfer energy, but only mechanical waves require a medium. Non-mechanical waves, such as electromagnetic waves, are also progressive. But they propagate without the need for a medium.

Examples of a Progressive Wave

The displacement relation in a progressive wave is a mathematical function of position (x) and time (t). It can describe the displacement of any particle in a medium through which a progressive wave travels. By calculating the displacement relation, we can capture both the progressive wave's shape in space and its movement through time.

The equation for a displacement relation in a progressive wave is typically represented as 

y(x, t) = a sin(kx - ωt + φ)

In this, a is the amplitude, k is the wave number, ω is the angular frequency, x is the position, t is the time, and φ is the phase constant.

This equation assumes that a particle in a medium undergoes simple harmonic motion (SHM) as a wave passes through it.

Also, the progressive wave’s displacement from the equilibrium position is a sinusoidal function of time. 

Here, we decode the parameters of the displacement relation equation. These should directly supplement your chapter’s subsections on this topic.

Amplitude (a)

The amplitude (a) tells that it’s the maximum displacement or distance moved by a point on the progressive wave from its equilibrium position.

The sine function in this varies between +1 and -1. So, the displacement y(x, t) varies between +a and -a.

While the instantaneous displacement y can vary from −a to +a, the amplitude 'a' remains a fixed, positive value.

• Crest: The point of maximum positive displacement (+a).
• Trough: The point of maximum negative displacement (-a).

Phase (kx - ωt + φ)

The entire argument of the sine function, (kx - ωt + φ), is called the phase of the wave.

The phase determines a particle's position in its oscillation cycle. At a fixed point in space, the phase changes with time. At a fixed time, the phase varies with position. 

The term φ is the initial phase or phase constant. It adjusts the starting position of the wave at x=0 and t=0. 

A non-zero φ tells us that the wave is a combination of sine and cosine functions, as shown by the relation:

a sin(kx - ωt + φ) = A sin(kx - ωt) + B cos(kx - ωt)

where a = √(A² + B²) and φ = tan⁻¹(B/A).

Wavelength (λ) and Angular Wave Number (k)

These terms describe the spatial characteristics of the progressive wave.

• Wavelength (λ): The wavelength is the minimum spatial distance between two consecutive points that are in the same phase. You can consider the distance between two crests, for instance. It defines the spatial period of the wave.
• Angular Wave Number (k): This is related to the wavelength by the formula k = 2π/λ. It represents how many radians of the wave's cycle exist per unit of distance.
• SI Unit: radians per metre (rad m⁻¹).

Period (T), Frequency (ν), and Angular Frequency (ω)

These terms describe the temporal (time-based) characteristics of the wave. They arise because each particle in the medium executes Simple Harmonic Motion (SHM) about its mean position. 

• Period (T): The period is the time it takes for one complete oscillation or cycle to pass a given point. The sine function repeats its value every 2π radians, so we have: ωT = 2π or T = 2π / ω
• Angular Frequency (ω): This describes how many radians of the wave's cycle pass per unit of time.
• SI Unit: radians per second (rad s⁻¹).
• Frequency (ν): The frequency is the number of complete oscillations that occur per second. It is the reciprocal of the period. ν = 1/T = ω / 2π
• SI Unit: Hertz (Hz).

Additionally, refer to periodic and oscillatory motion, which cover similar concepts. 

If you are preparing for JEE Main, here are some additional critical concepts to take note of these points. 

• The equation y = a sin(kx - ωt + φ) describes a wave moving in the +x direction. For -x direction, use (kx + ωt).
• φ sets the initial displacement at (x=0, t=0). For instance, if φ = π/2, y = a cos(kx - ωt). 
• The phase difference between two particles separated by distance Δx is Δϕ = kΔx.
• Wave number k (rad/m) is the spatial frequency, while angular frequency ω (rad/s) is the temporal frequency. Also, remember, wave speed v = λν = ω/k
• Also, the displacement relation assumes a linear regime of small amplitudes.  Beyond linear, the waveform may distort.

Here are some examples of questions that appear in JEE Main and other competitive tests. After practising these, you could freely download our compilation of previous JEE Main question papers.

Example 1: A progressive wave is given by $y=0.05\mathrm{s}\mathrm{i}\mathrm{n}(100\pi t-2\pi x)$ , where $y$  and $x$  are in meters, $t$ in seconds. Find the wave velocity and wavelength. 

Solution: Comparing with

$y=A\mathrm{s}\mathrm{i}\mathrm{n}(\omega t-kx)$  :

$\begin{array}{rr}\omega & =100\pi ,\mathrm{}k=2\pi \\ \nu =\frac{\omega }{k}=\frac{100\pi }{2\pi }& =50\mathrm{m}/\mathrm{s},\mathrm{}\lambda =\frac{2\pi }{k}=\frac{2\pi }{2\pi }=1\mathrm{m}\end{array}$

Example 2: For the wave $y=0.1\mathrm{s}\mathrm{i}\mathrm{n}(200t-4x)$ , calculate the phase difference between two points 0.25 m apart.

Solution: $\begin{array}{c}k=4{\mathrm{m}}^{-1},\mathrm{}\mathrm{\Delta }x=0.25\mathrm{m}\\ \mathrm{\Delta }\varphi =\frac{2\pi }{\lambda }\mathrm{\Delta }x=k\mathrm{\Delta }x=4\cdot 0.25=1\text{radian}\end{array}$

Example 3: A wave has a frequency of 50 Hz and speed $300\mathrm{m}/\mathrm{s}$ . Write its displacement equation with amplitude 0.02 m , assuming positive $x$ -direction propagation. 

Solution: $\begin{array}{c}v=n\lambda ⟹\lambda =\frac{300}{50}=6\mathrm{m},\mathrm{}k=\frac{2\pi }{\lambda }=\frac{2\pi }{6}=\frac{\pi }{3}{\mathrm{m}}^{-1},\mathrm{}\omega =2\pi n=2\pi \cdot 50=100\pi {\mathrm{s}}^{-1}\\ y=0.02\mathrm{s}\mathrm{i}\mathrm{n}\left(100\pi t-\frac{\pi }{3}x\right)\end{array}$

Check out the guides for each of these chapters and their topics, aligned perfectly with the latest Physics CBSE curriculum from NCERT.  

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