Displacement relation for a progressive wave

What is the equation of a progressive wave?

The minimum distance between two points having the same phase is called the wavelength of the wave, usually denoted by λ.

NCERT Physics Part II (Class 11), Section 14.3.2, page 283

In plain words: one formula holds the whole wave. Freeze time and it draws the wave’s shape along the string; stand at one place and it tells you how that bit of string bobs up and down; and the minus sign between kx and ωt says which way the pattern is sliding.

Why the equation describes a travelling wave

Look at NCERT Eq. 14.2 in three ways (pages 281–282):

1. Fix the time, t = t₀. The argument is kx + constant, so the shape of the string at that instant is a sine curve in x.
2. Fix the place, x = x₀. The argument is constant − ωt, so the element there moves sinusoidally in time: every element performs simple harmonic motion.
3. Follow a point of fixed phase, such as a crest. To keep kx − ωt + φ constant as t increases, x must increase. So the whole pattern moves in the +x direction.

The phase constant. φ is also called the phase constant. Including it is the same as using a combination of sine and cosine (NCERT Eq. 14.3), here with θ = kx − ωt:

with a = √(A^² + B^²) and tan φ = B/A. (NCERT writes φ = tan^⁻¹(B/A); that holds as written only when A is positive. In general choose φ so that cos φ has the sign of A and sin φ the sign of B.) By choosing the origin of x and t suitably, φ for one wave can be made zero (page 282). This does not remove the physically meaningful relative phase between two waves.

Amplitude and phase

Since the sine runs from −1 to +1, y runs from −a to +a. The amplitude a is taken as positive: it is the largest magnitude of displacement of the elements from their equilibrium positions, while y itself can be positive or negative (NCERT Section 14.3.1, page 282).

The argument kx − ωt + φ is the phase of the wave: given the amplitude, it fixes the displacement at any place and time. Its value at x = 0 and t = 0 is φ, the initial phase angle. (NCERT’s table of symbols, Fig. 14.5 on page 282, labels the whole expression kx − ωt + φ as the initial phase angle. The text on the same page gets it right: the whole expression is the phase, and only φ is the initial phase angle.)

Phase difference between two points. At one instant, two points a distance Δx apart differ in phase by kΔx = 2πΔx/λ. At one place, the signed change from t to t + Δt is −ωΔt for this +x wave. Its magnitude is ω|Δt| = 2π|Δt|/T (Example 3).

Wavelength and angular wave number

At t = 0, with φ = 0, the displacement is y(x, 0) = a sin kx (NCERT Eq. 14.5). The sine repeats when kx grows by 2π, so the shape repeats every 2π/k. That distance between consecutive crests (or troughs) is the wavelength (NCERT Eq. 14.6):

k is the angular wave number or propagation constant. Its SI unit is the radian per metre, often written simply as m^⁻¹ (page 283). Think of it as 2π times the number of waves that fit into one metre.

Careful: two points with the same displacement need not be a wavelength apart. Two points just either side of a crest have equal y but move in opposite directions. Only points in the same phase (same displacement and same direction of motion) are a whole number of wavelengths apart; points λ/2 apart are always in opposite phase. (NCERT’s derivation on page 283 speaks of points with the same displacement; it is points with the same phase that define λ.)

Period, angular frequency and frequency

Now watch one element, say at x = 0, with φ = 0: y(0, t) = a sin(−ωt) = −a sin ωt (NCERT page 283; plotted in NCERT Fig. 14.7). The period T is the time for that element to complete one full oscillation. The sine repeats when ωt grows by 2π, so ωT = 2π (NCERT Eqs. 14.7–14.8):

ω is the angular frequency of the wave, in rad s^⁻¹, like an angular velocity; ν is the frequency, in hertz (page 283).

Longitudinal waves

Everything above was worked out for a transverse wave on a string, but it applies equally to a longitudinal wave. There the displacement s of an element is along the direction of travel (NCERT Eq. 14.9):

where s = s(x, t) and a is now the displacement amplitude along x; k, ω and φ mean the same as before (page 283). Here x labels equilibrium position: the actual longitudinal position of that element is x + s(x,t).

Solved examples

Common mistakes to avoid

Quick revision summary

• A sinusoidal wave along +x: y(x, t) = a sin(kx − ωt + φ); along −x: kx + ωt.
• Fixed t: the shape is a sine curve. Fixed x: the element performs SHM. Fixed phase: the pattern moves along +x.
• Amplitude a > 0; phase kx − ωt + φ; initial phase angle φ.
• k = 2π/λ (rad m^⁻¹); ω = 2π/T = 2πν (rad s^⁻¹); ν = 1/T.
• Points Δx apart differ in phase by 2πΔx/λ.
• The same equation, with s for y, describes a longitudinal wave.

Questions students ask

What are the characteristics of a progressive wave?

A harmonic progressive wave has these characteristics: in a uniform, lossless one-dimensional medium, every particle performs SHM about its mean position with the same amplitude and frequency, with a phase delay along the direction of travel, so the phase changes by 2π over every wavelength and no point stays permanently at rest. The whole pattern of crests and troughs, or of compressions and rarefactions, moves forward at the constant speed v = ω/k = νλ, carrying energy along the medium in this nondispersive model. The phase speed ω/k need not equal the energy or group speed in a dispersive medium. A spherical wave can decrease in amplitude through geometric spreading even without dissipation.

What is the difference between wave number and angular wave number?

The wave number is the number of wavelengths in a unit length, 1/λ, while the angular wave number is 2π times that, k = 2π/λ, the phase change per unit length in radians, just as ω = 2πν is 2π times the frequency. Wave equations such as y = a sin(kx − ωt) use k, in rad m^⁻¹, and many books call k itself the wave number, so check which one a question means; 1/λ is used mostly in spectroscopy, often in cm^⁻¹. For example, a wave of wavelength 0.25 m has 1/λ = 4 m^⁻¹ and k = 8π ≈ 25.1 rad m^⁻¹.

How is the particle velocity related to the wave speed in a progressive wave?

In a wave travelling along +x, the particle velocity equals minus the wave speed times the slope of the wave at that point: ∂y/∂t = −v ∂y/∂x (a step beyond the NCERT text). For y = a sin(kx − ωt), ∂y/∂t = −aω cos(kx − ωt) and ∂y/∂x = ak cos(kx − ωt), with v = ω/k; for a wave along −x the minus sign drops out. The largest particle speed, aω, is ak = 2πa/λ times the wave speed: for an amplitude of 4 mm and a wavelength of 0.8 m, it is 2π × 0.004/0.8 ≈ 0.031 times the wave speed.

Can any function of x and t represent a travelling wave?

No, a function of x and t represents a wave travelling without change of shape only if x and t appear in the combination x − vt (a wave along +x) or x + vt (along −x), as in y = f(x − vt); the sine form a sin(kx − ωt) is the special case with v = ω/k. For example, y = 4/[1 + (2x − 3t)^²], with x in metres and t in seconds, is a pulse moving along +x at 3/2 = 1.5 m s^⁻¹ (the coefficients carry the units needed to make the squared argument dimensionless). A translating mathematical profile must also satisfy the governing wave equation and boundary conditions to be a physical solution. By contrast, y = a sin kx cos ωt is a standing wave, not a travelling one, and y = sin(x^² − t^²) does not keep a fixed shape at all.

How do you write the equation of a progressive wave from its amplitude, frequency and speed?

To write the equation of a progressive wave, find ω = 2πν and k = ω/v = 2π/λ, then put them into y = a sin(kx − ωt + φ), using kx + ωt for a wave moving along −x. For example, a wave of amplitude 2 mm and frequency 200 Hz travelling along −x at 40 m s^⁻¹ has ω = 400π rad s^⁻¹ and k = 400π/40 = 10π rad m^⁻¹ (λ = 0.2 m). If the element at x = 0 is at its mean position and moving in the +y direction at t = 0, then φ = 0 and the wave is y = 0.002 sin(10πx + 400πt) in SI units.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. In y = a sin(kx − ωt + φ), what do a, k, ω and φ stand for?

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   a is the amplitude, k the angular wave number (2π/λ), ω the angular frequency (2π/T) and φ the initial phase angle, the phase at x = 0 and t = 0.

2. Q2. A wave has wavelength 2 m and frequency 50 Hz. Find k and ω.

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   k = 2π/2 = π, about 3.14 rad m^⁻¹; ω = 2π × 50 = 100π, about 314 rad s^⁻¹.

3. Q3. How can you tell from its equation whether a wave travels along +x or −x?

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   If the phase contains kx − ωt, a point of fixed phase moves to larger x as t grows, so the wave travels along +x. If it contains kx + ωt, it travels along −x.