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A question gives you y = 0.05 sin(2.0x − 40t) and asks for the wave speed, the maximum speed of a particle and the tension in the string. Many students know the formulas but mix up which number goes where, or confuse the speed of the wave with the speed of the string. This post shows you how to read the equation term by term, works one full example, and lists the slips that cost marks.
What do the symbols in y = A sin(kx − ωt) mean?
NCERT Class 11 Physics, Chapter 14 Waves, writes a harmonic travelling wave as y(x, t) = a sin(kx − ωt + φ). Each symbol has one job:
A (or a), the amplitude: the largest displacement of any particle from its rest position.
k, the angular wave number: in rad/m. It tells you how quickly the phase changes along the string, and λ = 2π/k.
ω, the angular frequency: in rad/s. It tells you how quickly the phase changes in time, so T = 2π/ω and f = ω/2π.
φ, the phase constant: the starting phase at x = 0, t = 0.
The chapter then follows a crest. A point of fixed phase has kx − ωt constant, so as t grows, x must grow too. Differentiating gives the wave speed v = ω/k, which is the same as v = fλ.
Both the JEE (Main) 2026 syllabus and the NEET (UG) 2026 syllabus list "speed of travelling wave" and the "displacement relation for a progressive wave" in the oscillations and waves unit, so this reading skill is tested in both exams. Check the syllabus for your own exam year, as NTA publishes a new one each cycle.
How do you tell which way the wave moves?
Look at the sign between the x term and the t term.
kx − ωt: the wave moves towards +x.
kx + ωt: the wave moves towards −x.
A quick check: for kx − ωt to stay constant while t increases, x must increase. If the equation is written as sin(ωt − kx), the wave still moves towards +x, because sin(ωt − kx) = −sin(kx − ωt). Only the relative sign of the two terms matters.
Worked example: one equation, five answers
Question. A wave on a string is y = 0.05 sin(2.0x − 40t), with x and y in metres and t in seconds. The string has a mass per unit length of 0.05 kg/m. Find (a) the amplitude, wavelength and frequency, (b) the wave speed and direction, (c) the maximum particle speed, (d) the tension and (e) the phase difference between two points 0.5 m apart.
Compare with the standard form. A = 0.05 m, k = 2.0 rad/m, ω = 40 rad/s.
Wavelength and frequency. λ = 2π/k = 2π/2.0 = π m, about 3.14 m. f = ω/2π = 40/2π, about 6.37 Hz.
Wave speed. v = ω/k = 40/2.0 = 20 m/s, towards +x because of the minus sign. Check: fλ = (40/2π)(π) = 20 m/s.
Maximum particle speed. Each particle does simple harmonic motion with velocity ∂y/∂t = −Aω cos(kx − ωt), so the largest value is Aω = 0.05 × 40 = 2 m/s. That is ten times smaller than the wave speed here, and the two are not the same thing.
Tension. The chapter gives v = √(T/μ) for a stretched string, so T = μv² = 0.05 × 20² = 0.05 × 400 = 20 N.
Phase difference. Δφ = kΔx = 2.0 × 0.5 = 1 rad.
A useful link between them: since ∂y/∂t = −v (∂y/∂x) for any wave moving towards +x, the particle velocity at a point equals minus the wave speed times the slope of the string there. Questions that show a snapshot of the string and ask which way a point is moving can be answered with this one line.
What mistakes do students make most often?
Physics education researchers have studied this topic closely. Wittmann, Steinberg and Redish (1999) investigated student difficulties with mechanical waves and found that individual students use both correct and incorrect mental models to make sense of a single physical situation. In exam answers, that tends to show up as:
Treating k as the wavelength. k = 2π/λ, not λ. If k = 2.0 rad/m, λ is about 3.14 m, not 2 m.
Using ω as the frequency. f = ω/2π. Forgetting the 2π makes every later answer wrong by a factor of about 6.
Thinking a bigger flick makes a faster wave. Changing the amplitude changes the particle speed, not the wave speed, which depends only on T and μ.
Reading direction from the sign of the t term alone. Compare the signs of both terms.
Mixing units. Some questions give x in centimetres. Convert before taking λ = 2π/k.
If the movement is hard to picture, watching a wave run along a string helps. Our post on learning waves with simulations explains what to look for. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read, so you can change amplitude or frequency and watch which quantities move.
A practice routine for this week
Day 1: Take five wave equations from NCERT and its exemplar. For each, write A, k, ω, λ, f, v and direction in a single row before solving anything else.
Day 2: Solve five tension and mass-per-length questions using v = √(T/μ). Check each with dimensions.
Day 3: Five snapshot questions: given a string shape and direction of travel, find which points move up.
Day 4: Mixed set of eight from all three days, timed, with an error log marking each slip by type.
When you are ready for the next step, superposition, read our guide to standing waves in strings and pipes or beats in sound.
Tonight, write the standard form at the top of a page and decode three equations from your textbook without looking at the answers.
