PYQ Vault

Playbook

Waves

The wave equation, strings and pipes, beats and the Doppler effect. Half its questions have numeric answers.

Questions in the bank
80
q/paper in 2025–26
0.55
Numeric answer
50%
Notes pages
5

Tier: Long tail

When you’ll see it

A wave equation in x and t, a string or pipe at resonance, the speed of sound or of a wave on a string, beats, or a moving source or listener.

How this chapter is tested

Waves sits in the long tail, and many of its questions ask for a number. Most are short: read ω and k off an equation, pick the right harmonic formula, or write one Doppler ratio, and the answer follows in two or three lines.

The work is in the bookkeeping. Expand any common factor before reading ω and k, and keep k in m⁻¹ when the speed is asked in m/s. On strings and pipes, keep harmonics and overtones apart: a closed pipe has only odd harmonics, so its first overtone is the third harmonic. Speeds need T in kelvin and μ in kilograms per metre.

The Doppler page rewards a rule over a memorised sign pattern: motion towards raises the frequency, and an echo from a wall is shifted twice. The sine equation is the one from Oscillations, and superposition returns in Wave Optics.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • The wave equation

    In y = A sin(ωt − kx), f = ω/2π, λ = 2π/k and v = ω/k; same signs on t and x mean travel along −x; the largest particle speed is Aω.

  • Wave speed

    √(T/μ) on a string, √(Y/ρ) in a rod, √(γRT/M) in a gas; at a fixed temperature, pressure alone does not change the speed of sound.

  • Superposition and strings

    Resultant amplitude by the cosine rule; a string fixed at both ends has every harmonic nv/2L, and its frequency goes as √T.

  • Organ pipes and the resonance tube

    Open pipe nv/2L, closed pipe only (2n − 1)v/4L; the end correction is 0.3d, and v = 2f(l₂ − l₁) needs no end correction.

  • Beats and the Doppler effect

    Beat frequency |f₁ − f₂|; heard frequency f(v ± vₒ)/(v ∓ vₛ), with each sign set so that approach raises it; an echo is shifted twice.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • k left in cm⁻¹

    When x is in cm, ω/k comes out in cm/s, a hundred times larger than the value in m/s. Convert k to m⁻¹ first.

  • Overtone read as harmonic

    The first overtone of a closed pipe is the third harmonic, because the second harmonic does not exist there.

  • Beats in a time

    Ten beats in two seconds is a beat frequency of five hertz, not ten. Divide by the time before using the difference.

  • An echo shifted once

    A driver moving towards a wall hears the echo at f(v + u)/(v − u). Applying the formula once, for the source only, misses the second shift.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Waves notes

Drill every Waves question

80 questions from the bank, across 5 subtopics.

Drill the 80 questions

Drill one subtopic at a time

The 5 subtopics, in teaching order.

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