Waves and sound for JEE Main

Waves questions in JEE Main test two skills. The first is reading a wave equation at a glance: amplitude, wavelength, frequency, speed and direction. The second is fitting standing waves into strings and pipes: which harmonics are allowed and what the frequencies are. Master those two and the chapter becomes quick marks, because the arithmetic is light.

The syllabus covers wave motion, longitudinal and transverse waves, the speed of a travelling wave, the displacement relation for a progressive wave, superposition and reflection, standing waves in strings and organ pipes with their fundamental mode and harmonics, and beats.

Kinds of waves

• Transverse: particles vibrate perpendicular to the direction the wave travels. Waves on a string are the standard example.
• Longitudinal: particles vibrate along the direction of travel, forming compressions and rarefactions. Sound in air is longitudinal.

Mechanical waves need a medium. The wave carries energy and momentum forward; the particles of the medium only oscillate about their mean positions.

The progressive wave equation

y = A sin(kx − ωt + φ), with:

• wave number k = 2π/λ and angular frequency ω = 2πf
• wave speed v = ω/k = fλ
• (kx − ωt) means the wave moves in the +x direction; (kx + ωt) means the −x direction

The particle velocity is ∂y/∂t, with maximum value Aω. Do not confuse it with the wave speed ω/k.

A path difference Δx corresponds to a phase difference (2π/λ) × Δx.

Speed of waves

Newton assumed sound travels isothermally and got v = √(P/ρ), about 280 m/s in air, too low. Laplace pointed out that compressions and rarefactions are too quick for heat to flow, so the process is adiabatic, which adds the factor γ.

Consequences: in a given gas, v ∝ √T (absolute temperature). At constant temperature, changing the pressure does not change v, because P/ρ stays the same. Moist air is less dense than dry air, so sound travels slightly faster in it.

Superposition and reflection

When two waves meet, the displacements add. For two waves of the same frequency with amplitudes A₁ and A₂ and phase difference φ:

• Resultant amplitude: √(A₁² + A₂² + 2A₁A₂ cos φ)
• Intensity is proportional to amplitude squared.

On reflection from a fixed end (a rigid support, or a denser medium), the wave is inverted: a phase change of π. From a free end it returns without a phase change.

Standing waves

Two identical waves travelling in opposite directions produce y = 2A sin kx cos ωt. The pattern does not travel.

• Nodes (zero displacement) are λ/2 apart; so are the antinodes.
• A node and the next antinode are λ/4 apart.

For a string, f ∝ (1/L)√(T/μ): the laws of vibrating strings.

End correction. The antinode at an open end forms slightly outside the pipe, at about 0.6r beyond it (r = radius). A closed pipe's effective length is L + 0.6r; an open pipe's is L + 1.2r.

Resonance tube. A tuning fork is held over a tube whose air-column length can be changed. The first two resonances satisfy L₁ + e = λ/4 and L₂ + e = 3λ/4. Subtracting removes the end correction: L₂ − L₁ = λ/2.

Beats

Two sounds of slightly different frequencies f₁ and f₂ produce a loudness that rises and falls |f₁ − f₂| times per second.

• Loading a tuning fork with wax lowers its frequency.
• Filing its prongs raises it.

Use these to decide which of two possible frequencies is correct.

Worked problems

Problem 1: reading an equation. y = 0.02 sin(4πx − 200πt) in SI units. Find λ, f, the wave speed, the direction and the maximum particle speed.

k = 4π, so λ = 0.5 m. ω = 200π, so f = 100 Hz.
v = fλ = 50 m/s, in the +x direction.
Maximum particle speed = Aω = 0.02 × 200π = 4π ≈ 12.6 m/s.

Problem 2 (numerical answer): string. A 1 m string of mass 10 g is fixed at both ends under a tension of 400 N. Find its fundamental frequency in Hz.

μ = 0.01 kg/m, so v = √(400/0.01) = √40,000 = 200 m/s.
f₁ = v/2L = 200/2 = 100 Hz.

Problem 3: resonance tube. With a 500 Hz tuning fork, resonance occurs at air columns of 16 cm and 50 cm. Find the speed of sound and the end correction.

λ/2 = 50 − 16 = 34 cm, so λ = 68 cm and v = 500 × 0.68 = 340 m/s.
16 + e = λ/4 = 17 cm, so e = 1 cm. Check: 50 + 1 = 51 = 3λ/4.

Problem 4: beats and wax. Fork A is 256 Hz. Fork B gives 4 beats per second with A. When B is loaded with a little wax, the beats rise to 6 per second. Find B's frequency.

B is 252 Hz or 260 Hz. Wax lowers B's frequency.
If B were 260 Hz, lowering it would bring it closer to 256 Hz and reduce the beats.
If B is 252 Hz, lowering it moves it further from 256 Hz, so the beats increase. So B = 252 Hz.

Practice set

1. A closed pipe has a fundamental of 100 Hz. What is the next frequency it can produce?
2. Two tuning forks produce 4 beats per second. One is 256 Hz. What are the possible frequencies of the other?
3. The tension in a string is made four times as large. What happens to the wave speed and the fundamental frequency?
4. An open pipe and a closed pipe have the same fundamental frequency. Find the ratio of their lengths (open : closed).
5. Sound travels at 340 m/s in air at 27 °C. Find its speed at 127 °C.
6. A closed pipe is 85 cm long and v = 340 m/s. Ignoring end correction, how many of its natural frequencies are below 1,000 Hz?
7. Two points on a wave of wavelength 1 m are 25 cm apart along the direction of travel. Find their phase difference.
8. Two waves of equal intensity I meet with a phase difference of 2π/3. Find the resultant intensity.

Answers

1. Only odd harmonics exist, so the next is 300 Hz.
2. 252 Hz or 260 Hz.
3. v ∝ √T, so both double.
4. v/2L₀ = v/4L_c gives L₀ = 2L_c, so 2 : 1.
5. v ∝ √T: 340 × √(400/300) ≈ 340 × 1.155 ≈ 393 m/s.
6. f₁ = 340/(4 × 0.85) = 100 Hz. Allowed: 100, 300, 500, 700 and 900 Hz, so 5.
7. (2π/1) × 0.25 = π/2.
8. Resultant amplitude = √(A² + A² + 2A² cos 120°) = A, so the intensity is I.

What to do next

• Take ten wave equations and read off λ, f, v and the direction for each in under a minute.
• Draw the first three modes of a string, an open pipe and a closed pipe, marking nodes and antinodes.
• Solve 20 problems on strings, pipes, the resonance tube and beats.
• Revise simple harmonic motion, then use the physics formula sheet to review the mechanics and thermal chapters together.