Oscillations and waves for NEET

Oscillations and waves is really two linked chapters. The first is about one particle moving back and forth. The second is about many particles doing that in step, so that a disturbance travels. NEET questions from both are usually quick: a period, a maximum speed, a harmonic of a pipe, a beat frequency. The marks are lost on small slips, such as confusing an open pipe with a closed one or using degrees where the formula needs radians.

Simple harmonic motion

A particle moves in SHM when the restoring force is proportional to its displacement and directed towards the mean position:

F = −kx, so a = −ω²x, where ω² = k/m.

The displacement is x = A sin(ωt + φ), where A is the amplitude, ω the angular frequency and (ωt + φ) the phase. T = 2π/ω and f = 1/T.

Velocity leads displacement by a phase of π/2, and acceleration is opposite in phase to displacement.

Energy in SHM

• Potential energy: U = ½kx²
• Kinetic energy: K = ½k(A² − x²)
• Total energy: E = ½kA² = ½mω²A², constant throughout

KE equals PE where x² = A²/2, that is at x = A/√2. Both KE and PE oscillate at twice the frequency of the motion, because each goes through its maximum twice per cycle.

Springs and the simple pendulum

Spring–mass system: T = 2π√(m/k). The period does not depend on the amplitude, and it is the same whether the spring is horizontal or vertical.

• Springs in series: 1/k = 1/k₁ + 1/k₂ (softer).
• Springs in parallel: k = k₁ + k₂ (stiffer).
• Cutting a spring into n equal pieces makes each piece n times stiffer.

Simple pendulum. Pull the bob aside by a small angle θ. The restoring force is mg sin θ ≈ mgθ = mg(x/l). Comparing F = −(mg/l)x with F = −kx gives k = mg/l, so ω² = g/l and

T = 2π√(l/g)

• It holds only for small angles, because sin θ ≈ θ.
• It does not depend on the mass of the bob.
• A pendulum with T = 2 s is a seconds pendulum; its length is about 1 m on Earth.
• In a lift accelerating upward at a, use g + a; downward, use g − a. In free fall, g_eff = 0 and the pendulum does not oscillate.

Waves and wave speed

In a transverse wave, particles move perpendicular to the direction of travel (waves on a string). In a longitudinal wave, they move along it (sound in air, with compressions and rarefactions).

A progressive wave moving in the +x direction is y = A sin(kx − ωt), where k = 2π/λ is the wave number. Its speed is v = ω/k = fλ. A plus sign, sin(kx + ωt), means the wave travels in −x.

Newton assumed sound travels isothermally and used √(P/ρ), which gives about 280 m s⁻¹ in air. Laplace corrected this: the compressions are too quick for heat to flow, so they are adiabatic, and γ enters the formula.

Standing waves: strings and pipes

When two identical waves travel in opposite directions, they superpose to form a standing wave with fixed nodes and antinodes. Adjacent nodes are λ/2 apart; a node and the next antinode are λ/4 apart. A wave reflected from a rigid end is inverted (phase change of π); from a free end, it is not.

A closed pipe of length L has the same fundamental as an open pipe of length 2L. The "first overtone" of a closed pipe is its third harmonic, 3v/4L. This naming trips many students.

Resonance tube (an experimental-skills item). A tuning fork is held over an air column closed by water. The first resonance occurs at length l₁ ≈ λ/4 and the second at l₂ ≈ 3λ/4. So λ = 2(l₂ − l₁), and the end correction cancels out in the subtraction.

Beats

Two sounds of slightly different frequencies produce a rise and fall in loudness. The beat frequency is the difference of the two frequencies, f₁ − f₂ (taken as positive).

To find which fork is higher, use loading and filing:

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

Worked numericals

Example 1: speeds in SHM

A particle in SHM has A = 0.1 m and ω = 10 rad s⁻¹. Find v_max, a_max and the speed at x = 0.06 m.

• v_max = Aω = 0.1 × 10 = 1 m s⁻¹.
• a_max = Aω² = 0.1 × 100 = 10 m s⁻².
• v = ω√(A² − x²) = 10 × √(0.01 − 0.0036) = 10 × √0.0064 = 10 × 0.08 = 0.8 m s⁻¹.

Example 2: spring–mass energy

A 2 kg block on a spring of k = 200 N m⁻¹ oscillates with amplitude 0.1 m. Find T, the total energy and v_max.

• T = 2π√(2/200) = 2π × 0.1 ≈ 0.63 s.
• E = ½kA² = ½ × 200 × 0.01 = 1 J.
• v_max = √(2E/m) = √(2 × 1/2) = 1 m s⁻¹. Check: ω = √(k/m) = 10 rad s⁻¹, and Aω = 1 ✓.

Example 3: harmonics of a closed pipe

A pipe 0.85 m long is closed at one end. Take v = 340 m s⁻¹.

• Fundamental: v/4L = 340/3.4 = 100 Hz.
• First and second overtones: 300 Hz and 500 Hz (odd harmonics only).
• If the pipe is opened at both ends, its fundamental becomes v/2L = 340/1.7 = 200 Hz.

Example 4: beats with wax

Fork A has a frequency of 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 either 260 Hz or 252 Hz.
• Wax lowers B's frequency. If B were 260 Hz, lowering it would bring it closer to 256 Hz and reduce the beats.
• The beats increased, so B moved away from 256 Hz. B = 252 Hz.

Example 5: speed of sound from a resonance tube

With a 500 Hz fork, the first two resonance lengths are 16 cm and 50 cm.

• λ = 2(l₂ − l₁) = 2 × 34 cm = 68 cm = 0.68 m.
• v = fλ = 500 × 0.68 = 340 m s⁻¹.

Practice MCQs

1. Tuning forks of 256 Hz and 260 Hz sound together. The number of beats per second is: (a) 2 (b) 4 (c) 8 (d) 516
2. The mass on a spring is made 4 times larger. The period: (a) halves (b) stays the same (c) doubles (d) becomes 4 times
3. In SHM of amplitude A, kinetic and potential energy are equal at x = (a) A/2 (b) A/√2 (c) A/4 (d) A
4. A seconds pendulum is taken to a planet where g is a quarter of its value on Earth. Its period becomes: (a) 1 s (b) 2 s (c) 4 s (d) 8 s
5. The tension in a stretched string is made 4 times. Its fundamental frequency: (a) halves (b) doubles (c) becomes 4 times (d) is unchanged
6. The second overtone of a pipe closed at one end is how many times its fundamental? (a) 2 (b) 3 (c) 4 (d) 5
7. For a wave y = 0.05 sin(20x − 400t) in SI units, the wave speed is: (a) 20 m s⁻¹ (b) 0.05 m s⁻¹ (c) 400 m s⁻¹ (d) 8,000 m s⁻¹
8. A spring is cut into two equal halves, and the same mass is hung from one half. If the original period was T, the new period is: (a) 2T (b) √2 T (c) T/√2 (d) T/2

Answers

1. (b) 260 − 256 = 4.
2. (c) T ∝ √m, and √4 = 2.
3. (b) ½k(A² − x²) = ½kx² gives x = A/√2.
4. (c) T ∝ 1/√g, so the 2 s period doubles to 4 s.
5. (b) f ∝ v ∝ √T, and √4 = 2.
6. (d) Closed pipe harmonics are 1, 3, 5…; the second overtone is the fifth harmonic.
7. (a) v = ω/k = 400/20 = 20 m s⁻¹.
8. (c) Each half has spring constant 2k, and T ∝ 1/√k.

What to do next

• Make a one-page table of SHM quantities at the mean position and the extreme.
• Draw the first three modes for a string, an open pipe and a closed pipe, marking nodes and antinodes.
• Practise five beat questions with wax and filing until the logic is automatic.
• Go back to work, energy and power if the energy-in-SHM results feel shaky.

Next in the syllabus: electrostatics for NEET.