Waves and vibrations for NET: SHM formulas, traps and worked MCQs

Waves and vibrations in NET covers simple harmonic motion, the spring and pendulum, wave speed on strings and in air, standing waves in strings and pipes, beats and the Doppler effect. It is one of the heaviest physics chapters in the syllabus, and most of its questions are ratio problems you can answer without a calculator once the formulas are in your head.

Key facts

• Physics is 30% of NET-Engineering, a derived 60 of the 200 MCQs.
• Our syllabus data estimates about 8 questions from first-year Waves and Vibrations and about 4 from second-year Simple Harmonic Motion.
• Syllabus: SHM, spring-block and pendulum systems, waves on strings and in air, standing waves, interference, beats and the Doppler effect.

Source: NUST NET-Engineering composition and the SailToNUST physics syllabus, checked October 2026

What the chapter covers#

The first-year chapter starts with SHM and its two standard systems, then moves to travelling waves, superposition, standing waves, beats and Doppler. The second-year Simple Harmonic Motion chapter returns to SHM at more depth: the restoring-force condition, energy exchange, resonance and the sine-wave maths. Put together, they form the biggest block of oscillation questions in the paper.

Formulas you need#

SHM condition         a = −ω²x
Displacement          x = x₀ sin(ωt + φ)
Speed                 v = ω√(x₀² − x²);   v_max = ωx₀  (at x = 0)
Acceleration          a_max = ω²x₀        (at x = ±x₀)
ω = 2πf = 2π/T                            rad s⁻¹

Spring-block          T = 2π√(m/k)
Simple pendulum       T = 2π√(l/g)
Energy                E = ½kx₀² (constant);  KE = ½k(x₀² − x²)

Wave speed            v = fλ
Wave on a string      v = √(T/μ)   (μ = mass per metre, kg m⁻¹)
Speed of sound        v ∝ √T (kelvin);  about 332 m/s at 0 °C

Standing waves (length L)
String fixed both ends   fₙ = nv / 2L        n = 1, 2, 3 ...
Pipe open both ends      fₙ = nv / 2L        n = 1, 2, 3 ...
Pipe closed one end      fₙ = nv / 4L        n = 1, 3, 5 ... (odd only)
Node to node             λ/2

Beats                 f_beat = |f₁ − f₂|

Doppler (sound speed v)
Source moving toward still observer at uₛ    f' = f v / (v − uₛ)
Observer moving toward still source at uₒ    f' = f (v + uₒ) / v
(flip the signs for motion away)

Worked MCQs#

1. Doppler ratio. A sound source moves toward a stationary observer at one tenth of the speed of sound. What is the ratio of heard frequency to emitted frequency?

(a) 10/9 (b) 11/10 (c) (11/10)² (d) (9/10)²

f'/f = v / (v − v/10) = 1 / (9/10) = 10/9

Answer (a). If the observer moved toward a still source at v/10 instead, the ratio would be 11/10. Source and observer motion are not interchangeable.

2. Lengthening a pendulum. A pendulum has a period of 2.0 s. Its length is increased by 21%. What is the new period?

(a) 2.1 s (b) 2.2 s (c) 2.42 s (d) 2.0 s

T ∝ √l
T' = 2.0 × √1.21 = 2.0 × 1.1 = 2.2 s

Answer (b). Setters like 21% and 44% because 1.21 and 1.44 have clean square roots. Option (c) is what you get if you forget the square root.

3. Closed pipe. A pipe 0.5 m long is closed at one end. The speed of sound is 340 m/s. What is its fundamental frequency, and what is the next frequency it can produce?

f₁ = v / 4L = 340 / 2 = 170 Hz
next: 3f₁ = 510 Hz  (closed pipes skip even harmonics)

Open the closed end and the fundamental becomes v/2L = 340 Hz. That is the one-liner that shows up as "f_open = 2 f_closed" for pipes of equal length.

4. Beats and wax. Tuning fork A has frequency 256 Hz. It makes 4 beats per second with fork B. Loading B with wax reduces the beat frequency. What was B's frequency?

(a) 252 Hz (b) 254 Hz (c) 258 Hz (d) 260 Hz

B is 256 ± 4: either 252 Hz or 260 Hz
Wax lowers B's frequency
If B = 260, lowering it moves it closer to 256  →  fewer beats (matches)
If B = 252, lowering it moves it further away   →  more beats
B = 260 Hz

Answer (d). Filing a prong does the opposite: it raises the frequency.

5. Speed in SHM. A body moves in SHM with amplitude 0.1 m and period 0.2π s. What are its maximum speed and maximum acceleration?

ω = 2π / T = 2π / 0.2π = 10 rad/s
v_max = ωx₀ = 10 × 0.1 = 1 m/s
a_max = ω²x₀ = 100 × 0.1 = 10 m/s²

Speed is maximum at the centre, where acceleration is zero. Acceleration is maximum at the ends, where speed is zero.

Traps that cost marks#

• Spring period ignores g. A spring-block system has the same period on the Moon. A pendulum does not.
• Pendulum in free fall. If the support falls freely, effective g is zero and the pendulum does not oscillate. The period is infinite. Same for a pendulum inside an orbiting spacecraft.
• Pendulum clock on a hill. g is smaller higher up, so the period gets longer and the clock runs slow. You fix it by shortening the pendulum.
• Double m and k together. T = 2π√(m/k) stays the same.
• sin versus cos. x = x₀ sin ωt starts at the centre. x = x₀ cos ωt starts at the end, which is a phase difference of 90°.
• Sound speed and pressure. At fixed temperature, changing air pressure does not change the speed of sound. Temperature and humidity do.
• Nodes and antinodes in pipes. A displacement node is a pressure antinode. The closed end of a pipe is always a displacement node.

Speed tips for the exam#

Learn three square roots by heart: √1.21 = 1.1, √1.44 = 1.2, √0.81 = 0.9. Half of all pendulum and string questions are built on them.

For any "what happens to T if" question, write the proportionality and stop. T ∝ √l, T ∝ √m, T ∝ 1/√g, T ∝ 1/√k. Then scale.

For pipes, draw the standing wave. A closed pipe at its fundamental holds a quarter wavelength, an open pipe holds a half. The picture is faster than recalling which formula has 4L.

For wave interference and diffraction with light, see physical optics. All the formulas above are also on the physics formula sheet.

FAQ#