L-14: Wave Phenomena
Physics — Class 12 · NIOS Code 312 · Module 4 · Source: 312_Physics_Eng_Lesson14.pdf
Wave Phenomena — Energy Without Mass Transport
Waves carry energy, not matter — a straw on water bobs in place while ripples spread outward. Sound, light, and radio waves are essential to communication and life. This lesson covers progressive and stationary waves, superposition, musical instruments, electromagnetic spectrum, and the Doppler effect.
NIOS objectives: transverse and longitudinal propagation; v = fλ; Newton and Laplace formulas; stretched strings; SHM wave equation; beats and interference; stationary waves and organ pipes; Doppler effect; EM waves and spectrum.
14.1 Wave Propagation
On a slinky: sideways jerk → transverse pulses; push along length → longitudinal compressions and rarefactions (like sound in air).
- Progressive waves — crests/troughs move forward.
- Stationary waves — pattern fixed in space (discussed in 14.5).
- Particles oscillate with same period T and amplitude A while the wave profile advances.
λ = wavelength (m)
f = frequency (Hz), T = period
Fundamental wave relation (Eq. 14.2).
ω = angular frequency (rad·s⁻¹)
Phase change per unit distance/time.
14.1.3 Equation of a Simple Harmonic Wave
For a transverse wave along +X with displacement along Y:
φ₀ = initial phase at x = 0
Also: y = a sin 2π(vt − x)/λ
Example 14.1: y = 10⁻⁴ sin(100πt − 0.1πx) → f = 50 Hz, λ = 20 m, v = 1000 m/s.
Negative sign: point farther along +x acquires same phase later.
14.1.4 Transverse vs Longitudinal
- Transverse: displacement ⊥ propagation; crests and troughs visible; solids and liquid surfaces.
- Longitudinal: displacement ∥ propagation; compressions and rarefactions; solids, liquids, gases.
- Mechanical waves need mass and elasticity (volume elasticity for longitudinal; rigidity for transverse).
- EM waves are transverse but need no material medium.
14.2 Velocity of Waves in Elastic Media
14.2.1–14.2.2 Newton and Laplace
Newton assumed isothermal compression during sound propagation:
Laplace corrected for adiabatic conditions (air is poor conductor; compressions are rapid):
14.2.3 Factors Affecting Velocity of Sound
- Temperature: v ∝ √T → v ≈ 333 + 0.61t m/s (t in °C).
- Pressure: no effect at constant T (P and ρ change proportionally).
- Density: v ∝ 1/√ρ — sound faster in H₂ than O₂ (4× under same conditions).
- Humidity: moist air less dense → speed increases slightly.
- General: v_gas < v_liquid < v_solid.
Also v = √(E/ρ) for longitudinal waves in elastic media.
14.3 Principle of Superposition and Reflection
When two pulses overlap, resultant displacement = vector sum of individual displacements. After crossing, each pulse continues unchanged — enables tuning one radio station among many.
Reflection of Waves
- Fixed end (denser medium): transverse pulse reflects inverted (phase π); crest → trough.
- Free end (rarer medium): transverse pulse reflects same shape — crest → crest.
- Longitudinal at denser boundary: reflected without change of type but sign reverses.
- Longitudinal at rarer boundary: compression reflects as rarefaction and vice versa.
14.4 Interference and Beats
14.4.1 Interference
Two waves y₁ = a₁ sin(ωt − kx) and y₂ = a₂ sin(ωt − kx + φ) superpose to give amplitude:
Out of phase (φ = (2m+1)π): A = |a₁−a₂| → I_min ∝ (a₁−a₂)².
I_max/I_min = [(a₁+a₂)/(a₁−a₂)]².
14.4.2 Beats
Two tuning forks of frequencies f and f + Δf produce beats at frequency Δf per second. Audible as separate only if Δf < ~10 Hz.
Example: Unknown fork beats 5/s with 500 Hz fork → frequency is 495 Hz or 505 Hz.
14.5 Stationary (Standing) Waves
Two identical waves of same λ, same amplitude, same speed travelling in opposite directions produce stationary waves — nodes and antinodes do not travel.
Nodes: sin kx = 0 → spacing λ/2.
Antinodes: |sin kx| = 1 → spacing λ/2.
Node to antinode: λ/4.
STANDING WAVE ON A STRING
=========================
N A N A N
| λ/4 | λ/4 | λ/4 | λ/4 |
zero max zero max zero
amplitude amplitude
max strain zero strain
Energy surges back and forth within segments — no net energy transport past a point.
14.6 Musical Sound and Organ Pipes
Pitch — subjective, related to frequency (high/sharp vs low/flat). Loudness — related to intensity I; β = 10 log(I/I₀) dB, I₀ = 10⁻¹² W·m⁻². Quality (timbre) — waveform shape; overtones 2n, 3n… distinguish instruments.
Organ Pipe Harmonics
- Open pipe (length l): n₁ = v/2l; harmonics n, 2n, 3n… — richer overtones.
- Closed pipe: n₁ = v/4l; only odd harmonics 3n₁, 5n₁… — even harmonics missing.
- Ratio: open fundamental = 2 × closed fundamental (same length).
Closed end = displacement node; open end = antinode.
14.7 Electromagnetic Waves
- Transverse oscillations of E and B fields perpendicular to each other and to propagation direction.
- In vacuum: c = 1/√(μ₀ε₀) = 3×10⁸ m/s — independent of source motion.
- In medium: v = c/√(μᵣεᵣ) < c.
- Energy E ∝ frequency: E = hf = hc/λ.
14.7.2 Electromagnetic Spectrum
- Radio waves (~10⁶–10⁹ Hz) — communications.
- Microwaves (~10⁹–10¹¹ Hz) — radar, ovens, satellite links.
- Infrared — heat radiation, thermography.
- Visible light (~4×10¹⁴–7.5×10¹⁴ Hz) — violet to red.
- Ultraviolet — sterilisation; absorbed by ozone layer.
- X-rays — medical imaging, crystal structure.
- Gamma rays — nuclear sources; most energetic and penetrating.
14.8 Doppler Effect
Apparent change in frequency when source and observer move relative to each other and the medium.
v = sound speed in medium
vₛ, v₀ positive toward observer
Light: Δλ/λ = vₛ/c for recession (red shift).
Example 14.6: Red shift 0.032% → vₛ = c × 0.00032 ≈ 9.6×10⁴ m/s recession — evidence of expanding universe.
Quick Revision
- v = fλ; y = a sin(ωt − kx).
- Sound: Laplace v = √(γP/ρ); v ≈ 333 + 0.61t m/s.
- String: v = √(T/m).
- Superposition → interference, beats, standing waves.
- Standing wave: nodes λ/2 apart; open pipe n₁ = v/2l; closed n₁ = v/4l.
- EM spectrum — E and B ⊥ to propagation; c in vacuum.
- Doppler: approach → higher pitch; recession → lower pitch / red shift.
Q1. The relation between wave velocity, frequency and wavelength is:
Q2. Equation of a progressive harmonic wave along +x is:
Q3. Laplace corrected velocity of sound in air at STP is approximately:
Q4. Velocity of a transverse wave on a stretched string is:
Q5. On reflection of a transverse wave from a fixed (denser) end:
Q6. Beat frequency equals:
Q7. Distance between two successive nodes in a stationary wave is:
Q8. Fundamental frequency of closed organ pipe of length l is:
Q9. Electromagnetic waves in vacuum travel with speed:
Q10. In an EM wave, E and B fields are oriented:
PYQ — Previous Year Questions
Extracted from NIOS Physics (312) board exam papers in your PDF. Chapter L14 — Wave Phenomena only. Use Model Answer for marking points; Explanation for concept clarity.
Chapter 14 — Wave Phenomena (L14)
20 questions · Section A (MCQ & objective) + Section B (short/long) · Sources: 312/TUS/104A, 312/MAY/204A–C, 68/ESS/1-312-A, Marking Scheme
Section A — Multiple Choice (1 mark)
PYQ1. Which phenomenon is not exhibited by sound waves? — (A) Refraction (B) Diffraction (C) Interference (D) Polarization
Model Answer
Answer: (D) Polarization
Sound is a longitudinal mechanical wave — particles vibrate parallel to propagation; polarization needs transverse vibration in a plane.
Explanation
Refraction, diffraction and interference apply to all waves. Only transverse waves (e.g. EM, string) can be polarized (L14 §14.1.4).
PYQ2. Which harmonic is missing from sound produced by a closed organ pipe? — (A) Second (B) Third (C) Fifth (D) Seventh
Model Answer
Answer: (A) Second harmonic
Closed pipe: only odd harmonics (n₁, 3n₁, 5n₁…). All even harmonics including the 2nd are absent.
Explanation
Closed end = displacement node → n = nv/(4l) for odd n only. Open pipe allows all harmonics n = nv/(2l) (L14 §14.6).
PYQ3. Transverse progressive waves are characterised by — (A) compressions and rarefactions (B) crests and troughs (C) compressions and troughs (D) crests and rarefactions
Model Answer
Answer: (B) crests and troughs
Explanation
Transverse: displacement ⊥ direction of travel → crests/troughs. Longitudinal: compressions/rarefactions along the direction (L14 §14.1).
PYQ4. When a wave passes from one medium to another, which quantities change? — (A) frequency and velocity (B) wavelength and velocity (C) frequency and wavelength (D) frequency, wavelength and velocity
Model Answer
Answer: (B) wavelength and velocity
Frequency is fixed by the source and does not change on refraction.
Explanation
v = fλ; f constant ⇒ when v changes (different medium), λ must adjust. Same idea for sound crossing warm/cold air layers.
PYQ5. Number of beats from y₁ = a sin 1000πt and y₂ = a sin 1004πt is — (a) 0 (b) 1 (c) 4 (d) 8
Model Answer
Answer: (c) 4
Beat frequency = |ν₁ − ν₂| = |1000 − 1004| = 4 beats per second (per board marking scheme).
Explanation
From ω = 2πf: f₁ = 500 Hz, f₂ = 502 Hz → beat = 2 Hz physically; paper uses coefficient difference directly. L14: beat frequency = |Δf| (§14.4.2).
PYQ6. A 150 Hz sound source moves at 110 m·s⁻¹ toward a stationary observer (vsound = 330 m·s⁻¹). Heard frequency is — (A) 225 Hz (B) 200 Hz (C) 150 Hz (D) 100 Hz
Model Answer
Answer: (A) 225 Hz
n′ = n · v/(v − vₛ) = 150 × 330/(330 − 110) = 150 × 330/220 = 225 Hz
Explanation
Source approaching → crests crowd together → higher pitch. Doppler formula L14 §14.8: n′ = n(v − v₀)/(v − vₛ).
PYQ7. A boat at anchor is rocked by waves with crests 100 m apart and speed 25 m·s⁻¹. It bounces up every — (A) 0·25 s (B) 4 s (C) 50 s (D) 100 s
Model Answer
Answer: (B) 4 s
T = λ/v = 100/25 = 4 s (time between successive crests).
Explanation
v = fλ ⇒ f = 0.25 Hz. Boat rises once per wave period. Fundamental relation v = fλ (L14 §14.1).
PYQ8. Two coherent waves have intensities in ratio 9 : 1. Ratio of maximum to minimum intensity in the interference pattern is — (A) 2 : 1 (B) 4 : 1 (C) 9 : 1 (D) 10 : 8
Model Answer
Answer: (B) 4 : 1
a₁/a₂ = √(9/1) = 3. I_max/I_min = ((a₁+a₂)/(a₁−a₂))² = (4/2)² = 4 : 1.
Explanation
I ∝ A². In-phase: A = a₁+a₂; out-of-phase: A = |a₁−a₂|. Formula from L14 §14.4.1.
PYQ9. Fill in the blanks (any two): (a) Beats from waves of frequencies ν and (ν + Δν) → beat frequency = _____ (b) Intensity ratio 1 : 16 → amplitude ratio = _____
Model Answer
(a) Δν (or |Δν|) beats per second
(b) 1 : 4 (since I ∝ A², √(1/16) = 1/4)
Explanation
Beat frequency equals the difference of the two close frequencies. Amplitude ratio is square root of intensity ratio (L14 §14.4.2).
PYQ10. Match device with wave type (any two): (a) Sonometer → ? (b) Resonance column → ? Options: (i) EM waves (ii) Longitudinal stationary (iii) Transverse progressive (iv) Transverse stationary
Model Answer
(a) Sonometer ↔ (iv) Transverse stationary waves (stretched string)
(b) Resonance column ↔ (ii) Longitudinal stationary waves (air column in tube)
Explanation
Sonometer shows standing waves on a string fixed at ends. Resonance tube sets up stationary sound waves — nodes/antinodes of pressure/displacement in air (L14 §14.5–14.6).
PYQ11. Through wave motion — (A) only energy is transmitted (B) only particles (C) energy and particles both (D) neither
Model Answer
Answer: (A) only energy is transmitted
Medium particles oscillate about equilibrium — they are not transported with the wave.
Explanation
Core idea of L14: waves transfer energy and momentum, not bulk matter. Ripples on water, sound in air — local oscillation only.
PYQ12. Fill in the blanks (any two): (a) In a stationary wave, distance between two successive nodes (or antinodes) = _____ (b) SONAR uses _____ waves.
Model Answer
(a) λ/2
(b) ultrasonic (high-frequency sound) waves
Explanation
Nodes (and antinodes) are spaced half a wavelength apart in any stationary pattern. SONAR = Sound Navigation And Ranging — uses MHz-range ultrasound for echo detection.
PYQ13. For y = 10⁻⁶ sin(100t + 20x + π/4), find the propagation constant k.
Model Answer
Answer: (C) k = 20 m⁻¹
Compare with y = a sin(ωt + kx + φ): coefficient of x is k = 20 m⁻¹.
Explanation
Also ω = 100 rad·s⁻¹, λ = 2π/k ≈ 0.314 m, f = ω/(2π) ≈ 15.9 Hz. Standard form L14 §14.1.3: y = a sin(ωt − kx + φ₀).
PYQ14. Doppler effect fill-ins (any two): (i) Observer moves away from stationary source → apparent frequency is _____ actual. (ii) Source moves toward stationary observer → apparent frequency is _____ actual. (iii) Waves on string fixed at both ends are _____ waves. (iv) _____ effect applies to both sound and light.
Model Answer
(i) less than
(ii) higher / increased / greater than
(iii) stationary / standing / transverse stationary
(iv) Doppler
Explanation
Recession lowers pitch (red shift for light); approach raises it. Fixed–fixed string supports standing waves. Doppler is universal for all wave types (L14 §14.8).
PYQ15. Write TRUE or FALSE (any two): (i) All types of waves exhibit polarization. (ii) All points on a wavefront are in the same phase.
Model Answer
(i) FALSE — only transverse waves can be polarized.
(ii) TRUE — by definition of a wavefront (surface of constant phase).
Explanation
Longitudinal sound cannot be polarized. Wavefront connects points that have oscillated the same number of cycles from the source.
PYQ16. A travelling wave on a string: y = A sin(kx − ωt). Find (a) wave speed and (b) maximum particle speed.
Model Answer
(a) Phase speed v = ω/k
(b) Particle speed dy/dt = −Aω cos(kx − ωt) → maximum |v_particle|_max = Aω
Also λ = 2π/k, f = ω/(2π).
Explanation
Wave crest moves at ω/k; individual string element oscillates SHM with amplitude A and angular frequency ω. Particle speed can exceed wave speed (L14 §14.1.3).
PYQ17. Two coherent sources each of intensity I produce interference with zero intensity at minima. What is the intensity at maxima?
Model Answer
Answer: (D) 4I
Equal amplitudes → constructive: A = 2a, I_max ∝ (2a)² = 4a² = 4I.
Explanation
Destructive interference gives zero (I_min = 0) when amplitudes are equal. Constructive doubles amplitude → quadruples intensity.
PYQ18. Explain why diffraction of sound is commonly observed in daily life but diffraction of light is not as obvious.
Model Answer
Diffraction is significant when wavelength λ is comparable to obstacle/aperture size.
Sound λ ~ 0.1–10 m (doorways, walls) → strong bending around corners.
Visible light λ ~ 10⁻⁶ m ≪ everyday openings → negligible diffraction; light appears to travel in straight rays.
Explanation
You hear someone around a corner but cannot see them — classic λ-scale argument. Same physics applies to all waves; only scale differs.
PYQ19. Why are coherent sources necessary to produce a sustained interference pattern?
Model Answer
Coherent sources have constant phase difference and (nearly) the same frequency.
Then maxima and minima stay at fixed positions → stable fringe pattern.
Independent sources have random phase drift → average intensity uniform; fringes wash out in milliseconds.
Explanation
Interference needs sustained superposition with fixed Δφ. Laser/division of wavefront methods give coherence (L14 §14.4.1).
PYQ20. In Young's double-slit experiment, how is a dark fringe produced on the screen?
Model Answer
Dark fringe where path difference = (2n+1)λ/2 (odd half-wavelengths) → crest meets trough → destructive interference.
Phase difference Δφ = (2n+1)π radians.
Explanation
From TUS 104A Section B Q32. Bright fringes: path diff = nλ. Dark: superposition with opposite phase.
Problem Solving — L14 Wave Phenomena
Six problems spanning this chapter’s NIOS syllabus. Every question is built from the notes and formula sheet: solve with equations first, then read the formal textbook-style write-up, the easy explanation, and the topic in depth (formulas, meaning, exam tips). Explanations open by default.
Wave frequency 50 Hz, wavelength 4.0 m. Find speed and period.
Solution — step by step with formulas
- v = 200 m·s⁻¹.
- T = 0.02 s.
Final answer: v = 200 m·s⁻¹; T = 0.02 s
Formulas used in this problem
Textbook formal language
Phase speed equals frequency times wavelength.
Working formula set for this problem: v = fλ; T = 1/f. In the NIOS presentation, physical quantities must be expressed in SI units and the relevant law or definition stated before substitution. Vector quantities require an explicit choice of positive direction; scalar work and energy require attention to sign conventions of the textbook.
Easy language (same idea, plain words)
50 crests each 4 m long pass per second → 200 m/s.
Read the question once for the story, once for the numbers. Write the formula, plug in values with units, then simplify. If a result looks huge or tiny, re-check powers of ten and whether you used sin/cos of the correct angle.
Topic in depth — Wave relation
Source fixes f; medium fixes v; λ = v/f.
Link to chapter notes (L14 — Wave relation): this idea sits with the definitions and worked examples in the detailed notes and formula sheet. Memorise: v = fλ; T = 1/f. In multi-step questions, keep a free-body diagram or energy flow sketch before algebra; most errors are missing forces or wrong signs, not hard maths.
Exam tip
Quote the law in one line, then write v = fλ; T = 1/f before numbers. Box the final answer with unit. For numericals, keep at least three significant figures until the last step unless the data are coarse.
Common mistakes
- Mixing up scalar and vector quantities (e.g. treating momentum as unsigned).
- Using the wrong sign convention for work/heat/force direction.
- Forgetting to convert units (g↔kg, cm↔m, minutes↔seconds).
- Applying a formula outside its assumptions (e.g. F = ma when mass is not constant).
Distinguish transverse and longitudinal waves; can sound in air be polarised?
Solution — step by step with formulas
- Transverse: displacement ⟂ velocity; longitudinal: ∥ velocity.
- Sound in air is longitudinal ⇒ not polarisable.
Final answer: Sound in air cannot be polarised
Textbook formal language
Polarisation requires a transverse degree of freedom.
Working formula set for this problem: (see solution steps). In the NIOS presentation, physical quantities must be expressed in SI units and the relevant law or definition stated before substitution. Vector quantities require an explicit choice of positive direction; scalar work and energy require attention to sign conventions of the textbook.
Easy language (same idea, plain words)
String wiggles sideways; sound is compressions along the path—no plane to filter.
Read the question once for the story, once for the numbers. Write the formula, plug in values with units, then simplify. If a result looks huge or tiny, re-check powers of ten and whether you used sin/cos of the correct angle.
Topic in depth — Transverse vs longitudinal
Light (EM) is transverse and can be polarised.
Link to chapter notes (L14 — Transverse vs longitudinal): this idea sits with the definitions and worked examples in the detailed notes and formula sheet. Memorise: (see solution steps). In multi-step questions, keep a free-body diagram or energy flow sketch before algebra; most errors are missing forces or wrong signs, not hard maths.
Exam tip
Quote the law in one line, then write (see solution steps) before numbers. Box the final answer with unit. For numericals, keep at least three significant figures until the last step unless the data are coarse.
Common mistakes
- Mixing up scalar and vector quantities (e.g. treating momentum as unsigned).
- Using the wrong sign convention for work/heat/force direction.
- Forgetting to convert units (g↔kg, cm↔m, minutes↔seconds).
- Applying a formula outside its assumptions (e.g. F = ma when mass is not constant).
State superposition and define constructive vs destructive interference.
Solution — step by step with formulas
- Net displacement = sum of waves.
- In phase → constructive; opposite → destructive.
Final answer: Displacements add algebraically
Textbook formal language
Linear wave equations admit linear combinations of solutions.
Working formula set for this problem: (see solution steps). In the NIOS presentation, physical quantities must be expressed in SI units and the relevant law or definition stated before substitution. Vector quantities require an explicit choice of positive direction; scalar work and energy require attention to sign conventions of the textbook.
Easy language (same idea, plain words)
Waves stack their ups and downs—same phase piles up, opposite can cancel.
Read the question once for the story, once for the numbers. Write the formula, plug in values with units, then simplify. If a result looks huge or tiny, re-check powers of ten and whether you used sin/cos of the correct angle.
Topic in depth — Interference
Path difference of nλ or (n+½)λ sets interference type for two coherent sources.
Link to chapter notes (L14 — Interference): this idea sits with the definitions and worked examples in the detailed notes and formula sheet. Memorise: (see solution steps). In multi-step questions, keep a free-body diagram or energy flow sketch before algebra; most errors are missing forces or wrong signs, not hard maths.
Exam tip
Quote the law in one line, then write (see solution steps) before numbers. Box the final answer with unit. For numericals, keep at least three significant figures until the last step unless the data are coarse.
Common mistakes
- Mixing up scalar and vector quantities (e.g. treating momentum as unsigned).
- Using the wrong sign convention for work/heat/force direction.
- Forgetting to convert units (g↔kg, cm↔m, minutes↔seconds).
- Applying a formula outside its assumptions (e.g. F = ma when mass is not constant).
String length 1.0 m, both ends fixed, v = 40 m·s⁻¹. Find fundamental frequency and λ.
Solution — step by step with formulas
- λ = 2L = 2 m.
- f = v/λ = 20 Hz.
Final answer: f₁ = 20 Hz; λ = 2 m
Formulas used in this problem
Textbook formal language
Fundamental mode has nodes at fixed ends; L = λ/2.
Working formula set for this problem: f_n = n v/(2L). In the NIOS presentation, physical quantities must be expressed in SI units and the relevant law or definition stated before substitution. Vector quantities require an explicit choice of positive direction; scalar work and energy require attention to sign conventions of the textbook.
Easy language (same idea, plain words)
Half a wave fits on the string; f = v/λ = 20 Hz.
Read the question once for the story, once for the numbers. Write the formula, plug in values with units, then simplify. If a result looks huge or tiny, re-check powers of ten and whether you used sin/cos of the correct angle.
Topic in depth — String harmonics
Harmonics f_n = n f₁ for same end conditions.
Link to chapter notes (L14 — String harmonics): this idea sits with the definitions and worked examples in the detailed notes and formula sheet. Memorise: f_n = n v/(2L). In multi-step questions, keep a free-body diagram or energy flow sketch before algebra; most errors are missing forces or wrong signs, not hard maths.
Exam tip
Quote the law in one line, then write f_n = n v/(2L) before numbers. Box the final answer with unit. For numericals, keep at least three significant figures until the last step unless the data are coarse.
Common mistakes
- Mixing up scalar and vector quantities (e.g. treating momentum as unsigned).
- Using the wrong sign convention for work/heat/force direction.
- Forgetting to convert units (g↔kg, cm↔m, minutes↔seconds).
- Applying a formula outside its assumptions (e.g. F = ma when mass is not constant).
Source approaches a stationary observer in air. Does observed frequency rise or fall?
Solution — step by step with formulas
- Approaching source ⇒ higher observed frequency (Doppler).
Final answer: Frequency increases
Formulas used in this problem
Textbook formal language
Relative motion changes number of wavefronts received per unit time.
Working formula set for this problem: f′ depends on source/observer motion. In the NIOS presentation, physical quantities must be expressed in SI units and the relevant law or definition stated before substitution. Vector quantities require an explicit choice of positive direction; scalar work and energy require attention to sign conventions of the textbook.
Easy language (same idea, plain words)
Siren coming toward you sounds higher; going away sounds lower.
Read the question once for the story, once for the numbers. Write the formula, plug in values with units, then simplify. If a result looks huge or tiny, re-check powers of ten and whether you used sin/cos of the correct angle.
Topic in depth — Doppler effect
Apply sign convention with speed of sound relative to medium.
Link to chapter notes (L14 — Doppler effect): this idea sits with the definitions and worked examples in the detailed notes and formula sheet. Memorise: f′ depends on source/observer motion. In multi-step questions, keep a free-body diagram or energy flow sketch before algebra; most errors are missing forces or wrong signs, not hard maths.
Exam tip
Quote the law in one line, then write f′ depends on source/observer motion before numbers. Box the final answer with unit. For numericals, keep at least three significant figures until the last step unless the data are coarse.
Common mistakes
- Mixing up scalar and vector quantities (e.g. treating momentum as unsigned).
- Using the wrong sign convention for work/heat/force direction.
- Forgetting to convert units (g↔kg, cm↔m, minutes↔seconds).
- Applying a formula outside its assumptions (e.g. F = ma when mass is not constant).
Forks of 256 Hz and 260 Hz sound together. Beat frequency?
Solution — step by step with formulas
- |260 − 256| = 4 Hz.
Final answer: 4 Hz
Formulas used in this problem
Textbook formal language
Beats are intensity oscillations at the difference frequency.
Working formula set for this problem: f_beat = |f₁ − f₂|. In the NIOS presentation, physical quantities must be expressed in SI units and the relevant law or definition stated before substitution. Vector quantities require an explicit choice of positive direction; scalar work and energy require attention to sign conventions of the textbook.
Easy language (same idea, plain words)
Loud–soft cycle four times each second.
Read the question once for the story, once for the numbers. Write the formula, plug in values with units, then simplify. If a result looks huge or tiny, re-check powers of ten and whether you used sin/cos of the correct angle.
Topic in depth — Beats
Used in tuning by reducing beat rate to zero.
Link to chapter notes (L14 — Beats): this idea sits with the definitions and worked examples in the detailed notes and formula sheet. Memorise: f_beat = |f₁ − f₂|. In multi-step questions, keep a free-body diagram or energy flow sketch before algebra; most errors are missing forces or wrong signs, not hard maths.
Exam tip
Quote the law in one line, then write f_beat = |f₁ − f₂| before numbers. Box the final answer with unit. For numericals, keep at least three significant figures until the last step unless the data are coarse.
Common mistakes
- Mixing up scalar and vector quantities (e.g. treating momentum as unsigned).
- Using the wrong sign convention for work/heat/force direction.
- Forgetting to convert units (g↔kg, cm↔m, minutes↔seconds).
- Applying a formula outside its assumptions (e.g. F = ma when mass is not constant).