Waves Basics — Set 5

Correct Answer: A. About 17 m one-way to the reflector

• **About 17 m one-way to the reflector** = sound travels to the reflector (17 m) and back (17 m) = 34 m total; at 340 m/s, this takes 34/340 = 0.1 s — the minimum delay needed for the ear to distinguish the echo from the original sound. • **Formula**: minimum reflector distance = v × t_min / 2 = 340 × 0.1 / 2 = 17 m. • 💡 Wrong-option analysis: About 340 m one-way: this would mean a round trip of 680 m → 2 s delay — far more than needed; About 3.4 m one-way: round trip = 6.8 m → 0.02 s — too short for the ear to separate the echo; About 17 m round-trip path only: 17 m round-trip at 340 m/s = 0.05 s — still below the 0.1 s threshold.

Correct Answer: A. A periodic variation in loudness

• **A periodic variation in loudness** = beats are the result of two slightly different frequencies alternately reinforcing and cancelling, making the combined amplitude (and hence loudness) rise and fall periodically. • **Beat period** = 1/|f₁ − f₂| — if two forks differ by 4 Hz, loudness pulses 4 times per second. • 💡 Wrong-option analysis: A stopping of the wave: beats are a periodic modulation — neither wave stops; A sudden increase in wave speed: speed depends on the medium — not altered by two-frequency superposition; A permanent change in pitch only: pitch depends on frequency — beats do not permanently change frequency.

Correct Answer: A. β = 10 log10(I/I0)

• **β = 10 log₁₀(I/I₀)** = sound intensity level in decibels; I₀ = 10⁻¹² W/m² (threshold of hearing); each 10 dB increase = 10× the intensity. • **Scale**: 0 dB (threshold) → 60 dB (conversation) → 120 dB (pain threshold) → intensity ratio of 10¹². • 💡 Wrong-option analysis: β = I₀/I: inverted ratio — gives large values for quiet sounds, zero for loud — backwards; β = I/I0: no logarithm — gives just a dimensionless ratio in Bels × 10 step missing; β = 20 log₁₀(I/I₀): factor 20 is for amplitude ratios (pressure, voltage), not intensity.

Correct Answer: B. 340 m/s

• **340 m/s** = a commonly used approximate value for the speed of sound in air at room temperature (~20°C); the precise value is ≈ 343 m/s. • **Temperature dependence**: v ≈ 331 + 0.6T m/s — at 0°C it is 331 m/s; the 340 m/s figure corresponds to ~15°C. • 💡 Wrong-option analysis: 34 m/s: 10× too slow — approximately cycling speed; 3 × 10⁸ m/s: this is the speed of light — nearly a million times faster than sound; 3000 m/s: this is closer to the speed of sound in steel (~5000 m/s) — not in air.

Correct Answer: D. S-wave

• **S-wave** = secondary (shear) seismic waves are transverse — rock particles vibrate perpendicular to the direction of wave propagation; they cannot travel through liquids. • **Contrast**: P-waves (primary/pressure) are longitudinal — particles vibrate parallel to propagation; travel in solids AND liquids. • 💡 Wrong-option analysis: P-wave: primary waves are longitudinal, not transverse — fastest seismic waves; Surface wave only: surface waves (Love, Rayleigh) are at the Earth's surface — not the same as S-waves; Shock wave: a non-linear wave formed by supersonic motion — not a seismic wave type.

Correct Answer: B. Longitudinal waves

• **Longitudinal waves** = P-waves (primary/pressure waves) compress and expand the medium along the direction of travel — a longitudinal motion. • **P-wave speed** in granite ≈ 5000–6000 m/s; P-waves arrive before S-waves because they travel faster. • 💡 Wrong-option analysis: Transverse waves only: S-waves are transverse, not P-waves — P-waves can travel through liquid (outer core) but S-waves cannot; Electromagnetic waves: seismic waves are mechanical — they need the Earth as medium; Waves that need no medium: all seismic waves are mechanical — they require the solid Earth.

Correct Answer: D. Perpendicular to each other and perpendicular to direction of travel

• **Perpendicular to each other and perpendicular to direction of travel** = in an electromagnetic wave, E and B oscillate in planes perpendicular to each other and to the propagation direction — this makes EM waves transverse. • **Poynting vector** S = (E × B)/μ₀ points in the direction of propagation — E × B confirms both fields are perpendicular to travel. • 💡 Wrong-option analysis: Parallel to each other and parallel to direction of travel: a longitudinal EM wave — Maxwell's equations do not allow this in vacuum; Parallel to each other and perpendicular to direction of travel: E and B must be perpendicular to each other, not parallel; Randomly oriented always: EM waves are ordered, not random — random orientation would not allow propagation.

Correct Answer: B. E = cB

• **E = cB** = in an electromagnetic wave in vacuum, the magnitudes of E and B are related by E = cB, where c = 3 × 10⁸ m/s. • **Derivation**: from Maxwell's equations, the wave equations for E and B give E₀/B₀ = c — E field is always c times stronger in SI units. • 💡 Wrong-option analysis: E = c/B: dimensions wrong — c/B has units m/s / T = m/s / (V·s/m²) = m³/Vs² — not V/m; E = B: dimensionally impossible — E in V/m, B in T (Wb/m²) = V·s/m² — different units; E = B/c: gives V/m = T·m/s — inverted, wrong.

Correct Answer: D. 300 m/s

• **300 m/s** = v = distance/time = 120 m / 0.4 s = 300 m/s. • **Check units**: metres/seconds = m/s ✓ — a speed close to the speed of sound in air at room temperature. • 💡 Wrong-option analysis: 480 m/s: this would be 120/0.25 — wrong time used; 30 m/s: this would be 120/4 — 10× too slow; 120 m/s: this confuses distance (120 m) with speed — applies v = d, missing the time divisor.

Correct Answer: C. 2 m

• **2 m** = from λ = v/f: λ = 50/25 = 2 m. • **Verification**: v = fλ = 25 × 2 = 50 m/s ✓ — consistent with given values. • 💡 Wrong-option analysis: 4 m: this would give v = 25 × 4 = 100 m/s — not 50 m/s; 1 m: this would give v = 25 × 1 = 25 m/s — not 50 m/s; 0.5 m: this would give v = 25 × 0.5 = 12.5 m/s — not 50 m/s.