Waves Basics — Set 3

Correct Answer: A. k = 2π/λ

• **k = 2π/λ** = wave number (spatial analogue of angular frequency ω = 2π/T); it measures how many radians of phase change occur per metre. • **SI unit**: rad/m or simply m⁻¹; higher k means shorter wavelength (more oscillations per metre). • 💡 Wrong-option analysis: k = f: frequency is cycles per second — not the spatial wave number; k = 1/T: this is frequency (f = 1/T) — a temporal quantity; k = λ: wavelength itself — not its inverse.

Correct Answer: C. ω = 2πf

• **ω = 2πf** = angular frequency is the rate of change of phase in radians per second; one complete cycle covers 2π radians in T = 1/f seconds. • **SI unit**: rad/s; ω = 2π × 50 = ~314 rad/s for a 50 Hz wave. • 💡 Wrong-option analysis: ω = 1/f: this equals T (period) — a time, not angular frequency; ω = f/2π: this would give ω units of Hz/rad — less natural; ω = f²: dimensionally wrong — Hz² ≠ rad/s.

Correct Answer: D. 10 m/s

• **10 m/s** = from v = fλ: v = 5 Hz × 2 m = 10 m/s. • **Check units**: Hz × m = (cycles/s) × m = m/s — dimensionally correct. • 💡 Wrong-option analysis: 7 m/s: this would be f + λ = 5 + 2 — incorrect formula; 20 m/s: this would be f × λ² = 5 × 4 — no physical basis; 2.5 m/s: this would be f/λ = 5/2 — incorrect formula.

Correct Answer: A. 2 m

• **2 m** = from λ = v/f: λ = 340/170 = 2 m. • **Check**: v = fλ = 170 × 2 = 340 m/s ✓ — the relationship is consistent. • 💡 Wrong-option analysis: 170 m: this equals v/1 — confusing wavelength with the speed value; 510 m: this equals v × (1/f) × 1.5 — no physical basis; 1 m: this would require v = 170 m/s — not the given speed.

Correct Answer: C. Solids

• **Solids** = v = √(E/ρ); solids have very high elastic moduli (e.g., steel E ≈ 200 GPa), so vibrations propagate rapidly — typically 3000–6000 m/s. • **Order**: solids (~5000 m/s) > liquids (~1500 m/s) > gases (~340 m/s); vacuum: sound cannot travel at all. • 💡 Wrong-option analysis: Free space only: 'free space' implies vacuum — sound cannot travel without a medium; Vacuum: no particles to transmit vibrations — speed is undefined/zero; Gases: lowest elastic modulus and compressible — slowest propagation.

Correct Answer: A. Less than 20 Hz

• **Less than 20 Hz** = infrasound lies below the lower threshold of human hearing (~20 Hz); particles vibrate too slowly for the ear to detect. • **Natural sources**: earthquakes, volcanic eruptions, ocean waves — all produce infrasound that can travel thousands of kilometres. • 💡 Wrong-option analysis: Between 20 Hz and 20 kHz: this is the audible range — not infrasound; Greater than 20 kHz: this is ultrasound — the opposite extreme; Exactly 1 kHz: this is a mid-audible tone — not infrasound.

Correct Answer: C. per meter (m⁻¹)

• **Per metre (m⁻¹)** = wave number k = 2π/λ has units of radians per metre (rad/m), often simplified to m⁻¹. • **Physical meaning**: k tells how many complete wave cycles (×2π radians) fit in one metre; larger k = shorter wavelength. • 💡 Wrong-option analysis: Metre (m): this is the unit of wavelength — not wave number; Hertz (Hz): this is the unit of frequency — temporal, not spatial; Second (s): unit of time period — not wave number.

Correct Answer: A. Speed in the new medium

• **Speed in the new medium** = refraction occurs because the wave speed changes at the boundary between two media; Snell's law relates the angles of incidence and refraction to the speed ratio. • **Snell's law for waves**: n₁ sin θ₁ = n₂ sin θ₂, where n ∝ 1/v — bending toward the normal when entering a slower medium. • 💡 Wrong-option analysis: Frequency only: frequency is set by the source and does not change at a boundary; Phase only: phase changes continuously along the wave — not the cause of bending; Amplitude only: amplitude may decrease due to reflection — not the cause of directional change.

Correct Answer: D. Amplitude of the wave

• **Amplitude of the wave** = for linear (small-amplitude) waves in a uniform medium, wave speed depends only on medium properties (tension, density, elasticity) — not on how large the disturbance is. • **String wave**: v = √(T/μ) — tension T and linear density μ determine speed, not amplitude; amplitude affects energy, not speed. • 💡 Wrong-option analysis: Tension in a string: higher tension → higher speed (v ∝ √T) — directly affects speed; Density of medium: higher density → lower speed (v ∝ 1/√ρ); Elastic property of medium: higher elasticity → higher speed.

Correct Answer: C. Louder sound

• **Louder sound** = amplitude determines the maximum displacement of particles, which is directly related to the energy carried; higher amplitude → higher intensity → louder perceived sound. • **I ∝ A²**: doubling amplitude quadruples intensity (+6 dB in decibels). • 💡 Wrong-option analysis: Slower sound: amplitude does not affect wave speed in linear waves; Lower pitch: pitch depends on frequency — not amplitude; Higher pitch: pitch depends on frequency — not amplitude.