Waves Basics — Set 6

Correct Answer: D. 4 Hz

• **4 Hz** = beat frequency = |f₁ − f₂| = |256 − 260| = 4 Hz — the two tuning forks alternately reinforce and cancel 4 times per second. • **Practical use**: musicians listen for these 4 beats per second and adjust tension/position until beats disappear (f₁ = f₂). • 💡 Wrong-option analysis: 260 Hz: this is just f₂ — not the beat frequency; 6 Hz: this would be |256 − 262| — wrong calculation; 516 Hz: this is f₁ + f₂ = 256 + 260 — sum, not difference.

Correct Answer: D. λ/4

• **λ/4** = in a standing wave, nodes and antinodes alternate; each is separated from the nearest node or antinode by exactly one quarter wavelength (λ/4). • **Node-to-node spacing = λ/2** — one full half-wavelength between adjacent nodes; node-to-antinode = λ/4. • 💡 Wrong-option analysis: 2λ: this is twice the wavelength — much larger than the node–antinode spacing; λ/2: this is the node-to-node (or antinode-to-antinode) distance — not node-to-nearest-antinode; λ: this is the full wavelength — two node–antinode spacings.

Correct Answer: D. f_n = n(v/2L)

• **f_n = n(v/2L)** = for a string fixed at both ends, n half-wavelengths fit in length L; f₁ = v/2L (fundamental), f₂ = 2v/2L, f₃ = 3v/2L, … — integer multiples of f₁. • **Harmonic series**: all integer harmonics are supported (n = 1, 2, 3, …), unlike a closed pipe which supports only odd harmonics. • 💡 Wrong-option analysis: f_n = v/4L: fundamental of a closed pipe — not a string; f_n = v/2nL: this gives decreasing frequency with n — wrong (harmonics increase with n); f_n = v/L: this would be f₂ (2nd harmonic) only — not the general nth harmonic formula.

Correct Answer: A. Only odd harmonics

• **Only odd harmonics** = a closed pipe (node at closed end, antinode at open end) supports modes where λ = 4L/n for n = 1, 3, 5, … — only odd integers satisfy the boundary condition. • **Missing even harmonics** gives the closed pipe a hollow, woody tone compared to an open pipe (all harmonics present). • 💡 Wrong-option analysis: No harmonics: all real pipes support resonant modes above the fundamental; All harmonics: only open pipes (both ends open) support all integer harmonics; Only even harmonics: the closed pipe suppresses even harmonics, not odd ones.

Correct Answer: B. Comparable to wavelength

• **Comparable to wavelength** = significant diffraction occurs when the aperture or obstacle size d ≈ λ; if d >> λ, the wave barely bends (geometric shadow); if d << λ, the wave passes as if the obstacle is absent. • **Example**: sound (λ ≈ 0.03–17 m) diffracts strongly around typical doorways (width ~1 m); visible light (λ ≈ 500 nm) requires very narrow slits for significant diffraction. • 💡 Wrong-option analysis: Independent of wavelength always: diffraction depends critically on the wavelength-to-aperture ratio; Much larger than wavelength: d >> λ means geometric optics applies — little diffraction; Much smaller than wavelength: d << λ means the obstacle barely affects the wave — no prominent diffraction.

Correct Answer: A. v = f × λ

• **v = f × λ** = the fundamental wave equation: wave speed equals frequency multiplied by wavelength; this applies to all mechanical and electromagnetic waves. • **Example**: at f = 100 Hz and v = 340 m/s, λ = 3.4 m; at f = 10,000 Hz, λ = 0.034 m — frequency and wavelength are inversely proportional at constant speed. • 💡 Wrong-option analysis: v = f + λ: dimensionally wrong — m/s ≠ Hz + m; v = f / λ: gives Hz/m = s⁻¹/m — not m/s; v = f − λ: dimensionally inconsistent and physically meaningless.

Correct Answer: B. Solids and on liquid surfaces

• **Solids and on liquid surfaces** = transverse waves require the medium to support shear (restoring force perpendicular to propagation); solids have shear modulus; liquids support transverse waves only at their free surface (e.g., ripples). • **Contrast**: longitudinal waves travel in all three states — solid, liquid, and gas — since all resist compression. • 💡 Wrong-option analysis: Solids only: transverse waves also exist at liquid surfaces (water ripples) — solids only is too restrictive; Gases only: gases support only longitudinal waves — shear forces are zero in gases; Liquids only: bulk liquids do not support transverse waves — only surface waves.

Correct Answer: C. Diffraction

• **Diffraction** = the bending of waves around edges or through small openings into the geometric shadow region. • **Condition for strong diffraction**: aperture or obstacle size ≈ wavelength; explains why we can hear sound around corners but cannot see light around them. • 💡 Wrong-option analysis: Refraction: direction change when wave enters a new medium due to speed change — not bending around edges; Reflection: bouncing of waves off a surface — no spreading into shadow; Interference: superposition of two or more waves producing pattern of constructive/destructive regions — different from bending around obstacles.

Correct Answer: D. Longitudinal

• **Longitudinal** = sound waves cause particles to vibrate parallel to the direction of propagation, creating regions of compression (high pressure) and rarefaction (low pressure). • **Requires a medium**: air, water, or solid — sound cannot propagate in vacuum because there are no particles to compress. • 💡 Wrong-option analysis: Transverse: particle vibration perpendicular to propagation — this is light and string waves, not sound; Surface: confined to an interface between two media — sound propagates in bulk media; Electromagnetic: self-propagating electric/magnetic fields, needs no medium — sound is mechanical.

Correct Answer: A. Beats

• **Beats** = when two waves of slightly different frequencies f₁ and f₂ superpose, the amplitude varies periodically at the rate |f₁ − f₂| — heard as a periodic rise and fall in loudness. • **Beat period** = 1/|f₁ − f₂|; used in tuning instruments by adjusting until beats disappear (f₁ = f₂). • 💡 Wrong-option analysis: Overtones: harmonics above the fundamental — produced by a single source, not two-frequency superposition; Resonance: maximum energy transfer at the natural frequency — not a superposition of two different frequencies; Harmonics: integer multiples of the fundamental — a property of a single periodic wave, not beat superposition.