This MCQ module is based on: Speed Waves String Gas
TOPIC 30 OF 33
Speed Waves String Gas
🎓 Class 11
Physics
CBSE
Theory
Ch 14 – Waves
⏱ ~14 min
🌐 Language: [gtranslate]
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Speed Waves String Gas
14.4 The Speed of a Travelling Wave
The speed of a wave depends on the properties of the medium in which it propagates, not on properties of the source. A wave speeds up if the medium is more "elastic" (stiffer) or less dense.
General rule:
\[v = \sqrt{\frac{\text{elastic property}}{\text{inertial property}}}\]
The elastic property is what tries to restore the medium; the inertial property is what resists changes in motion.
14.5 Speed of a Transverse Wave on a Stretched String
For a string under tension T with linear mass density μ (mass per unit length, kg/m), the transverse wave speed is:
Wave speed on a string:
\[v = \sqrt{\frac{T}{\mu}}\]
where T = tension (N) and μ = mass per unit length (kg/m).
The elastic property is the tension T (a higher T snaps the string back faster); the inertial property is μ (heavier strings respond slower).
14.6 Speed of a Longitudinal Wave (Sound)
Sound is a longitudinal pressure wave. In any medium of bulk modulus B and density ρ:
\[v = \sqrt{\frac{B}{\rho}}\]
For solids, the relevant elastic modulus is Young's modulus Y for thin rods:
\[v_{\text{solid rod}} = \sqrt{\frac{Y}{\rho}}\]
| Medium | Speed of sound (m/s) |
|---|---|
| Air (0°C) | 331 |
| Air (20°C) | 343 |
| Helium (0°C) | 972 |
| Hydrogen (0°C) | 1284 |
| Water (20°C) | 1482 |
| Iron | 5130 |
| Diamond | 12,000 |
14.6.1 Newton's Formula
Newton (1687) assumed that sound propagation is an isothermal process — heat exchange keeps temperature constant. For an ideal gas under isothermal conditions, the bulk modulus equals the pressure: B_iso = P. So:
\[v_{\text{Newton}} = \sqrt{\frac{P}{\rho}}\]
For air at STP (P = 1.013×10⁵ Pa, ρ = 1.29 kg/m³): v_Newton ≈ 280 m/s. But experiments showed v_exp ≈ 332 m/s — a 16% discrepancy. Something was wrong.
14.6.2 Laplace's Correction
In 1816 Laplace pointed out that sound oscillations are too rapid for heat to flow in and out — the process is actually adiabatic, not isothermal. For an adiabatic process in an ideal gas: PV^γ = constant, giving B_adi = γP where γ = C_p/C_v is the ratio of specific heats.
Laplace's corrected formula:
\[v = \sqrt{\frac{\gamma P}{\rho}}\]
For air γ = 1.40, so v = √(1.40 × 1.013×10⁵ / 1.29) ≈ 332 m/s, matching experiment.
14.6.3 Effect of Temperature
Using ideal gas law PV = nRT and ρ = nM/V:
\[v = \sqrt{\frac{\gamma RT}{M}}\]
So v ∝ √T (in kelvin). Empirically, sound speed in air increases by ~0.6 m/s for every 1°C rise. At 20°C → 343 m/s; at 30°C → 349 m/s.
14.6.4 Effect of Humidity and Pressure
Sound speed is independent of pressure for an ideal gas at constant temperature (because P/ρ stays constant). It does depend on humidity: water vapour is lighter than dry air, so humid air has lower ρ → faster sound. This is why monsoon afternoons sound "alive" with slightly faster echoes.
Interactive Simulation: String Wave Speed Calculator
Adjust the tension and linear density to see how the wave speed changes on a string.
v = 100.0 m/s
Worked Example 1: Sitar string speed
A sitar string of mass 0.5 g/cm is under tension 200 N. Find the speed of transverse waves on it.
μ = 0.5 g/cm = 0.05 kg/m
\[v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{200}{0.05}} = \sqrt{4000} \approx \boxed{63.2\,\text{m/s}}\]
Worked Example 2: Speed of sound in air at 30°C
Speed of sound at 0°C is 332 m/s. Find the speed at 30°C.
v ∝ √T:
\[\frac{v_{30}}{v_0} = \sqrt{\frac{T_{30}}{T_0}} = \sqrt{\frac{303}{273}}\]
\[v_{30} = 332\times 1.054 \approx \boxed{350\,\text{m/s}}\]
About 18 m/s faster — noticeable for ultrasound medical imaging which depends on accurate timing.
Worked Example 3: Sound in helium vs air
Compare speed of sound in helium (M = 4 g/mol, γ = 1.67) to air (M = 29 g/mol, γ = 1.40) at the same temperature.
\[\frac{v_{He}}{v_{air}} = \sqrt{\frac{\gamma_{He} M_{air}}{\gamma_{air} M_{He}}} = \sqrt{\frac{1.67\times 29}{1.40\times 4}} = \sqrt{8.65}\approx \boxed{2.94}\]
Helium gives ~3× faster sound. Inhaling helium raises voice pitch because vocal-tract resonances scale with sound speed.
Activity 14.2 — Measuring Speed of Sound (Resonance Tube)L4 Analyse
Materials: Tuning fork (known frequency), tall glass cylinder, water reservoir at adjustable height.
- Strike the tuning fork and hold above the cylinder.
- Raise/lower water level until the sound becomes loudest (first resonance).
- Note water-level height. Continue raising until next loudest (second resonance).
- Difference between the two heights ≈ λ/2.
- Calculate v = νλ.
Predict: What value should the experiment yield in your classroom (~25°C)?
Around 346 m/s for room temperature. The standard set-up gets within 1-2 m/s of textbook value. The first resonance occurs at length ≈ λ/4 (end correction adds ~0.6r for tube radius r). Subtracting end-correction effects, two successive resonances differ by exactly λ/2.
Competency-Based Questions
A sound engineer is designing acoustics for an open-air concert in Mumbai. She needs to predict how sound speed varies with temperature, humidity, and altitude. She also studies the strings on the violin section.
Q1. Speed of a transverse wave on a stretched string depends on:L1 Remember
Answer: (c). v = √(T/μ). Frequency depends on the source; the medium dictates the speed.
Q2. The actual speed of sound in air matches Newton's formula well at sub-audio frequencies (e.g., a slowly moving piston) but fails at audible frequencies. Explain why.L4 Analyse
At very low frequencies, compressions are slow enough for heat to equalize — the process is essentially isothermal, so Newton's v = √(P/ρ) holds. At audio frequencies, each compression-rarefaction cycle is too fast for heat conduction — the process is adiabatic, requiring Laplace's γ factor. This was the historic puzzle Laplace solved.
Q3. Fill in the blank: When temperature rises by 1°C, speed of sound in air increases by approximately ____ m/s.L1 Remember
~0.6 m/s. Derived from v ∝ √T: at 273 K, dv/dT ≈ v/(2T) ≈ 332/546 ≈ 0.6 m/s per K.
Q4. True/False: Sound travels faster in humid air than in dry air.L5 Evaluate
TRUE. Water vapour (M_H2O = 18) is lighter than air (M_air = 29). Humid air has lower density, so v = √(γP/ρ) increases. At 30°C with 100% humidity, sound is ~1-2 m/s faster than in dry air.
Q5. HOT: Design a low-cost method to estimate the bulk modulus of water using sound, given access to a swimming pool, two divers, a stopwatch, and an underwater hammer.L6 Create
Plan: Divers stand at known distance L (e.g., 50 m) along the pool length, fully submerged. Diver A strikes the hammer underwater. Diver B starts a stopwatch on hearing the click and stops it after a fixed return signal. Calculate v_water = L/t. With v ≈ 1500 m/s and ρ ≈ 1000 kg/m³, B = v²ρ = 1500² × 1000 ≈ 2.25×10⁹ Pa = 2.25 GPa (matches the textbook value of ~2.2 GPa). Improvements: use synchronised waterproof microphones with electronic timing instead of human reaction.
Assertion–Reason Questions
(A) Both true, R explains A. (B) Both true, R does NOT explain A. (C) A true, R false. (D) A false, R true.
A: Newton's formula for the speed of sound in air gives a value lower than the experimental one.
R: Newton assumed sound propagation in air is an adiabatic process.
(C). A is TRUE. R is FALSE — Newton assumed isothermal. Laplace later corrected this with the adiabatic assumption.
A: Speed of sound in a gas is independent of pressure at constant temperature.
R: Both P and ρ change together for an ideal gas under isothermal pressure change, keeping P/ρ constant.
(A). Both true; R explains A. Doubling P doubles ρ, so v = √(γP/ρ) is unchanged.
A: Tightening a guitar string raises its pitch.
R: Wave speed v = √(T/μ) increases with tension, and higher v on a fixed-length string gives higher fundamental frequency.
(A). Both true and R explains A perfectly. ν = v/(2L) for fixed L.
Frequently Asked Questions - Speed Waves String Gas
What is the main concept covered in Speed Waves String Gas?
In NCERT Class 11 Physics Chapter 14 (Waves), "Speed Waves String Gas" covers core principles and equations needed for board exam success. The MyAiSchool lesson explains the topic with definitions, derivations, worked examples, and interactive simulations. Key formulas and dimensional analysis are included to build conceptual depth and problem-solving skills aligned with the CBSE 2025-26 syllabus.
How is Speed Waves String Gas useful in real-life applications?
Real-life applications of Speed Waves String Gas from NCERT Class 11 Physics Chapter 14 include engineering design, satellite mechanics, sports biomechanics, transportation safety, and electrical/electronic devices. The MyAiSchool lesson links every concept to a tangible example so students see physics as a problem-solving framework for the physical world, not as abstract formulas.
What are the key formulas in Speed Waves String Gas?
Key formulas in Speed Waves String Gas (NCERT Class 11 Physics Chapter 14 Waves) are derived step-by-step in the MyAiSchool lesson. Students should memorize the final formula AND understand its derivation for full board marks. Each formula is listed with its dimensional formula, SI unit, applicability range, and common pitfalls. The Summary section at the end of each part includes a quick-reference formula card.
How does this part connect to other parts of Chapter 14?
NCERT Class 11 Physics Chapter 14 (Waves) is structured so each part builds on the previous one. Speed Waves String Gas connects directly to neighbouring parts via shared definitions, units, and methodology. The MyAiSchool lesson cross-references related concepts with internal links so students can navigate the whole chapter as one connected story rather than disconnected fragments.
What types of CBSE board questions come from Speed Waves String Gas?
CBSE board questions from Speed Waves String Gas typically include: (1) 1-mark MCQs on definitions and formulas, (2) 2-mark short-answer derivations or applications, (3) 3-mark numerical problems with units, (4) 5-mark long-answer derivations followed by application. The MyAiSchool lesson tags each Competency-Based Question (CBQ) with Bloom level (L1-L6) so students know how to study for each weight.
How can students use the interactive simulation effectively?
The interactive simulation in the Speed Waves String Gas lesson allows students to adjust input parameters (sliders or selectors) and see physical quantities update in real time. To use it effectively: (1) try extreme values to understand limiting cases, (2) compare with the analytical formula, (3) check unit consistency, (4) test special configurations from worked examples. The simulation reinforces conceptual intuition that pure formula manipulation cannot.
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