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Waves Formula Sheet and PYQs

Complete Class 11 Physics Waves Formula Sheet, NCERT Examples, NCERT Exercises, PYQs, Numericals and Quick Revision Notes.

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Map

Formula Families Covered

Wave Formulae

Progressive waves, phase, velocity, acceleration, superposition, beats and standing waves.

Sound Formulae

Speed of sound, Laplace correction, intensity, loudness, decibel level and echo.

Organ Pipe Formulae

Open pipe, closed pipe, harmonics, end correction, displacement and pressure nodes.

Exam Bank

NCERT-style solved examples, exercises, 70+ PYQ-style questions and revision notes.

Complete Formula Sheet

All Important Waves Formulae

Wave Speedv = fλ

Symbols: v: wave speed, f: frequency, λ: wavelength

Units: m s^-1, Hz, m

Exam Tip: Use for periodic waves in a uniform medium.

Common Mistake: Do not use particle speed in place of wave speed.

Frequency and Periodf = 1/T

Symbols: f: frequency, T: time period

Units: Hz, s

Exam Tip: One oscillation per second means 1 Hz.

Common Mistake: Do not confuse T with total time for many oscillations.

Angular Frequencyω = 2πf = 2π/T

Symbols: ω: angular frequency

Units: rad s^-1

Exam Tip: Coefficient of t in y = A sin(ωt - kx).

Common Mistake: Do not write ω = 2πT.

Wave Numberk = 2π/λ

Symbols: k: angular wave number

Units: rad m^-1

Exam Tip: Coefficient of x in wave equation.

Common Mistake: Do not confuse k with spring constant.

Wave Equation+x direction: y = A sin(ωt - kx)

Symbols: A: amplitude, ω: angular frequency, k: wave number

Units: m, rad s^-1, rad m^-1

Exam Tip: Minus before kx means propagation in +x direction.

Common Mistake: Do not decide direction from the sign of t alone.

Wave Equation Alternatey = A sin[(2π/λ)(vt - x)]

Symbols: v: wave velocity, x: position, t: time

Units: m, m s^-1, s

Exam Tip: Form vt - x represents +x direction.

Common Mistake: x - vt also represents +x direction with opposite phase convention.

Particle Velocityv_p = dy/dt = Aω cos(ωt - kx)

Symbols: v_p: transverse particle velocity

Units: m s^-1

Exam Tip: Differentiate displacement with respect to time.

Common Mistake: Do not call this wave velocity.

Maximum Particle Velocityv_p,max = Aω

Symbols: A: amplitude, ω: angular frequency

Units: m s^-1

Exam Tip: Maximum value of cosine is 1.

Common Mistake: Use A in metre, not cm, unless answer is in cm/s.

Particle Accelerationa = d²y/dt² = -ω²y

Symbols: a: transverse acceleration of particle

Units: m s^-2

Exam Tip: SHM relation applies to each medium particle.

Common Mistake: Negative sign means acceleration is towards mean position.

Wave Slopedy/dx = -Ak cos(ωt - kx)

Symbols: dy/dx: slope of string/wave profile

Units: dimensionless

Exam Tip: Differentiate with respect to x at fixed time.

Common Mistake: Do not mix slope with velocity.

Phaseφ = ωt - kx + φ₀

Symbols: φ: phase, φ₀: initial phase

Units: radian

Exam Tip: Same phase points differ by 2πn.

Common Mistake: Phase is not displacement.

Phase DifferenceΔφ = 2πΔx/λ = 2πΔt/T

Symbols: Δx: path difference, Δt: time difference

Units: radian

Exam Tip: Use distance form or time form as per data.

Common Mistake: Do not mix path difference and time difference units.

Path DifferenceΔx = (λ/2π)Δφ

Symbols: Δx: path difference

Units: m

Exam Tip: Useful in interference and phase questions.

Common Mistake: Use radians for phase difference.

Constructive InterferenceΔx = nλ, Δφ = 2nπ

Symbols: n: integer

Units: m, rad

Exam Tip: Waves arrive in phase.

Common Mistake: Do not use half wavelength for maxima.

Destructive InterferenceΔx = (2n+1)λ/2

Symbols: n: integer

Units: m

Exam Tip: Waves arrive in opposite phase.

Common Mistake: This is for equal-amplitude complete cancellation.

Resultant AmplitudeR = √(A₁² + A₂² + 2A₁A₂cosφ)

Symbols: A₁, A₂: amplitudes, φ: phase difference

Units: m

Exam Tip: Use when two SHM/waves of same frequency superpose.

Common Mistake: Do not add amplitudes directly unless phase is zero.

Equal Amplitude ResultantR = 2A cos(φ/2)

Symbols: A: each amplitude, φ: phase difference

Units: m

Exam Tip: Fast way for two equal-amplitude waves.

Common Mistake: Use magnitude if cosine is negative.

Standing Wavey = 2A sin kx cos ωt

Symbols: nodes from sin kx = 0

Units: m

Exam Tip: Standing waves do not transport energy along the medium.

Common Mistake: Do not call antinode spacing one wavelength.

Node Spacingdistance between adjacent nodes = λ/2

Symbols: λ: wavelength

Units: m

Exam Tip: Same for adjacent antinodes.

Common Mistake: Node to nearest antinode is λ/4.

String Fundamentalf₁ = v/2L

Symbols: L: string length fixed at both ends

Units: Hz

Exam Tip: Same pattern as open pipe displacement condition.

Common Mistake: Use L as vibrating length.

String Harmonicsf_n = nv/2L

Symbols: n = 1, 2, 3, ...

Units: Hz

Exam Tip: All harmonics are allowed.

Common Mistake: Closed pipe has only odd harmonics.

Wave Speed on Stringv = √(T/μ)

Symbols: T: tension, μ: mass per unit length

Units: m s^-1

Exam Tip: Higher tension increases speed.

Common Mistake: Do not use total mass unless divided by length.

Power on StringP_avg = (1/2)μω²A²v

Symbols: μ: linear density, A: amplitude

Units: W

Exam Tip: Shows power proportional to A².

Common Mistake: Amplitude doubling makes power four times.

Beat Frequencyf_b = |f₁ - f₂|

Symbols: f₁, f₂: close frequencies

Units: Hz

Exam Tip: Number of loud beats per second.

Common Mistake: Unknown frequency may be above or below standard.

Sound Formulae

Sound Waves, Intensity, Loudness and Laplace Correction

Speed of Sound in Gasv = √(γP/ρ)

Symbols: γ: ratio of specific heats, P: pressure, ρ: density

Units: m s^-1

Exam Tip: Laplace correction uses adiabatic bulk modulus.

Common Mistake: Do not use isothermal formula for sound in air.

Elastic Wave Speedv = √(E/ρ)

Symbols: E: elastic modulus, ρ: density

Units: m s^-1

Exam Tip: General elastic wave speed form.

Common Mistake: Choose correct modulus for medium.

Newton Formulav = √(P/ρ)

Symbols: P: pressure, ρ: density

Units: m s^-1

Exam Tip: Historical formula before Laplace correction.

Common Mistake: It underestimates speed in air.

Laplace Correctionv_Laplace = √(γ) v_Newton

Symbols: γ: C_p/C_v

Units: m s^-1

Exam Tip: Sound compression/rarefaction is nearly adiabatic.

Common Mistake: Do not take γ = 1 for air.

Temperature Formulav = v₀ + 0.61T

Symbols: T: temperature in degree Celsius

Units: m s^-1

Exam Tip: Approximate speed in air near room temperature.

Common Mistake: Do not put Kelvin in this linear Celsius formula.

Kelvin Dependencev ∝ √T_K

Symbols: T_K: absolute temperature

Units: m s^-1

Exam Tip: Use for ratio problems.

Common Mistake: Temperature must be in Kelvin.

IntensityI = P/A

Symbols: P: power, A: area

Units: W m^-2

Exam Tip: Intensity is power per unit area.

Common Mistake: Do not confuse intensity with loudness.

Spherical Wave IntensityI = P/(4πr²)

Symbols: r: distance from point source

Units: W m^-2

Exam Tip: Intensity follows inverse square law.

Common Mistake: Doubling distance makes intensity one-fourth.

Sound Levelβ = 10 log₁₀(I/I₀)

Symbols: I₀ = 10^-12 W m^-2

Units: dB

Exam Tip: Use base-10 logarithm.

Common Mistake: 10 dB increase means intensity becomes 10 times.

Intensity Ratioβ₂ - β₁ = 10 log₁₀(I₂/I₁)

Symbols: I₁, I₂: intensities

Units: dB

Exam Tip: Good for comparing two sound levels.

Common Mistake: Do not subtract intensities directly.

Loudness IdeaLoudness depends on intensity and ear response

Symbols: I: objective intensity; loudness: subjective sensation

Units: phon/sone in acoustics

Exam Tip: For school exams, intensity is physical quantity.

Common Mistake: Loudness is not measured in W m^-2.

Pressure Amplitude RelationI ∝ p₀²

Symbols: p₀: pressure amplitude

Units: W m^-2

Exam Tip: If pressure amplitude doubles, intensity becomes four times.

Common Mistake: Do not use direct proportionality with p₀.

Mach NumberM = u/v

Symbols: u: source speed, v: sound speed

Units: dimensionless

Exam Tip: M > 1 means supersonic.

Common Mistake: Classical Doppler formula fails at v_s approaching v.

Echo Distanced = vt/2

Symbols: t: echo time, v: sound speed

Units: m

Exam Tip: Sound travels to obstacle and back.

Common Mistake: Do not forget factor 2.

Organ Pipe Formulae

Open Pipe, Closed Pipe, Nodes and Antinodes

Open Pipe Fundamentalf₁ = v/2L

Symbols: L: pipe length, v: sound speed

Units: Hz

Exam Tip: Both ends are displacement antinodes.

Common Mistake: End correction may be needed in precision problems.

Open Pipe Harmonicsf_n = nv/2L

Symbols: n = 1, 2, 3, ...

Units: Hz

Exam Tip: All harmonics are present.

Common Mistake: Do not omit even harmonics in open pipe.

Open Pipe Wavelengthλ_n = 2L/n

Symbols: n: harmonic number

Units: m

Exam Tip: For n loops in open pipe.

Common Mistake: Use effective length if end correction is given.

Closed Pipe Fundamentalf₁ = v/4L

Symbols: Closed end: displacement node; open end: displacement antinode

Units: Hz

Exam Tip: Lowest mode has quarter wavelength in pipe.

Common Mistake: Closed pipe fundamental is half of open pipe same length.

Closed Pipe Odd Harmonicsf_n = nv/4L, n = 1, 3, 5, ...

Symbols: n: odd harmonic number

Units: Hz

Exam Tip: Only odd harmonics are allowed.

Common Mistake: Do not include n = 2, 4, 6 for closed pipe.

Closed Pipe Wavelengthλ_n = 4L/n, n odd

Symbols: L: pipe length

Units: m

Exam Tip: Use odd n only.

Common Mistake: Third harmonic is 3v/4L, not 2v/4L.

Pressure NodesOpen end: pressure node

Symbols: Pressure variation is minimum at open end

Units: --

Exam Tip: Pressure node corresponds to displacement antinode.

Common Mistake: Do not confuse pressure and displacement patterns.

Pressure AntinodeClosed end: pressure antinode

Symbols: Pressure variation is maximum at closed end

Units: --

Exam Tip: Closed end has displacement node.

Common Mistake: Pressure and displacement nodes are interchanged.

End CorrectionL_eff = L + e for one open end; L + 2e for two open ends

Symbols: e: end correction

Units: m

Exam Tip: Use effective length in organ pipe formulas.

Common Mistake: Do not add end correction at closed end.

Resonance TubeL + e = λ/4 for first resonance

Symbols: L: air column length

Units: m

Exam Tip: Second resonance in closed pipe differs by λ/2.

Common Mistake: Difference of successive resonances gives λ/2.

Doppler Formulae

Observer Motion, Source Motion, Beats and Sign Convention

General Dopplerf' = f(v ± v₀)/(v ∓ v_s)

Symbols: v: sound speed, v₀: observer speed, v_s: source speed

Units: Hz

Exam Tip: Choose signs so approach increases f' and separation decreases f'.

Common Mistake: Do not memorize signs without drawing arrows.

Observer Towards Sourcef' = f(v + v₀)/v

Symbols: stationary source, moving observer

Units: Hz

Exam Tip: Observer meets more wavefronts per second.

Common Mistake: Observer speed belongs in numerator.

Observer Away From Sourcef' = f(v - v₀)/v

Symbols: stationary source, observer receding

Units: Hz

Exam Tip: Receiving rate decreases.

Common Mistake: Do not put v₀ in denominator.

Source Towards Observerf' = fv/(v - v_s)

Symbols: moving source, stationary observer

Units: Hz

Exam Tip: Wavelength in front becomes smaller.

Common Mistake: Approaching source uses minus in denominator.

Source Away From Observerf' = fv/(v + v_s)

Symbols: source receding from stationary observer

Units: Hz

Exam Tip: Wavelength reaching observer is longer.

Common Mistake: Receding source uses plus in denominator.

Both Approachf' = f(v + v₀)/(v - v_s)

Symbols: source and observer move towards each other

Units: Hz

Exam Tip: Frequency increases strongly.

Common Mistake: Do not use relative velocity alone in denominator.

Both Recedef' = f(v - v₀)/(v + v_s)

Symbols: source and observer separate

Units: Hz

Exam Tip: Frequency decreases strongly.

Common Mistake: Check whether separation really increases.

Beat Frequencyf_b = |f₁ - f₂|

Symbols: nearby frequencies

Units: Hz

Exam Tip: Used in tuning instruments.

Common Mistake: Beat frequency is not average frequency.

Reflected Doppler ApproximationΔf ≈ 2uf/v

Symbols: u: reflector speed, f: emitted frequency

Units: Hz

Exam Tip: Radar/ultrasound reflection gives double shift.

Common Mistake: Use wave speed of the relevant wave.

Sign RuleApproach: f' increases; separation: f' decreases

Symbols: Draw S and O arrows first

Units: --

Exam Tip: This rule prevents most sign errors.

Common Mistake: Do not rely only on plus/minus symbols.

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NCERT Examples

Important NCERT Class 11 Solved Examples

Example 1Wave Speed

A wave has frequency 50 Hz and wavelength 0.8 m. Find wave speed.

Show Solution

Given/Idea: f=50 Hz, λ=0.8 m

Solution: v=fλ=50×0.8=40 m s^-1

Final Answer: 40 m s^-1

Example 2Wave Equation

For y=0.02 sin(100πt - 4πx), find amplitude, angular frequency and wave number.

Show Solution

Given/Idea: A=0.02 m, coefficient of t=100π, coefficient of x=4π

Solution: ω=100π rad s^-1, k=4π rad m^-1

Final Answer: A=0.02 m, ω=100π, k=4π

Example 3Wave Velocity from Equation

For y=5 sin(20t - 4x), find wave velocity.

Show Solution

Given/Idea: ω=20, k=4

Solution: v=ω/k=20/4=5 m s^-1

Final Answer: 5 m s^-1

Example 4Particle Velocity

A wave has A=0.01 m and ω=200 rad s^-1. Find maximum particle speed.

Show Solution

Given/Idea: A=0.01 m, ω=200

Solution: v_p,max=Aω=0.01×200=2 m s^-1

Final Answer: 2 m s^-1

Example 5Phase Difference

Two points are separated by λ/6. Find phase difference.

Show Solution

Given/Idea: Δx=λ/6

Solution: Δφ=2πΔx/λ=2π/6=π/3

Final Answer: π/3 rad

Example 6String Speed

A string has tension 100 N and linear density 0.01 kg m^-1. Find wave speed.

Show Solution

Given/Idea: T=100 N, μ=0.01 kg m^-1

Solution: v=√(T/μ)=√(100/0.01)=100 m s^-1

Final Answer: 100 m s^-1

Example 7Standing Wave

A string of length 1 m has speed 200 m/s. Find fundamental frequency.

Show Solution

Given/Idea: L=1 m, v=200 m/s

Solution: f₁=v/2L=200/2=100 Hz

Final Answer: 100 Hz

Example 8Sound in Air

Find speed of sound at 20 degree Celsius using v=331+0.61T.

Show Solution

Given/Idea: T=20

Solution: v=331+0.61×20=343.2 m/s

Final Answer: 343.2 m/s

Example 9Intensity Level

If I=10^-8 W m^-2, find sound level.

Show Solution

Given/Idea: I=10^-8, I₀=10^-12

Solution: β=10log(I/I₀)=10log(10⁴)=40 dB

Final Answer: 40 dB

Example 10Open Pipe

An open pipe of length 0.5 m has sound speed 340 m/s. Find fundamental frequency.

Show Solution

Given/Idea: L=0.5 m, v=340 m/s

Solution: f₁=v/2L=340/1=340 Hz

Final Answer: 340 Hz

Example 11Closed Pipe

A closed pipe of length 0.25 m has v=340 m/s. Find first resonant frequency.

Show Solution

Given/Idea: L=0.25 m

Solution: f₁=v/4L=340/1=340 Hz

Final Answer: 340 Hz

Example 12Closed Pipe Harmonic

Find third harmonic of a closed pipe of length 0.5 m if v=340 m/s.

Show Solution

Given/Idea: n=3, L=0.5 m

Solution: f₃=3v/4L=3×340/2=510 Hz

Final Answer: 510 Hz

Example 13Beats

Two tuning forks 256 Hz and 260 Hz are sounded together. Find beats per second.

Show Solution

Given/Idea: f₁=256, f₂=260

Solution: f_b=|256-260|=4 Hz

Final Answer: 4 beats/s

Example 14Beat Ambiguity

A 512 Hz fork gives 5 beats/s with another fork. Find possible frequencies.

Show Solution

Given/Idea: standard=512 Hz, beat=5 Hz

Solution: unknown=512±5

Final Answer: 507 Hz or 517 Hz

Example 15Doppler Source Approaches

A 500 Hz source moves towards stationary observer at 34 m/s. v=340 m/s. Find apparent frequency.

Show Solution

Given/Idea: f=500, v=340, v_s=34

Solution: f'=fv/(v-v_s)=500×340/306=555.6 Hz

Final Answer: 555.6 Hz

Example 16Doppler Observer Approaches

Observer moves towards 600 Hz stationary source at 20 m/s. v=340 m/s. Find f'.

Show Solution

Given/Idea: f=600, v₀=20

Solution: f'=f(v+v₀)/v=600×360/340=635.3 Hz

Final Answer: 635.3 Hz

Example 17Echo

An echo is heard after 2 s. Speed of sound is 340 m/s. Find distance of wall.

Show Solution

Given/Idea: t=2 s, v=340 m/s

Solution: d=vt/2=340×2/2=340 m

Final Answer: 340 m

Example 18Resonance Tube

Two successive resonances in a closed pipe differ by 17 cm. Find wavelength.

Show Solution

Given/Idea: L₂-L₁=17 cm

Solution: Difference=λ/2, so λ=34 cm

Final Answer: 0.34 m

NCERT Exercises

Exercise-style Questions With Solutions

Exercise 1Wave Direction

Identify direction of y=A sin(ωt-kx).

Show Solution

Given/Idea: Phase constant gives ωt-kx=constant. As t increases, x increases.

Solution: Wave travels in +x direction.

Final Answer: undefined

Exercise 2Negative Direction

Identify direction of y=A sin(ωt+kx).

Show Solution

Given/Idea: ωt+kx=constant. As t increases, x decreases.

Solution: Wave travels in -x direction.

Final Answer: undefined

Exercise 3Find Wavelength

In y=0.01 sin(50t-2x), find wavelength.

Show Solution

Given/Idea: k=2 rad/m, λ=2π/k=2π/2=π m.

Solution: π m

Final Answer: undefined

Exercise 4Find Frequency

In y=0.01 sin(100πt-5x), find frequency.

Show Solution

Given/Idea: ω=100π, f=ω/2π=50 Hz.

Solution: 50 Hz

Final Answer: undefined

Exercise 5Acceleration

If y=0.02 m and ω=10 rad/s at an instant, find acceleration.

Show Solution

Given/Idea: a=-ω²y=-100×0.02=-2 m/s².

Solution: -2 m/s²

Final Answer: undefined

Exercise 6Intensity Ratio

Sound level increases by 20 dB. Find intensity ratio.

Show Solution

Given/Idea: 20=10log(I₂/I₁), ratio=10².

Solution: 100

Final Answer: undefined

Exercise 7Temperature Ratio

Speed of sound ratio at 300 K and 1200 K.

Show Solution

Given/Idea: v ∝ √T, ratio=√(1200/300)=2.

Solution: 2

Final Answer: undefined

Exercise 8Open Pipe Second Harmonic

Open pipe has f₁=170 Hz. Find f₂.

Show Solution

Given/Idea: Open pipe has all harmonics, f₂=2f₁.

Solution: 340 Hz

Final Answer: undefined

Exercise 9Closed Pipe Next Frequency

Closed pipe fundamental is 100 Hz. Find next allowed frequency.

Show Solution

Given/Idea: Closed pipe allowed harmonics are 1,3,5... next is 3f₁.

Solution: 300 Hz

Final Answer: undefined

Exercise 10Beat Tuning

A fork produces 6 beats with standard. Filing increases fork frequency and beats decrease. Was original fork above or below standard?

Show Solution

Given/Idea: Filing increases frequency. If beats decrease, fork was below standard.

Solution: Below standard.

Final Answer: undefined

PYQ Bank

CBSE, NEET, JEE, IB, IGCSE and A-Level Questions

Question 1CBSE

Define wavelength and frequency of a progressive wave.

Show Answer

Wavelength is distance between consecutive same-phase particles; frequency is number of oscillations per second.

Question 2CBSE

Write relation between wave velocity, frequency and wavelength.

Show Answer

v=fλ.

Question 3CBSE

What is the SI unit of wave number?

Show Answer

rad m^-1 or m^-1 when radian is treated dimensionless.

Question 4CBSE

For y=A sin(ωt-kx), what is wave speed?

Show Answer

v=ω/k.

Question 5CBSE

Differentiate between particle velocity and wave velocity.

Show Answer

Particle velocity is dy/dt of medium particle; wave velocity is speed of disturbance, v=fλ.

Question 6CBSE

What is beat frequency?

Show Answer

f_b=|f₁-f₂|.

Question 7CBSE

Why is sound speed in gas given by Laplace correction?

Show Answer

Sound propagation is adiabatic, not isothermal.

Question 8CBSE

What is the distance between adjacent nodes in a standing wave?

Show Answer

λ/2.

Question 9CBSE

Why are only odd harmonics present in closed pipe?

Show Answer

Boundary conditions require displacement node at closed end and antinode at open end.

Question 10CBSE

What happens to apparent frequency when source approaches observer?

Show Answer

It increases.

Question 11NEET

A wave has f=100 Hz and λ=2 m. Find speed.

Show Answer

v=fλ=200 m/s.

Question 12NEET

A wave equation has ω=40 rad/s and k=5 rad/m. Find speed.

Show Answer

v=ω/k=8 m/s.

Question 13NEET

If amplitude doubles, maximum particle velocity becomes?

Show Answer

It doubles because v_p,max=Aω.

Question 14NEET

If intensity of sound becomes 100 times, sound level increases by?

Show Answer

20 dB.

Question 15NEET

At 27 degree Celsius approximate speed of sound in air is?

Show Answer

v=331+0.61×27=347.5 m/s approximately.

Question 16NEET

Open pipe length is L. Fundamental frequency is?

Show Answer

v/2L.

Question 17NEET

Closed pipe length is L. Fundamental frequency is?

Show Answer

v/4L.

Question 18NEET

Two forks 300 Hz and 305 Hz are sounded. Beat frequency?

Show Answer

5 Hz.

Question 19NEET

Distance between node and nearest antinode is?

Show Answer

λ/4.

Question 20NEET

Doppler Effect is due to change in what?

Show Answer

Apparent frequency due to relative motion.

Question 21JEE Main

For y=2 sin(10t-5x), find wave speed.

Show Answer

v=ω/k=10/5=2 units/s.

Question 22JEE Main

For y=A sin(100πt-2πx), find frequency and wavelength.

Show Answer

f=50 Hz, λ=1 m.

Question 23JEE Main

Find maximum transverse velocity for A=2 cm, f=5 Hz.

Show Answer

ω=10π, v_max=0.02×10π=0.2π m/s.

Question 24JEE Main

Two waves of equal amplitude A and phase difference 120 degree superpose. Resultant amplitude?

Show Answer

R=2A cos60 degree=A.

Question 25JEE Main

If two equal waves interfere destructively, phase difference is?

Show Answer

(2n+1)π.

Question 26JEE Main

String speed depends on tension how?

Show Answer

v ∝ √T.

Question 27JEE Main

If string tension is made four times, frequency becomes?

Show Answer

Twice, for same length and mode.

Question 28JEE Main

Open pipe third harmonic if fundamental is 200 Hz?

Show Answer

600 Hz.

Question 29JEE Main

Closed pipe first overtone if fundamental is 120 Hz?

Show Answer

Third harmonic = 360 Hz.

Question 30JEE Main

Source approaching stationary observer: formula?

Show Answer

f'=fv/(v-v_s).

Question 31JEE Advanced

A source and observer move towards each other. How should signs be chosen?

Show Answer

Use larger numerator and smaller denominator: f'=f(v+v₀)/(v-v_s).

Question 32JEE Advanced

For a standing wave y=2A sin kx cos ωt, where are nodes?

Show Answer

sin kx=0, so x=nπ/k=nλ/2.

Question 33JEE Advanced

Power carried by a sinusoidal wave on string depends on amplitude how?

Show Answer

P ∝ A².

Question 34JEE Advanced

If pressure amplitude doubles in sound wave, intensity becomes?

Show Answer

Four times.

Question 35JEE Advanced

Two close frequencies have average 500 Hz and beat frequency 6 Hz. Find frequencies.

Show Answer

497 Hz and 503 Hz.

Question 36JEE Advanced

A reflected Doppler shift is approximately why doubled?

Show Answer

Wave is shifted on incidence and reflection from moving target.

Question 37JEE Advanced

If radial relative velocity is zero, classical first-order Doppler shift?

Show Answer

Zero.

Question 38JEE Advanced

Why source velocity appears in denominator in Doppler formula?

Show Answer

Moving source changes wavelength.

Question 39JEE Advanced

Why observer velocity appears in numerator?

Show Answer

Moving observer changes rate of receiving wavefronts.

Question 40JEE Advanced

Closed pipe cannot have which harmonics?

Show Answer

Even harmonics.

Question 41IB

State the relation between period and frequency.

Show Answer

f=1/T.

Question 42IB

What is phase difference between two points separated by half wavelength?

Show Answer

π rad.

Question 43IB

What is a node?

Show Answer

A point of zero displacement in a standing wave.

Question 44IB

What is an antinode?

Show Answer

A point of maximum displacement in a standing wave.

Question 45IB

How does sound intensity vary with distance from point source?

Show Answer

I ∝ 1/r².

Question 46IB

Explain beats in one sentence.

Show Answer

Periodic loudness variation due to superposition of two close frequencies.

Question 47IB

What does a 10 dB increase mean for intensity?

Show Answer

Intensity becomes 10 times.

Question 48IGCSE

Sound is longitudinal or transverse in air?

Show Answer

Longitudinal.

Question 49IGCSE

Compression and rarefaction are related to what wave?

Show Answer

Longitudinal sound wave.

Question 50IGCSE

Can sound travel in vacuum?

Show Answer

No.

Question 51IGCSE

What changes when pitch increases?

Show Answer

Frequency increases.

Question 52IGCSE

What changes when loudness increases physically?

Show Answer

Intensity/amplitude increases.

Question 53IGCSE

Echo formula for distance?

Show Answer

d=vt/2.

Question 54A-Level

Write speed of transverse wave on a stretched string.

Show Answer

v=√(T/μ).

Question 55A-Level

State formula for sound intensity level.

Show Answer

β=10log₁₀(I/I₀).

Question 56A-Level

Why is temperature in Kelvin required in v ∝ √T?

Show Answer

Because proportionality uses absolute thermodynamic temperature.

Question 57A-Level

State the general two-source resultant amplitude formula.

Show Answer

R=√(A₁²+A₂²+2A₁A₂cosφ).

Question 58A-Level

What is end correction?

Show Answer

Small extra effective length added at open end of pipe.

Question 59A-Level

A closed pipe has first resonance at L. Next resonance occurs at?

Show Answer

L+λ/2, so difference is λ/2.

Question 60Assertion-Reason

Assertion: Wave speed equals particle speed. Reason: Particles move with the wave.

Show Answer

Both assertion and reason are false for progressive waves in a medium.

Question 61Assertion-Reason

Assertion: In an open pipe both ends are displacement antinodes. Reason: Air is free to vibrate at open ends.

Show Answer

Both true and reason explains assertion.

Question 62Assertion-Reason

Assertion: Closed pipe has odd harmonics only. Reason: One end is node and the other is antinode.

Show Answer

Both true and reason explains assertion.

Question 63Assertion-Reason

Assertion: Beat frequency is zero when two frequencies are equal. Reason: f_b=|f₁-f₂|.

Show Answer

Both true and reason explains assertion.

Question 64Assertion-Reason

Assertion: Laplace correction gives greater sound speed than Newton's formula. Reason: Sound propagation is adiabatic.

Show Answer

Both true and reason explains assertion.

Question 65True/False

Wave number k=2πλ.

Show Answer

False. k=2π/λ.

Question 66True/False

Angular frequency ω=2πf.

Show Answer

True.

Question 67True/False

Adjacent nodes are separated by λ.

Show Answer

False. They are separated by λ/2.

Question 68True/False

In a closed pipe, second harmonic is absent.

Show Answer

True.

Question 69True/False

Doppler Effect changes the emitted frequency of source.

Show Answer

False. It changes observed frequency.

Question 70Conceptual

Why is y=A sin(ωt-kx) a wave travelling in +x direction?

Show Answer

For constant phase, x increases as t increases.

Question 71Conceptual

Why does intensity decrease with distance from a point source?

Show Answer

Same power spreads over larger spherical area.

Question 72Conceptual

Why are beats useful in tuning?

Show Answer

Beat frequency becomes smaller as frequencies become closer.

Question 73Conceptual

Why is pressure node at open end of organ pipe?

Show Answer

Pressure must nearly equal atmospheric pressure at open end.

Question 74Conceptual

Why is sound speed higher at higher temperature?

Show Answer

Molecules move faster and pressure-density response is faster.

Question 75Numerical

A 1 m open pipe has v=340 m/s. Find first three frequencies.

Show Answer

170 Hz, 340 Hz, 510 Hz.

Question 76Numerical

A 1 m closed pipe has v=340 m/s. Find first three allowed frequencies.

Show Answer

85 Hz, 255 Hz, 425 Hz.

Question 77Numerical

If sound level changes from 40 dB to 70 dB, intensity ratio?

Show Answer

10³.

Question 78Numerical

Two forks produce 8 beats/s. One is 256 Hz. Possible other frequencies?

Show Answer

248 Hz or 264 Hz.

Question 79Numerical

Wave has T=0.02 s and λ=4 m. Find speed.

Show Answer

f=50 Hz, v=200 m/s.

Question 80Case Study

A guitar string is tuned using a 440 Hz tuning fork. Beats reduce from 5/s to 1/s as tension is adjusted. What does this show?

Show Answer

The string frequency is approaching 440 Hz.

Question 81Case Study

A student sees y=0.03 sin(60t+5x). Which way does the wave travel?

Show Answer

Negative x direction.

Question 82Case Study

A resonance tube has consecutive resonances at 18 cm and 52 cm. Find wavelength.

Show Answer

Difference=34 cm=λ/2, so λ=68 cm.

Question 83Case Study

An ambulance approaches and then recedes. How does pitch change?

Show Answer

Higher during approach, lower during recession.

Question 84Case Study

A sound source power is constant. Listener moves from 2 m to 4 m. Intensity becomes?

Show Answer

One-fourth.

Revision Notes

Quick Revision Sheet

One-page Revision Sheet

v=fλ links speed, frequency and wavelength.
ω=2πf and k=2π/λ are read from wave equation.
For y=A sin(ωt-kx), wave travels in +x direction.
Particle velocity is dy/dt; wave velocity is ω/k.
Acceleration of particle is a=-ω²y.
Standing wave nodes are λ/2 apart.
Open pipe has all harmonics; closed pipe has odd harmonics only.

Most Important Formulas

v=fλ
v=√(T/μ) for string waves.
v=√(γP/ρ) for sound in gas.
I=P/A and β=10log(I/I₀).
Open pipe: f_n=nv/2L.
Closed pipe: f_n=nv/4L, n odd.
Doppler: f'=f(v±v₀)/(v∓v_s).

Most Important Concepts

Waves transfer energy without net transport of medium.
Phase decides position in oscillation.
Superposition means algebraic addition of displacements.
Beats need close frequencies.
Sound is longitudinal in air.
Pressure and displacement nodes are interchanged in organ pipes.
Doppler shift depends on radial relative motion.

Exam Tips

Read coefficients of t and x carefully in wave equation.
Convert cm to m before using Aω or wave speed formulas.
Use Kelvin in temperature ratio formula.
In closed pipe, first overtone is third harmonic.
In beats, unknown frequency can be above or below standard.
Draw arrows before Doppler formula.
For sound level, intensity ratio is logarithmic.

Most Common Mistakes

Writing k=2πλ instead of 2π/λ.
Confusing particle velocity with wave velocity.
Using even harmonics for closed pipe.
Forgetting end correction in resonance tube questions when given.
Using Celsius in v∝√T ratio problems.
Taking beat frequency as sum of frequencies.
Putting source speed in numerator in Doppler formula.

Last Day Revision Notes

Memorize formula families, not isolated formulas.
Direction: ωt-kx means +x, ωt+kx means -x.
Node-antinode distance is λ/4.
Sound intensity varies as amplitude squared.
A 10 dB rise means intensity becomes 10 times.
Open-open pipe behaves like string fixed at both ends for frequency pattern.
Approach increases Doppler frequency; separation decreases it.

Last Minute

Most Important Formulas in One View

Wave Speed: v = fλ
Frequency and Period: f = 1/T
Angular Frequency: ω = 2πf = 2π/T
Wave Number: k = 2π/λ
Wave Equation: +x direction: y = A sin(ωt - kx)
Wave Equation Alternate: y = A sin[(2π/λ)(vt - x)]
Particle Velocity: v_p = dy/dt = Aω cos(ωt - kx)
Maximum Particle Velocity: v_p,max = Aω
Particle Acceleration: a = d²y/dt² = -ω²y
Wave Slope: dy/dx = -Ak cos(ωt - kx)
Phase: φ = ωt - kx + φ₀
Phase Difference: Δφ = 2πΔx/λ = 2πΔt/T
Path Difference: Δx = (λ/2π)Δφ
Constructive Interference: Δx = nλ, Δφ = 2nπ
Destructive Interference: Δx = (2n+1)λ/2
Resultant Amplitude: R = √(A₁² + A₂² + 2A₁A₂cosφ)
Equal Amplitude Resultant: R = 2A cos(φ/2)
Standing Wave: y = 2A sin kx cos ωt
Node Spacing: distance between adjacent nodes = λ/2
String Fundamental: f₁ = v/2L
String Harmonics: f_n = nv/2L
Wave Speed on String: v = √(T/μ)
Power on String: P_avg = (1/2)μω²A²v
Beat Frequency: f_b = |f₁ - f₂|
Speed of Sound in Gas: v = √(γP/ρ)
Elastic Wave Speed: v = √(E/ρ)
Newton Formula: v = √(P/ρ)
Laplace Correction: v_Laplace = √(γ) v_Newton
Temperature Formula: v = v₀ + 0.61T
Kelvin Dependence: v ∝ √T_K
Intensity: I = P/A
Spherical Wave Intensity: I = P/(4πr²)
Sound Level: β = 10 log₁₀(I/I₀)
Intensity Ratio: β₂ - β₁ = 10 log₁₀(I₂/I₁)
Loudness Idea: Loudness depends on intensity and ear response
Pressure Amplitude Relation: I ∝ p₀²
Mach Number: M = u/v
Echo Distance: d = vt/2
Open Pipe Fundamental: f₁ = v/2L
Open Pipe Harmonics: f_n = nv/2L
Open Pipe Wavelength: λ_n = 2L/n
Closed Pipe Fundamental: f₁ = v/4L
Closed Pipe Odd Harmonics: f_n = nv/4L, n = 1, 3, 5, ...
Closed Pipe Wavelength: λ_n = 4L/n, n odd
Pressure Nodes: Open end: pressure node
Pressure Antinode: Closed end: pressure antinode
End Correction: L_eff = L + e for one open end; L + 2e for two open ends
Resonance Tube: L + e = λ/4 for first resonance
General Doppler: f' = f(v ± v₀)/(v ∓ v_s)
Observer Towards Source: f' = f(v + v₀)/v
Observer Away From Source: f' = f(v - v₀)/v
Source Towards Observer: f' = fv/(v - v_s)
Source Away From Observer: f' = fv/(v + v_s)
Both Approach: f' = f(v + v₀)/(v - v_s)
Both Recede: f' = f(v - v₀)/(v + v_s)
Beat Frequency: f_b = |f₁ - f₂|
Reflected Doppler Approximation: Δf ≈ 2uf/v
Sign Rule: Approach: f' increases; separation: f' decreases

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Waves Formula Sheet and PYQs

Formula Families Covered

All Important Waves Formulae

Sound Waves, Intensity, Loudness and Laplace Correction

Open Pipe, Closed Pipe, Nodes and Antinodes

Observer Motion, Source Motion, Beats and Sign Convention

Need Personal Help in Waves?

Important NCERT Class 11 Solved Examples

A wave has frequency 50 Hz and wavelength 0.8 m. Find wave speed.

For y=0.02 sin(100πt - 4πx), find amplitude, angular frequency and wave number.

For y=5 sin(20t - 4x), find wave velocity.

A wave has A=0.01 m and ω=200 rad s-1. Find maximum particle speed.

Two points are separated by λ/6. Find phase difference.

A string has tension 100 N and linear density 0.01 kg m-1. Find wave speed.

A string of length 1 m has speed 200 m/s. Find fundamental frequency.

Find speed of sound at 20 degree Celsius using v=331+0.61T.

If I=10-8 W m-2, find sound level.

An open pipe of length 0.5 m has sound speed 340 m/s. Find fundamental frequency.

A closed pipe of length 0.25 m has v=340 m/s. Find first resonant frequency.

Find third harmonic of a closed pipe of length 0.5 m if v=340 m/s.

Two tuning forks 256 Hz and 260 Hz are sounded together. Find beats per second.

A 512 Hz fork gives 5 beats/s with another fork. Find possible frequencies.

A 500 Hz source moves towards stationary observer at 34 m/s. v=340 m/s. Find apparent frequency.

Observer moves towards 600 Hz stationary source at 20 m/s. v=340 m/s. Find f'.

An echo is heard after 2 s. Speed of sound is 340 m/s. Find distance of wall.

Two successive resonances in a closed pipe differ by 17 cm. Find wavelength.

Exercise-style Questions With Solutions

Identify direction of y=A sin(ωt-kx).

Identify direction of y=A sin(ωt+kx).

In y=0.01 sin(50t-2x), find wavelength.

In y=0.01 sin(100πt-5x), find frequency.

If y=0.02 m and ω=10 rad/s at an instant, find acceleration.

Sound level increases by 20 dB. Find intensity ratio.

Speed of sound ratio at 300 K and 1200 K.

Open pipe has f1=170 Hz. Find f2.

Closed pipe fundamental is 100 Hz. Find next allowed frequency.

A fork produces 6 beats with standard. Filing increases fork frequency and beats decrease. Was original fork above or below standard?

CBSE, NEET, JEE, IB, IGCSE and A-Level Questions

Define wavelength and frequency of a progressive wave.

Write relation between wave velocity, frequency and wavelength.

What is the SI unit of wave number?

For y=A sin(ωt-kx), what is wave speed?

Differentiate between particle velocity and wave velocity.

What is beat frequency?

Why is sound speed in gas given by Laplace correction?

What is the distance between adjacent nodes in a standing wave?

Why are only odd harmonics present in closed pipe?

What happens to apparent frequency when source approaches observer?

A wave has f=100 Hz and λ=2 m. Find speed.

A wave equation has ω=40 rad/s and k=5 rad/m. Find speed.

If amplitude doubles, maximum particle velocity becomes?

If intensity of sound becomes 100 times, sound level increases by?

At 27 degree Celsius approximate speed of sound in air is?

Open pipe length is L. Fundamental frequency is?

Closed pipe length is L. Fundamental frequency is?

Two forks 300 Hz and 305 Hz are sounded. Beat frequency?

Distance between node and nearest antinode is?

Doppler Effect is due to change in what?

For y=2 sin(10t-5x), find wave speed.

For y=A sin(100πt-2πx), find frequency and wavelength.

Find maximum transverse velocity for A=2 cm, f=5 Hz.

Two waves of equal amplitude A and phase difference 120 degree superpose. Resultant amplitude?

If two equal waves interfere destructively, phase difference is?

String speed depends on tension how?

If string tension is made four times, frequency becomes?

Open pipe third harmonic if fundamental is 200 Hz?

Closed pipe first overtone if fundamental is 120 Hz?

Source approaching stationary observer: formula?

A source and observer move towards each other. How should signs be chosen?

For a standing wave y=2A sin kx cos ωt, where are nodes?

Power carried by a sinusoidal wave on string depends on amplitude how?

If pressure amplitude doubles in sound wave, intensity becomes?

Two close frequencies have average 500 Hz and beat frequency 6 Hz. Find frequencies.

A reflected Doppler shift is approximately why doubled?

If radial relative velocity is zero, classical first-order Doppler shift?

Why source velocity appears in denominator in Doppler formula?

Why observer velocity appears in numerator?

Closed pipe cannot have which harmonics?

State the relation between period and frequency.

What is phase difference between two points separated by half wavelength?

A wave has A=0.01 m and ω=200 rad s^-1. Find maximum particle speed.

A string has tension 100 N and linear density 0.01 kg m^-1. Find wave speed.

If I=10^-8 W m^-2, find sound level.

Open pipe has f₁=170 Hz. Find f₂.

Assertion: Beat frequency is zero when two frequencies are equal. Reason: f_b=|f₁-f₂|.