Waves ISC Class 11: Notes, Standing Waves and Organ Pipes

What Waves is about

Weightage: Waves belongs to the unit Oscillations and Waves, which CISCE weights at 10 of the 70 marks in the ISC Class XI Physics Paper I (Theory). Class XI is examined by the school, so your school sets the paper within this weightage.Source: CISCE ISC Physics syllabus, examination year 2028, Class XI unit weightage

Key concepts

Wave motion: transverse and longitudinal waves

A wave is a disturbance that travels through a medium (or space) transferring energy, while each particle of the medium only oscillates about its mean position. Mechanical waves need a material medium with elasticity and inertia; electromagnetic waves do not. Transverse waves: particles vibrate perpendicular to the direction of propagation, forming crests and troughs; they need a medium with rigidity, so they travel in solids and on liquid surfaces but not through gases (waves on a string, ripples). Longitudinal waves: particles vibrate along the direction of propagation, forming compressions (high pressure and density) and rarefactions; they travel in solids, liquids and gases (sound, waves on a spring pushed along its length). Sound is a longitudinal mechanical wave; it cannot travel through a vacuum.

Worked example

Classify each as transverse or longitudinal: (a) a wave on a sitar string, (b) sound in air, (c) a pulse sent along a slinky by pushing one end along its axis, (d) light.

Try it first, then show the solutionSolution

1. (a) Transverse: string particles move sideways. (b) Longitudinal: air layers move to and fro along the direction of travel.
2. (c) Longitudinal: compressions travel along the spring. (d) Transverse electromagnetic wave (not mechanical).

Answer: (a) transverse (b) longitudinal (c) longitudinal (d) transverse (electromagnetic)

Characteristics of a harmonic wave; v = fλ

Amplitude A: the maximum displacement of a particle. Wavelength λ: the distance between two nearest particles in the same phase (crest to crest, or compression to compression). Period T: the time for one oscillation of a particle, which is also the time for the wave to move one wavelength. Frequency f = 1/T, fixed by the source. Since the wave advances λ in time T, its speed is $v=\frac{\lambda }{T}=f\lambda$. When a wave passes into another medium, its frequency stays the same while its speed and wavelength change. Two graphs describe a harmonic wave: the displacement-distance graph (a snapshot at one instant, showing λ) and the displacement-time graph (one particle over time, showing T).

Worked example

A tuning fork of frequency 256 Hz sounds in air where the speed of sound is 340 m s⁻¹. Find the wavelength. What would it be in water, where sound travels at 1480 m s⁻¹?

Try it first, then show the solutionSolution

1. λ = v/f = 340/256 = 1.33 m
2. In water f is unchanged: λ = 1480/256 = 5.78 m

Answer: 1.33 m in air; 5.78 m in water

Displacement relation for a progressive wave

A harmonic wave travelling along +x with amplitude A is $y(x,t)=A\sin (kx-\omega t+\varphi )$; along −x it is $A\sin (kx+\omega t+\varphi )$. Here $k=\frac{2\pi }{\lambda }$ is the angular wave number (rad m⁻¹) and ω = 2πf the angular frequency, so the wave speed is $v=\frac{\omega }{k}=f\lambda$. At a fixed x the particle performs SHM; at a fixed t the wave is a sine curve. Phase difference between two points Δx apart at the same instant: $\Delta \varphi =\frac{2\pi }{\lambda }\Delta x$; between two instants Δt apart at one point: $\frac{2\pi }{T}\Delta t$. Points λ apart are in phase; points λ/2 apart are in opposite phase.

Worked example

A wave is described by y = 0.05 sin(20x − 400t), with x and y in metres and t in seconds. Find its amplitude, wavelength, frequency, speed and direction of travel.

Try it first, then show the solutionSolution

1. A = 0.05 m; k = 20 rad m⁻¹, so λ = 2π/20 = 0.314 m
2. ω = 400 rad s⁻¹, so f = 400/2π = 63.7 Hz
3. v = ω/k = 400/20 = 20 m s⁻¹, along +x (minus sign between the terms)

Answer: 0.05 m; 0.314 m; 63.7 Hz; 20 m s⁻¹ along +x

Speed of sound: Newton's formula and Laplace's correction

The speed of a mechanical wave depends on the elasticity and inertia of the medium. The general formula (no derivation) is $v=\sqrt{\frac{E}{\rho }}$, where E is the appropriate modulus. In solids (thin rods) $v=\sqrt{\frac{Y}{\rho }}$; in liquids and gases $v=\sqrt{\frac{B}{\rho }}$, with B the bulk modulus. Newton assumed that compressions and rarefactions in air are isothermal, so B = p: $v=\sqrt{\frac{p}{\rho }}$ = $\sqrt{\frac{1.013\times 10^5}{1.293}}$ ≈ 280 m s⁻¹ at STP, about 16% below the measured 332 m s⁻¹. Laplace's correction: the changes are so fast that there is no time for heat exchange, so they are adiabatic, B = γp: $v=\sqrt{\frac{\gamma p}{\rho }}$ = $\sqrt{1.4}\times 280$ ≈ 331 m s⁻¹, in agreement with experiment. Sound is faster in liquids (water ≈ 1480 m s⁻¹) and in solids (steel ≈ 5000 m s⁻¹) because their moduli are much larger relative to their densities (brief treatment).

Worked example

Find the speed of sound in air at STP using (a) Newton's formula and (b) Laplace's correction. (p = 1.013 × 10⁵ Pa, ρ = 1.293 kg m⁻³, γ = 1.4)

Try it first, then show the solutionSolution

1. (a) $v=\sqrt{\frac{1.013\times 10^5}{1.293}}=280m{s}^{-1}$
2. (b) $v=\sqrt{\frac{1.4\times 1.013\times 10^5}{1.293}}=331m{s}^{-1}$

Answer: 280 m s⁻¹ (too low); 331 m s⁻¹ (matches experiment)

Factors affecting the speed of sound; supersonic and ultrasonic

From $v=\sqrt{\frac{\gamma p}{\rho }}$ and p/ρ = RT/M: $v=\sqrt{\frac{\gamma RT}{M}}$. Pressure: at constant temperature p/ρ is constant (Boyle's law), so v does not depend on pressure. Temperature: v ∝ √T (kelvin); speed rises by about 0.6 m s⁻¹ per °C near room temperature. Density: at the same pressure, v ∝ 1/√ρ, so sound is faster in hydrogen than in oxygen (4 times). Humidity: water vapour is less dense than dry air, so sound travels slightly faster in humid air. Wind adds its component along the direction of travel. Supersonic speeds exceed the speed of sound (Mach number = speed/speed of sound > 1), producing shock waves and a sonic boom. Ultrasonic waves have frequencies above 20 kHz (beyond human hearing): used in medical scanning, SONAR and cleaning; infrasonic waves are below 20 Hz.

Worked example

The speed of sound in air at 0 °C is 331 m s⁻¹. Find it at 27 °C, and the temperature at which it would be twice its value at 0 °C.

Try it first, then show the solutionSolution

1. $v_{27}=331\sqrt{\frac{300}{273}}=347m{s}^{-1}$
2. Doubling needs four times T: 4 × 273 = 1092 K = 819 °C

Answer: 347 m s⁻¹; 819 °C

Superposition, interference and reflection

Principle of superposition: when two or more waves meet at a point, the resultant displacement is the algebraic (vector) sum of the displacements due to each wave; after crossing, each wave continues unchanged. Interference (simple ideas): when two coherent waves (same frequency, constant phase difference) overlap, the intensity is greatest where they arrive in phase (constructive, path difference nλ) and least where they arrive in opposite phase (destructive, path difference (n + ½)λ). Reflection: a wave meeting a boundary is reflected. At a rigid boundary (fixed end of a string, closed end of a pipe for displacement) the reflected wave is inverted, a phase change of π; at a free boundary (free end, open end of a pipe) it is reflected without phase change. Reflected sound gives echoes; the reflected and incident waves together set up standing waves.

Worked example

Two loudspeakers driven by the same source emit sound of wavelength 0.5 m. A listener is 3.0 m from one and 3.25 m from the other. Will the sound be loud or faint there?

Try it first, then show the solutionSolution

1. Path difference = 0.25 m = λ/2
2. The waves arrive in opposite phase: destructive interference

Answer: Faint (a point of destructive interference)

Standing (stationary) waves

Two identical waves travelling in opposite directions, $y_1=A\sin (kx-\omega t)$ and $y_2=A\sin (kx+\omega t)$, superpose to give $y=[2A\sin kx]\cos \omega t$. Each particle oscillates with frequency f but with an amplitude $2A\sin kx$ that depends on its position. Nodes (zero amplitude) are at sin kx = 0: x = 0, λ/2, λ, …; antinodes (amplitude 2A) at x = λ/4, 3λ/4, …. Adjacent nodes (or antinodes) are λ/2 apart; a node and the next antinode are λ/4 apart. The pattern does not travel and transports no net energy. All particles between two adjacent nodes move in phase; those in neighbouring loops move in opposite phase.

Worked example

A standing wave on a string has adjacent nodes 25 cm apart, and the string vibrates at 400 Hz. Find the wavelength and the speed of the travelling waves that form it.

Try it first, then show the solutionSolution

1. Node to node = λ/2 = 25 cm, so λ = 0.5 m
2. v = fλ = 400 × 0.5 = 200 m s⁻¹

Answer: 0.5 m; 200 m s⁻¹

Vibrations of a stretched string; the sonometer

A string of length l fixed at both ends has nodes at the ends, so it fits a whole number of half-wavelengths: $l=\frac{n\lambda }{2}$. With $v=\sqrt{\frac{T}{m}}$ (T tension, m mass per unit length): $f_n=\frac{n}{2l}\sqrt{\frac{T}{m}}$, n = 1, 2, 3, … The lowest, $f_0=\frac{1}{2l}\sqrt{\frac{T}{m}}$, is the fundamental (first harmonic); $f_n=n{f}_0$ are the harmonics, and those above the fundamental are overtones (the second harmonic is the first overtone). Laws of vibration of a stretched string: (1) law of length, f ∝ 1/l (T, m constant); (2) law of tension, f ∝ √T; (3) law of mass, f ∝ 1/√m. The sonometer (a wire stretched over a hollow box between two movable bridges, with weights giving the tension) verifies these laws by resonance with tuning forks; a small paper rider flies off at resonance.

Worked example

A sonometer wire 0.5 m long with mass per unit length 1 × 10⁻³ kg m⁻¹ is under a tension of 100 N. Find its fundamental frequency and its second harmonic.

Try it first, then show the solutionSolution

1. $v=\sqrt{\frac{100}{10^{-3}}}=316m{s}^{-1}$
2. $f_0=\frac{v}{2l}=\frac{316}{1.0}=316Hz$; second harmonic = 632 Hz

Answer: 316 Hz; 632 Hz

Organ pipes: closed and open

In an air column, the closed end is a displacement node and an open end an antinode (ignoring end correction). Closed pipe (closed at one end) of length L: $L=\frac{\lambda }{4},\frac{3\lambda }{4},\frac{5\lambda }{4}$ …, so $f=\frac{v}{4L},\frac{3v}{4L},\frac{5v}{4L}$ …: only odd harmonics (1 : 3 : 5); the first overtone is the third harmonic. Open pipe (open at both ends): $L=\frac{\lambda }{2},\lambda ,\frac{3\lambda }{2}$ …, so $f=\frac{v}{2L},\frac{2v}{2L},\frac{3v}{2L}$ …: all harmonics (1 : 2 : 3). An open pipe's fundamental is twice that of a closed pipe of the same length, and its notes are richer. In practice the antinode forms slightly beyond the open end: the end correction e ≈ 0.3D (D = diameter) is added to L.

Worked example

Find the fundamental and the first two overtones of a pipe 25 cm long when (a) closed at one end, (b) open at both ends. (v = 340 m s⁻¹)

Try it first, then show the solutionSolution

1. (a) $f_1=\frac{340}{4\times 0.25}=340Hz$; overtones 3f₁ = 1020 Hz, 5f₁ = 1700 Hz
2. (b) $f_1=\frac{340}{2\times 0.25}=680Hz$; overtones 1360 Hz, 2040 Hz

Answer: Closed: 340, 1020, 1700 Hz; open: 680, 1360, 2040 Hz

Beats (qualitative)

When two sound waves of slightly different frequencies $f_1$ and $f_2$ travel together, they are alternately in phase and out of phase, so the loudness rises and falls periodically. One rise and fall is one beat; the beat frequency is $|f_1-f_2|$. The ear can distinguish beats up to about 10 per second. Uses: tuning musical instruments (adjust until the beats vanish) and finding an unknown frequency. Loading a tuning-fork prong with wax lowers its frequency; filing it raises the frequency. If loading the unknown fork increases the number of beats, its frequency was below the standard one.

Worked example

Tuning forks of 256 Hz and 260 Hz are sounded together. How many beats are heard per second, and how long is the interval between successive loud sounds?

Try it first, then show the solutionSolution

1. Beat frequency = 260 − 256 = 4 Hz
2. Interval between loud sounds = 1/4 = 0.25 s

Answer: 4 beats per second; 0.25 s

Physics in real life: Waves

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Progressive Waves and Speed of Sound
• Standing Waves, Organ Pipes and Beats

Waves formula sheet

Wave speed

$v=f\lambda =\frac{\omega }{k},k=\frac{2\pi }{\lambda }$

f set by the source.

Progressive wave

$y=A\sin (kx-\omega t)(+x);y=A\sin (kx+\omega t)(-x)$

Phase difference $\frac{2\pi }{\lambda }\Delta x$.

Speed on a string

$v=\sqrt{\frac{T}{m}}$

m = mass per unit length.

Speed of sound

$v=\sqrt{\frac{E}{\rho }};\text{Newton }\sqrt{\frac{p}{\rho }};\text{Laplace }\sqrt{\frac{\gamma p}{\rho }}=\sqrt{\frac{\gamma RT}{M}}$

v ∝ √T; independent of p at constant T.

Standing wave

$y=[2A\sin kx]\cos \omega t$

Nodes λ/2 apart; node to antinode λ/4.

Stretched string

$f_n=\frac{n}{2l}\sqrt{\frac{T}{m}}$

All harmonics.

Closed pipe

$f=\frac{(2n-1)v}{4L}$

Odd harmonics only.

Open pipe

$f=\frac{nv}{2L}$

All harmonics.

Beats

$f_{beat}=|f_1-f_2|$

Qualitative treatment.

Waves: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Waves

The question patterns to expect, and the method that works for each.

Tricks to remember Waves

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Laplace adds γ

  Newton √(p/ρ) is too slow; Laplace multiplies by √γ (sound is quick, so adiabatic).

  Use it when: Explaining the correction.

• Closed is odd

  Closed pipe: odd harmonics only (1, 3, 5); open pipe: all (1, 2, 3).

  Use it when: Organ-pipe questions.

• Node to node is half

  N-N = λ/2, N-A = λ/4.

  Use it when: Standing-wave spacing.

• String laws: L-T-M

  f ∝ 1/L, f ∝ √T, f ∝ 1/√m.

  Use it when: Sonometer and law questions.

• Wax lowers, file raises

  Waxing a prong adds mass (f down); filing removes mass (f up).

  Use it when: Beat questions with unknown forks.

Waves: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section D style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 5-mark question split into 1 + 1 + 2 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Waves

The slips examiners see most often, and what to write instead.

• ✗Mistake: Saying pressure changes the speed of sound

  Students see p in √(γp/ρ) and conclude v rises with pressure.

  Wrong working

  Doubling the pressure increases v by √2.

  Right working

  At constant temperature, ρ doubles with p, so p/ρ and v are unchanged.

  ✓Do this: Write v = √(γRT/M) to see that only T (and M) matter.

  Usually costs: 1 mark.

• ✗Mistake: Giving a closed pipe even harmonics

  Students apply the string formula f = nv/2L to every pipe.

  Wrong working

  Closed pipe 25 cm: 340, 680, 1020 Hz.

  Right working

  Closed pipe: 340, 1020, 1700 Hz (odd harmonics only).

  ✓Do this: Draw the node at the closed end before writing L in terms of λ.

  Usually costs: 1-2 marks.

• ✗Mistake: Mixing up harmonics and overtones

  Students call the second harmonic the second overtone.

  Wrong working

  Second overtone of an open pipe = 2f₁.

  Right working

  First overtone = second harmonic (2f₁); second overtone = third harmonic (3f₁). For a closed pipe the first overtone is 3f₁.

  ✓Do this: Overtone number = harmonic number − 1 (for all-harmonic systems).

  Usually costs: 1 mark.

• ✗Mistake: Not explaining why Laplace's correction is needed

  Students write the corrected formula without the physical reason.

  Wrong working

  Laplace changed p to γp.

  Right working

  Compressions and rarefactions are rapid and air is a poor conductor, so there is no time for heat exchange: the process is adiabatic, B = γp.

  ✓Do this: Always give the reason: isothermal versus adiabatic.

  Usually costs: 1 mark.

• ✗Mistake: Wrong sign reading for wave direction

  Students think a minus sign in sin(kx − ωt) means travel to the left.

  Wrong working

  y = A sin(kx − ωt) moves along −x.

  Right working

  It moves along +x: a crest at kx − ωt = constant moves to larger x as t increases.

  ✓Do this: Minus between the terms means +x; plus means −x.

  Usually costs: ½-1 mark.

Practice questions on Waves

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Sound waves in air are:
   (a) transverse
   (b) longitudinal
   (c) electromagnetic
   (d) both transverse and longitudinal

   Hint

   Compressions and rarefactions.

   Answer

   (b)

2. Q2 · 1 mark · MCQExam level

   In a standing wave, the distance between a node and the next antinode is:
   (a) λ
   (b) λ/2
   (c) λ/4
   (d) 2λ

   Hint

   Half of node-to-node.

   Answer

   (c)

3. Q3 · 1 mark · Very short answerExam level

   What are beats?

   Hint

   Rise and fall of loudness.

   Answer

   The periodic rise and fall in loudness heard when two sounds of slightly different frequencies are heard together; beat frequency = |f₁ − f₂|.

4. Q4 · 1 mark · Very short answerExam level

   What are ultrasonic waves? Give one use.

   Hint

   Above 20 kHz.

   Answer

   Sound waves of frequency above 20 kHz (beyond human hearing); used in medical scanning (ultrasonography) or SONAR.

5. Q5 · 2 marks · Short answerExam level

   Distinguish between transverse and longitudinal waves (two points).

   Hint

   Direction of vibration; media.

   Answer

   Transverse: particles vibrate perpendicular to the direction of travel, forming crests and troughs; cannot travel through gases. Longitudinal: particles vibrate along the direction of travel, forming compressions and rarefactions; travel through solids, liquids and gases.

6. Q6 · 2 marks · Short answerHarder exam level

   A wave is y = 0.03 sin(0.5πx + 100πt) in SI units. Find its wavelength, frequency, speed and direction of travel.

   Hint

   λ = 2π/k, v = ω/k; a + sign means travel along −x.

   Answer

   k = 0.5π, so λ = 2π/0.5π = 4 m; ω = 100π, so f = 50 Hz; v = ω/k = 200 m s⁻¹ along −x.

7. Q7 · 2 marks · Short answerHarder exam level

   Calculate the speed of sound in hydrogen at STP by Newton's formula and by Laplace's formula. (p = 1.013 × 10⁵ Pa, ρ = 0.0899 kg m⁻³, γ = 1.41)

   Hint

   Laplace multiplies Newton's value by √γ.

   Answer

   Newton: √(1.013 × 10⁵/0.0899) = 1062 m s⁻¹; Laplace: √1.41 × 1062 = 1260 m s⁻¹ (close to the measured value, nearly four times the speed in air).

8. Q8 · 2 marks · Short answerHarder exam level

   The speed of sound in air at 15 °C is 340 m s⁻¹. Find it on a hot day at 35 °C and on a cold day at −13 °C.

   Hint

   v ∝ √T, with T in kelvin.

   Answer

   35 °C: 340 × √(308/288) = 352 m s⁻¹; −13 °C: 340 × √(260/288) = 323 m s⁻¹.

9. Q9 · 2 marks · Short answerHarder exam level

   An open pipe 0.5 m long is sounded. Find the frequencies of its first three harmonics. (v = 340 m s⁻¹)

   Hint

   f = nv/2L.

   Answer

   340 Hz, 680 Hz, 1020 Hz.

10. Q10 · 3 marks · Short answerHarder exam level

    By what percentage does the fundamental frequency of a stretched wire increase if its tension is increased by 44%?

    Hint

    f ∝ √T.

    Answer

    f' = f√1.44 = 1.2f: an increase of 20%.

11. Q11 · 3 marks · Short answerHarder exam level

    A string 1.2 m long, fixed at both ends, vibrates in three loops at 300 Hz. Find the wavelength, the wave speed and the fundamental frequency.

    Hint

    Three loops = 3λ/2.

    Answer

    3λ/2 = 1.2 m, so λ = 0.8 m; v = 300 × 0.8 = 240 m s⁻¹; f₁ = 300/3 = 100 Hz.

12. Q12 · 5 marks · Long answerHarder exam level

    (a) Write Newton's formula for the speed of sound in a gas and explain why Laplace modified it. [3]
    (b) Explain the effect of temperature and of humidity on the speed of sound in air. [2]

    Hint

    Isothermal versus adiabatic.

    Answer

    (a) Newton: v = √(p/ρ), assuming isothermal changes (B = p); it gives about 280 m s⁻¹ for air, about 16% too low. Laplace: the compressions and rarefactions are very rapid and air conducts heat poorly, so the changes are adiabatic, B = γp and v = √(γp/ρ) ≈ 331 m s⁻¹, agreeing with experiment. (b) v = √(γRT/M) ∝ √T, so sound is faster in warm air (about 0.6 m s⁻¹ per °C). Water vapour is lighter than dry air, so humid air has lower density and sound travels slightly faster.

13. Q13 · 3 marks · Short answerHOTS / competency

    A fork of 256 Hz gives 4 beats per second with an unknown fork. When the unknown fork's prongs are filed slightly, the beats drop to 2 per second. Find its original frequency.

    Hint

    Filing raises the frequency.

    Answer

    Possible values 260 Hz or 252 Hz. Filing raises the frequency; beats fall only if it moves towards 256 Hz, so it was 252 Hz (now 254 Hz).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Relation between v, f and λ?Show answer

   v = fλ.

2. 2.Equation of a wave travelling along +x?Show answer

   y = A sin(kx − ωt).

3. 3.Laplace's formula?Show answer

   v = √(γp/ρ).

4. 4.Does the speed of sound depend on pressure?Show answer

   Not at constant temperature.

5. 5.Equation of a standing wave?Show answer

   y = [2A sin kx] cos ωt.

6. 6.Fundamental frequency of a stretched string?Show answer

   f₀ = (1/2l)√(T/m).

7. 7.Harmonics of a closed pipe?Show answer

   Odd only: v/4L, 3v/4L, 5v/4L …

8. 8.Beat frequency?Show answer

   |f₁ − f₂|.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you distinguish transverse and longitudinal waves?
• ☐Can you read λ, f and v from a wave equation?
• ☐Can you explain Laplace's correction?
• ☐Can you state how temperature, pressure and humidity affect the speed of sound?
• ☐Can you derive y = 2A sin kx cos ωt and locate nodes and antinodes?
• ☐Can you state the laws of vibration of a stretched string?
• ☐Can you find the harmonics of closed and open pipes?
• ☐Can you find an unknown frequency using beats?

Waves: frequently asked questions