Class 11 Physics Unit 8: Waves – Important Questions & Answers with Solved Numericals. 
This post covers all important topics of waves, including wave motion, transverse and longitudinal waves, stationary waves, sound waves, Doppler effect, and organ pipes. Perfect for exam preparation, quick revision, board exams, and competitive test practice. Learn and master all key concepts with step-by-step solutions and explanations."
SHORT QUESTIONS WITH BRIEF ANSWERS (Unit-7 Oscillation)
Q1. Define wave motion.
Wave motion is the transfer of energy through a medium or space without the transfer of matter.
2. Define wavelength and frequency.
Wavelength is the distance between two consecutive points in the same phase, and frequency is the number of oscillations per second.
3. What is the relation between wave speed, frequency, and wavelength?
v = fλ
4. What is a progressive wave?
A progressive wave is a wave that transfers energy from one place to another.
5. What is a stationary wave?
A stationary wave is formed due to the superposition of two identical waves traveling in opposite directions, with no net energy transfer.
6. Define crest and trough.
Crest is the highest point of a transverse wave, and trough is the lowest point.
7. Define compression and rarefaction.
Compression is a region of high pressure, and rarefaction is a region of low pressure in a longitudinal wave.
8. What are nodes and antinodes?
Nodes are points of zero displacement, while antinodes are points of maximum displacement in stationary waves.
9. What is interference?
Interference is the superposition of two waves of the same frequency producing regions of reinforcement and cancellation.
10. What are beats?
Beats are periodic variations in sound intensity due to interference of two waves of slightly different frequencies.
11. What is the beat frequency?
Beat frequency = |f₁ − f₂|
12. Define Doppler effect.
Doppler effect is the apparent change in frequency due to relative motion between source and observer.
13. What is an organ pipe?
An organ pipe is a tube in which standing waves are produced in an air column.
14. Why is an open pipe richer in harmonics than a closed pipe?
Because an open pipe supports all harmonics, while a closed pipe supports only odd harmonics.
15. Why does sound travel faster in solids than in gases?
Because solids have higher elasticity and closely packed particles.
16. On what factors does the speed of sound in air depend?
Temperature, humidity, and nature of the medium.
17. Why does sound not travel in vacuum?
Because sound requires a material medium to propagate.
18. What is ultrasound?
Sound waves having frequency greater than 20 kHz.
19. Give one medical use of ultrasound.
Ultrasound is used in medical imaging (sonography) to diagnose internal organs and fetus.
20. What is the principle of superposition?
When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements.
LONG / COMPREHENSIVE QUESTIONS WITH BRIEF EXAM ANSWERS
Q1. Explain wave motion. Define wavelength and frequency, and derive the relation v = fλ.
Wave motion is the propagation of a disturbance through a medium or space that transfers energy from one place to another without the transfer of matter. In wave motion, particles of the medium vibrate about their mean positions while the disturbance travels forward.
Wavelength is defined as the distance between two consecutive points in the same phase of vibration, such as two crests, two troughs, two compressions, or two rarefactions. Frequency is the number of oscillations or vibrations produced by the source in one second and is measured in hertz (Hz).
If a wave travels one wavelength λ in one time period T, then the speed of the wave is given by v = λ/T. Since frequency f = 1/T, substituting T = 1/f gives v = fλ. This equation relates wave speed, frequency, and wavelength.
2. Describe longitudinal and transverse waves with examples and clearly explain the difference between them.
In longitudinal waves, the particles of the medium vibrate parallel to the direction of wave propagation. These waves consist of compressions, where particles are close together, and rarefactions, where particles are far apart. Sound waves in air and waves in a slinky spring are examples of longitudinal waves.
In transverse waves, the particles of the medium vibrate perpendicular to the direction of wave propagation. These waves consist of crests and troughs. Waves on a stretched string and water surface waves are examples of transverse waves.
The main difference between the two types of waves is the direction of vibration of particles relative to the direction of wave travel. Longitudinal waves can travel through solids, liquids, and gases, while transverse mechanical waves can travel only through solids and liquids.
3. Explain crest, trough, compression, rarefaction, node, and antinode.
Crest is the highest point of a transverse wave above the equilibrium position, while trough is the lowest point below the equilibrium position. Compression is a region in a longitudinal wave where particles are crowded together and pressure is high, while rarefaction is a region where particles are spread apart and pressure is low.
Node is a point in a stationary wave where the displacement of particles is zero, and antinode is a point where the displacement is maximum. Nodes and antinodes are formed due to the interference of two identical waves traveling in opposite directions.
4. What are stationary waves? Show that as the string vibrates in more loops, its frequency increases and wavelength decreases.
Stationary waves are formed when two identical waves of the same frequency and amplitude travel in opposite directions and superpose. In stationary waves, nodes and antinodes are formed, and there is no net transfer of energy along the medium.
For a stretched string of length L, the fundamental mode of vibration has one loop, and its wavelength is λ₁ = 2L. The fundamental frequency is f₁ = v/2L.
For the second harmonic, the string vibrates in two loops, and λ₂ = L, so f₂ = 2f₁.
For the third harmonic, λ₃ = 2L/3, and f₃ = 3f₁.
Thus, as the number of loops increases, the wavelength decreases and the frequency increases.
5. Explain Newton’s formula for the speed of sound and show how Laplace corrected it.
Newton assumed that sound propagation in air is an isothermal process and derived the speed of sound as v = √(P/ρ), where P is pressure and ρ is density of air. This formula gave a value lower than the experimental value.
Laplace corrected Newton’s assumption by considering the process to be adiabatic rather than isothermal. He introduced the ratio of specific heats γ and modified the formula to v = √(γP/ρ). This corrected formula gives a value close to the experimentally observed speed of sound in air.
6. Explain the speed of sound in a gas and state the factors affecting the speed of sound in air.
The speed of sound in a gas is given by v = √(γRT/M), where γ is the ratio of specific heats, R is the gas constant, T is absolute temperature, and M is molar mass. This shows that the speed depends on elasticity and density of the medium.
In air, the speed of sound depends on temperature, humidity, and the nature of the gas. It does not depend on pressure if temperature remains constant.
7. Show that the speed of sound increases by 0.61 m/s for each 1°C rise in temperature.
The speed of sound in air is proportional to the square root of absolute temperature, i.e., v ∝ √T. For small temperature changes,
Δv / v = ½ (ΔT / T).
At 0°C, T = 273 K and v ≈ 331 m/s. For ΔT = 1 K,
Δv = 331 / (2 × 273) ≈ 0.61 m/s.
Hence, for each 1°C rise in temperature, the speed of sound increases by about 0.61 m/s.
8. What are beats? Explain their formation and derive the expression for beat frequency.
Beats are periodic variations in the intensity of sound produced when two waves of slightly different frequencies interfere with each other. When the waves are in phase, constructive interference occurs and sound is loud, and when they are out of phase, destructive interference occurs and sound becomes faint.
If the two frequencies are f₁ and f₂, the beat frequency is given by
f_beats = |f₁ − f₂|.
This means the number of beats per second is equal to the difference in frequencies of the two sources.
9. Explain Doppler effect and derive expressions for apparent frequency.
Doppler effect is the apparent change in frequency of a wave due to relative motion between the source and the observer.
When the source approaches a stationary observer, the observed frequency is
f' = f × v / (v − vₛ).
When the observer moves towards a stationary source, the observed frequency is
f' = f × (v + vₒ) / v.
The frequency increases when the source or observer approaches and decreases when they move away.
10. What are organ pipes? Show that an open organ pipe is richer in harmonics than a closed organ pipe.
Organ pipes are hollow tubes in which air columns vibrate to produce sound. In an open organ pipe, both ends are open and act as antinodes, so all harmonics are present. The fundamental frequency is f = v/2L.
In a closed organ pipe, one end is closed and the other is open, so only odd harmonics are present. The fundamental frequency is f = v/4L.
Since open pipes support both odd and even harmonics, they produce a richer and more musical sound than closed pipes.
11. Explain vibrations in a closed organ pipe and show that the frequency of the third harmonic is 5v/4L.
In a closed organ pipe, the closed end is a node and the open end is an antinode. The fundamental mode has wavelength λ₁ = 4L and frequency f₁ = v/4L.
The next allowed mode is the third harmonic with wavelength λ₃ = 4L/3 and frequency f₃ = 3v/4L. The fifth harmonic has wavelength λ₅ = 4L/5 and frequency f₅ = 5v/4L.
Thus, the frequency of the third overtone (fifth harmonic) is 5v/4L.Miscellaneous Numerical Problems with Proper Solutions
Numerical No. 1
Problem:
A wave travels with a speed of 600 m/s and has a frequency of 300 Hz. Calculate its wavelength.
Given:
Frequency of wave, f = 250 Hz
Speed of wave, v = 500 m/s
To find:
Wavelength (λ)
Solution:
v = fλ
λ = v / f = 500 / 250 = 2 m
Answer:
Wavelength = 2 m
Numerical No. 2
Problem:
A sound wave has a wavelength of 0.75 m and a frequency of 400 Hz. Find the speed of the wave.
Given:
Wavelength, λ = 0.75 m
Frequency, f = 400 Hz
To find:
Wave speed (v)
Solution:
v = fλ = 400 × 0.75 = 300 m/s
Answer:
Speed of wave = 300 m/s
Numerical No. 3
Problem:
In a ripple tank, 600 waves pass through a point in 15 seconds. Find the frequency of the waves.
Given:
Number of waves = 600
Time = 15 s
To find:
Frequency (f)
Solution:
f = N / t = 600 / 15 = 40 Hz
Answer:
Frequency = 40 Hz
Numerical No. 4
Problem:
A sound wave travels with a speed of 340 m/s and has a frequency of 1700 Hz. Calculate its wavelength.
Given:
Speed of sound = 340 m/s
Frequency = 1700 Hz
To find:
Wavelength
Solution:
λ = v / f = 340 / 1700 = 0.2 m
Answer:
Wavelength = 0.2 m
Numerical No. 5
Problem:
An open organ pipe has a length of 0.75 m. Calculate its fundamental frequency if the speed of sound is 300 m/s.
Given:
Length of open organ pipe, L = 0.75 m
Speed of sound = 300 m/s
To find:
Fundamental frequency
Solution:
For open pipe: f₁ = v / 2L
f₁ = 300 / (2 × 0.75) = 200 Hz
Answer:
Fundamental frequency = 200 Hz
Numerical No. 6
Problem:
A closed organ pipe has a length of 0.4 m. Find its fundamental frequency if the speed of sound is 320 m/s.
Given:
Length of closed pipe = 0.4 m
Speed of sound = 320 m/s
To find:
Fundamental frequency
Solution:
For closed pipe: f₁ = v / 4L
f₁ = 320 / (4 × 0.4) = 200 Hz
Answer:
Fundamental frequency = 200 Hz
Numerical No. 7
Problem:
If the amplitude of a wave is increased to 4 times its original value, how does its intensity change?
Given:
A₂ = 4A₁
To find:
Change in intensity
Solution:
Intensity ∝ A²
I₂ / I₁ = (4A)² / A² = 16
Answer:
Intensity becomes 16 times
Numerical No. 8
Problem:
Two tuning forks of frequencies 300 Hz and 308 Hz are sounded together. Find the number of beats produced per second.
Given:
f₁ = 300 Hz
f₂ = 308 Hz
To find:
Beat frequency
Solution:
Beat frequency = |f₁ − f₂| = |308 − 300| = 8 Hz
Answer:
Beats = 8 per second
Numerical No. 9
Problem:
A source emitting sound of frequency 800 Hz moves toward a stationary observer with a speed of 20 m/s. If the speed of sound is 340 m/s, find the observed frequency.
Given:
f = 800 Hz
v = 340 m/s
vₛ = 20 m/s
To find:
Observed frequency (f′)
Solution:
f′ = f v / (v − vₛ)
f′ = 800 × 340 / (340 − 20)
f′ = 850 Hz
Answer:
Observed frequency ≈ 850 Hz
Numerical No. 10
Problem:
An observer moves toward a stationary source with speed 10 m/s. The source frequency is 600 Hz and speed of sound is 330 m/s. Find the observed frequency.
Given:
f = 600 Hz
v = 330 m/s
vₒ = 10 m/s
To find:
Observed frequency
Solution:
f′ = f (v + vₒ) / v
f′ = 600 × (330 + 10) / 330
f′ = 618 Hz
Answer:
Observed frequency ≈ 618 Hz
Numerical No. 11
Problem:
A stretched string of length 1.5 m produces a wave with speed 600 m/s. Find its fundamental frequency.
Given:
L = 1.5 m
v = 600 m/s
To find:
Fundamental frequency
Solution:
f₁ = v / 2L
f₁ = 600 / (2 × 1.5) = 200 Hz
Answer:
Fundamental frequency = 200 Hz
Numerical No. 12 (Board favourite)
Problem:
The speed of sound in air at 0°C is 331 m/s. Find the speed of sound at 40°C.
Given:
v₀ = 331 m/s
T = 40°C
To find:
Speed of sound
Solution:
v = v₀ + 0.61T
v = 331 + 0.61 × 40
v = 331 + 24.4
v = 355.4 m/s
Answer:
Speed of sound ≈ 355 m/s
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