Waves—I Problems

This section provides 100 problems to test your understanding of wave motion, including wave characteristics, wave equations, superposition, interference, and standing waves. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering wave mechanics, a key topic for JEE/NEET success.

Numerical Problems

1. A wave on a string has tension $T=200\text{N}$ and linear mass density $\mu =0.04\text{kg/m}$. Calculate the wave speed.

   • (a) $69\text{m/s}$
   • (b) $70\text{m/s}$
   • (c) $71\text{m/s}$
   • (d) $72\text{m/s}$

2. A wave has frequency $f=150\text{Hz}$ and wavelength $\lambda =2\text{m}$. Calculate the wave speed.

   • (a) $290\text{m/s}$
   • (b) $300\text{m/s}$
   • (c) $310\text{m/s}$
   • (d) $320\text{m/s}$

3. A sound wave in air has $B=1.4\times {10}^5\text{Pa}$ and $\rho =1.2{\text{kg/m}}^3$. Calculate the speed of sound.

   • (a) $340\text{m/s}$
   • (b) $341\text{m/s}$
   • (c) $342\text{m/s}$
   • (d) $343\text{m/s}$

4. Two points on a wave have a path difference of $0.6\text{m}$, with wavelength $\lambda =1.2\text{m}$. Calculate the phase difference.

   • (a) $\pi \text{rad}$
   • (b) $\frac{\pi }{2}\text{rad}$
   • (c) $2\pi \text{rad}$
   • (d) $\frac{3\pi }{2}\text{rad}$

5. A wave is described by $y=0.03\sin (6\pi x-1200\pi t)$ (in SI units). Calculate the wave speed.

   • (a) $195\text{m/s}$
   • (b) $200\text{m/s}$
   • (c) $205\text{m/s}$
   • (d) $210\text{m/s}$

6. A wave has $k=3\text{rad/m}$ and $f=100\text{Hz}$. Calculate the wavelength.

   • (a) $2.08\text{m}$
   • (b) $2.09\text{m}$
   • (c) $2.10\text{m}$
   • (d) $2.11\text{m}$

7. A wave $y=0.05\sin (2x-400t)$ (in SI units) propagates. Calculate the particle velocity at $x=0$, $t=0$.

   • (a) $-20\text{m/s}$
   • (b) $-15\text{m/s}$
   • (c) $0\text{m/s}$
   • (d) $20\text{m/s}$

8. A wave has $\lambda =0.4\text{m}$ and $v=320\text{m/s}$. Calculate the frequency.

   • (a) $790\text{Hz}$
   • (b) $800\text{Hz}$
   • (c) $810\text{Hz}$
   • (d) $820\text{Hz}$

9. Two waves $y_1=0.02\sin (3\pi x-600\pi t)$ and $y_2=0.02\sin (3\pi x-600\pi t+\pi /3)$ (in SI units) interfere. Calculate the resultant amplitude.

   • (a) $0.035\text{m}$
   • (b) $0.036\text{m}$
   • (c) $0.037\text{m}$
   • (d) $0.038\text{m}$

10. Two sound waves have frequencies $f_1=300\text{Hz}$ and $f_2=306\text{Hz}$. Calculate the beat frequency.

    • (a) $5\text{Hz}$
    • (b) $6\text{Hz}$
    • (c) $7\text{Hz}$
    • (d) $8\text{Hz}$

11. Two wave sources are separated by $0.8\text{m}$, with $\lambda =1.6\text{m}$. Calculate the path difference for constructive interference at a point $0.8\text{m}$ from each source.

    • (a) $0\text{m}$
    • (b) $0.4\text{m}$
    • (c) $0.8\text{m}$
    • (d) $1.2\text{m}$

12. Two waves $y_1=0.04\sin (4x-800t)$ and $y_2=0.04\sin (4x-800t+\pi )$ (in SI units) interfere. Calculate the resultant amplitude at $x=0$.

    • (a) $0\text{m}$
    • (b) $0.02\text{m}$
    • (c) $0.04\text{m}$
    • (d) $0.08\text{m}$

13. A string ($L=0.6\text{m}$, $T=90\text{N}$, $\mu =0.01\text{kg/m}$) is fixed at both ends. Calculate the fundamental frequency.

    • (a) $149\text{Hz}$
    • (b) $150\text{Hz}$
    • (c) $151\text{Hz}$
    • (d) $152\text{Hz}$

14. A closed pipe ($L=0.34\text{m}$, $v=340\text{m/s}$) produces sound. Calculate the fundamental frequency.

    • (a) $248\text{Hz}$
    • (b) $250\text{Hz}$
    • (c) $252\text{Hz}$
    • (d) $254\text{Hz}$

15. A string ($L=1.2\text{m}$, $v=60\text{m/s}$) is fixed at both ends. Calculate the frequency of the second harmonic.

    • (a) $49\text{Hz}$
    • (b) $50\text{Hz}$
    • (c) $51\text{Hz}$
    • (d) $52\text{Hz}$

16. An open pipe ($L=0.17\text{m}$, $v=340\text{m/s}$) produces sound. Calculate the third harmonic frequency.

    • (a) $2985\text{Hz}$
    • (b) $2995\text{Hz}$
    • (c) $3000\text{Hz}$
    • (d) $3015\text{Hz}$

17. A wave on a string has $T=150\text{N}$ and $\mu =0.03\text{kg/m}$. Calculate the wave speed.

    • (a) $70\text{m/s}$
    • (b) $71\text{m/s}$
    • (c) $72\text{m/s}$
    • (d) $73\text{m/s}$

18. A wave has $f=250\text{Hz}$ and $\lambda =1.2\text{m}$. Calculate the wave speed.

    • (a) $290\text{m/s}$
    • (b) $300\text{m/s}$
    • (c) $310\text{m/s}$
    • (d) $320\text{m/s}$

19. A sound wave in water has $B=2.2\times {10}^9\text{Pa}$ and $\rho =1000{\text{kg/m}}^3$. Calculate the speed of sound.

    • (a) $1470\text{m/s}$
    • (b) $1480\text{m/s}$
    • (c) $1490\text{m/s}$
    • (d) $1500\text{m/s}$

20. Two points on a wave have a path difference of $0.3\text{m}$, with $\lambda =0.6\text{m}$. Calculate the phase difference.

    • (a) $\pi \text{rad}$
    • (b) $\frac{\pi }{2}\text{rad}$
    • (c) $2\pi \text{rad}$
    • (d) $\frac{3\pi }{2}\text{rad}$

21. A wave is described by $y=0.01\sin (8\pi x-1600\pi t)$ (in SI units). Calculate the wave speed.

    • (a) $195\text{m/s}$
    • (b) $200\text{m/s}$
    • (c) $205\text{m/s}$
    • (d) $210\text{m/s}$

22. A wave has $k=4\text{rad/m}$ and $f=80\text{Hz}$. Calculate the wavelength.

    • (a) $1.56\text{m}$
    • (b) $1.57\text{m}$
    • (c) $1.58\text{m}$
    • (d) $1.59\text{m}$

23. A wave $y=0.02\sin (5x-1000t)$ (in SI units) propagates. Calculate the particle velocity at $x=0$, $t=0$.

    • (a) $-20\text{m/s}$
    • (b) $-15\text{m/s}$
    • (c) $0\text{m/s}$
    • (d) $20\text{m/s}$

24. A wave has $\lambda =0.8\text{m}$ and $v=400\text{m/s}$. Calculate the frequency.

    • (a) $490\text{Hz}$
    • (b) $500\text{Hz}$
    • (c) $510\text{Hz}$
    • (d) $520\text{Hz}$

25. Two waves $y_1=0.05\sin (2\pi x-400\pi t)$ and $y_2=0.05\sin (2\pi x-400\pi t+\pi /4)$ (in SI units) interfere. Calculate the resultant amplitude.

    • (a) $0.094\text{m}$
    • (b) $0.095\text{m}$
    • (c) $0.096\text{m}$
    • (d) $0.097\text{m}$

26. Two sound waves have $f_1=440\text{Hz}$ and $f_2=446\text{Hz}$. Calculate the beat frequency.

    • (a) $5\text{Hz}$
    • (b) $6\text{Hz}$
    • (c) $7\text{Hz}$
    • (d) $8\text{Hz}$

27. Two wave sources are separated by $0.5\text{m}$, with $\lambda =1\text{m}$. Calculate the path difference for destructive interference at a point $0.5\text{m}$ from each source.

    • (a) $0\text{m}$
    • (b) $0.25\text{m}$
    • (c) $0.5\text{m}$
    • (d) $1.0\text{m}$

28. Two waves $y_1=0.03\sin (6x-1200t)$ and $y_2=0.03\sin (6x-1200t+2\pi /3)$ (in SI units) interfere. Calculate the resultant amplitude at $x=0$.

    • (a) $0.030\text{m}$
    • (b) $0.031\text{m}$
    • (c) $0.032\text{m}$
    • (d) $0.033\text{m}$

29. A string ($L=0.8\text{m}$, $T=160\text{N}$, $\mu =0.02\text{kg/m}$) is fixed at both ends. Calculate the fundamental frequency.

    • (a) $99\text{Hz}$
    • (b) $100\text{Hz}$
    • (c) $101\text{Hz}$
    • (d) $102\text{Hz}$

30. A closed pipe ($L=0.68\text{m}$, $v=340\text{m/s}$) produces sound. Calculate the third harmonic frequency.

    • (a) $747\text{Hz}$
    • (b) $748\text{Hz}$
    • (c) $749\text{Hz}$
    • (d) $750\text{Hz}$

31. A string ($L=1.5\text{m}$, $v=75\text{m/s}$) is fixed at both ends. Calculate the frequency of the fourth harmonic.

    • (a) $99\text{Hz}$
    • (b) $100\text{Hz}$
    • (c) $101\text{Hz}$
    • (d) $102\text{Hz}$

32. An open pipe ($L=0.34\text{m}$, $v=340\text{m/s}$) produces sound. Calculate the fourth harmonic frequency.

    • (a) $1995\text{Hz}$
    • (b) $2000\text{Hz}$
    • (c) $2005\text{Hz}$
    • (d) $2010\text{Hz}$

33. A rocket launch acoustic wave ($L=2\text{m}$, $v=340\text{m/s}$) forms standing waves. Calculate the fundamental frequency for an open pipe model.

    • (a) $84\text{Hz}$
    • (b) $85\text{Hz}$
    • (c) $86\text{Hz}$
    • (d) $87\text{Hz}$

34. Two waves $y_1=0.01\sin (10x-2000t)$ and $y_2=0.01\sin (10x-2000t+\pi /6)$ (in SI units) interfere. Calculate the resultant amplitude.

    • (a) $0.019\text{m}$
    • (b) $0.020\text{m}$
    • (c) $0.021\text{m}$
    • (d) $0.022\text{m}$

35. A wave $y=0.04\sin (8x-1600t)$ (in SI units) propagates. Calculate the particle velocity at $x=0$, $t=0$.

    • (a) $-64\text{m/s}$
    • (b) $0\text{m/s}$
    • (c) $64\text{m/s}$
    • (d) $128\text{m/s}$

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Conceptual Problems

36. What distinguishes a transverse wave from a longitudinal wave?

• (a) Transverse waves travel faster
• (b) Transverse waves have displacement perpendicular to propagation, longitudinal parallel
• (c) Longitudinal waves require a medium
• (d) Transverse waves have higher frequency

37. What does the wave speed on a string depend on?

• (a) Amplitude
• (b) Frequency
• (c) Tension and linear mass density
• (d) Wavelength

38. What condition leads to constructive interference?

• (a) Phase difference of $\pi$
• (b) Phase difference of $2\pi$
• (c) Path difference of $\lambda /2$
• (d) Different amplitudes

39. What does the beat frequency represent?

• (a) Sum of the frequencies
• (b) Difference between the frequencies
• (c) Average of the frequencies
• (d) Product of the frequencies

40. What is the unit of wave number $k$?

• (a) $\text{rad/m}$
• (b) $\text{Hz}$
• (c) $\text{m/s}$
• (d) $\text{rad/s}$

41. What happens to wave speed on a string if tension is quadrupled?

• (a) Doubles
• (b) Quadruples
• (c) Halves
• (d) Remains the same

42. What does a node in a standing wave indicate?

• (a) Maximum displacement
• (b) Zero displacement
• (c) Maximum velocity
• (d) Minimum energy

43. What is the physical significance of $v=f\lambda$?

• (a) Particle velocity
• (b) Wave speed
• (c) Phase difference
• (d) Beat frequency

44. What does a standing wave on a string require?

• (a) Two waves of different frequencies
• (b) Two waves traveling in opposite directions with the same frequency
• (c) A single wave
• (d) Destructive interference only

45. What is the dimension of wave speed?

• (a) $[\text{L}{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

46. What does a zero resultant amplitude in interference indicate?

• (a) Constructive interference
• (b) Destructive interference
• (c) No interference
• (d) Standing wave formation

47. What is the significance of $\frac{2\pi }{\lambda }$ in a wave equation?

• (a) Angular frequency
• (b) Wave number
• (c) Phase difference
• (d) Amplitude

48. What happens to the frequency of a standing wave if the string length is doubled?

• (a) Doubles
• (b) Halves
• (c) Quadruples
• (d) Remains the same

49. What does a closed pipe produce compared to an open pipe?

• (a) All harmonics
• (b) Only odd harmonics
• (c) Only even harmonics
• (d) No harmonics

50. How does wave speed relate to the medium’s properties for a sound wave?

• (a) Depends on frequency
• (b) Depends on amplitude
• (c) Depends on bulk modulus and density
• (d) Depends on wavelength

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Derivation Problems

51. Derive the wave speed on a string $v=\sqrt{\frac{T}{\mu }}$.

52. Derive the relationship $v=f\lambda$ for a wave.

53. Derive the wave speed of sound in a medium $v=\sqrt{\frac{B}{\rho }}$.

54. Derive the phase difference from path difference $\mathrm{\Delta }\varphi =\frac{2\pi }{\lambda }\mathrm{\Delta }x$.

55. Derive the general solution to the wave equation $y(x,t)=A\sin (kx-\omega t+\varphi )$.

56. Derive the resultant amplitude of two interfering waves with phase difference $\varphi$.

57. Derive the beat frequency for two waves $f_{\text{beat}}=|f_1-f_2|$.

58. Derive the condition for constructive and destructive interference using path difference.

59. Derive the standing wave equation on a string $y=2A\sin (kx)\cos (\omega t)$.

60. Derive the harmonic frequencies on a string fixed at both ends $f_n=\frac{nv}{2L}$.

61. Derive the frequencies in a closed pipe $f_n=\frac{(2n-1)v}{4L}$.

62. Derive the frequencies in an open pipe $f_n=\frac{nv}{2L}$.

63. Derive the particle velocity for a wave $v_{\text{particle}}=\frac{\partial y}{\partial t}$.

64. Derive the wave number and angular frequency from $y=A\sin (kx-\omega t)$.

65. Derive the node positions in a standing wave $x=\frac{n\lambda }{2}$.

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NEET-style Conceptual Problems

66. What is the unit of frequency in SI units?

• (a) $\text{Hz}$
• (b) $\text{rad/s}$
• (c) $\text{m/s}$
• (d) $\text{m}$

67. What does a zero particle velocity in a standing wave indicate?

• (a) Antinode
• (b) Node
• (c) Maximum amplitude
• (d) Wave speed

68. Which condition results in destructive interference?

• (a) Phase difference of $2\pi$
• (b) Phase difference of $\pi$
• (c) Same amplitude
• (d) Same frequency

69. What happens to wave speed in a denser medium for a sound wave?

• (a) Increases
• (b) Decreases
• (c) Remains the same
• (d) Becomes zero

70. What is the dimension of wavelength?

• (a) $[\text{L}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

71. What does the wave number $k$ represent?

• (a) Frequency of the wave
• (b) Spatial frequency of the wave
• (c) Speed of the wave
• (d) Amplitude of the wave

72. What is the role of tension in wave speed on a string?

• (a) Increases speed if increased
• (b) Decreases speed if increased
• (c) No effect
• (d) Reduces wavelength

73. What happens to the frequency of a standing wave if the wave speed doubles?

• (a) Doubles
• (b) Halves
• (c) Remains the same
• (d) Quadruples

74. Why does a closed pipe produce only odd harmonics?

• (a) Both ends are open
• (b) Boundary condition requires a node at the closed end
• (c) Wave speed is higher
• (d) Frequency is doubled

75. What is the unit of angular frequency?

• (a) $\text{rad/s}$
• (b) $\text{Hz}$
• (c) $\text{m/s}$
• (d) $\text{rad/m}$

76. What does a constant $kx-\omega t$ in a wave equation indicate?

• (a) Constant amplitude
• (b) Constant phase
• (c) Constant speed
• (d) Constant frequency

77. Which type of wave requires a medium to propagate?

• (a) Electromagnetic
• (b) Transverse only
• (c) Mechanical
• (d) Longitudinal only

78. What is the direction of particle displacement in a longitudinal wave?

• (a) Perpendicular to wave propagation
• (b) Parallel to wave propagation
• (c) Circular
• (d) Random

79. What does a pseudo-force do in a non-inertial frame for wave motion?

• (a) Affects wave speed
• (b) Affects the observed frequency
• (c) Creates interference
• (d) Reduces amplitude

80. What is the dimension of amplitude in a wave?

• (a) $[\text{L}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

81. What is the role of standing waves in rocket acoustic design?

• (a) Increases noise
• (b) Can cause resonance, affecting structural integrity
• (c) Reduces wave speed
• (d) Increases frequency

82. What happens to particle velocity at an antinode in a standing wave?

• (a) Zero
• (b) Maximum
• (c) Constant
• (d) Minimum but non-zero

83. Why does interference occur in wave motion?

• (a) Waves travel at different speeds
• (b) Waves superpose and add their displacements
• (c) Waves have different amplitudes
• (d) Waves have different frequencies

84. What is the significance of $\frac{v}{2L}$ for a string fixed at both ends?

• (a) Wave speed
• (b) Fundamental frequency
• (c) Wavelength
• (d) Beat frequency

85. What is the unit of linear mass density $\mu$?

• (a) $\text{kg/m}$
• (b) ${\text{kg/m}}^2$
• (c) $\text{N/m}$
• (d) $\text{Pa}$

86. What does a zero path difference in interference indicate?

• (a) Destructive interference
• (b) Constructive interference
• (c) No interference
• (d) Standing wave

87. What is the physical significance of $\frac{\partial y}{\partial t}$ in a wave?

• (a) Wave speed
• (b) Particle velocity
• (c) Phase difference
• (d) Amplitude

88. Why does an open pipe produce all harmonics?

• (a) One end is closed
• (b) Both ends are antinodes for displacement
• (c) Wave speed is lower
• (d) Frequency is halved

89. What is the dimension of frequency?

• (a) $[{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

90. How does interference affect sound in rocket launches?

• (a) Increases amplitude
• (b) Can reduce noise through destructive interference
• (c) Increases frequency
• (d) Reduces wave speed

91. What is the role of bulk modulus in sound wave speed?

• (a) Measures elasticity, increasing speed if higher
• (b) Reduces speed
• (c) Affects amplitude
• (d) Affects frequency

92. What does a standing wave’s wavelength depend on in a pipe?

• (a) Frequency
• (b) Pipe length and harmonic number
• (c) Amplitude
• (d) Wave speed only

93. What is the physical significance of $2A\sin (kx)$ in a standing wave?

• (a) Time-dependent amplitude
• (b) Position-dependent amplitude
• (c) Wave speed
• (d) Frequency

94. What is the dimension of particle velocity in a wave?

• (a) $[\text{L}{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

95. Why does wave speed not depend on frequency for a mechanical wave?

• (a) Frequency affects amplitude
• (b) Speed depends on medium properties only
• (c) Wavelength changes inversely
• (d) Frequency affects phase

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NEET-style Numerical Problems

96. A wave on a string has $T=50\text{N}$ and $\mu =0.01\text{kg/m}$. What is the wave speed?

• (a) $69\text{m/s}$
• (b) $70\text{m/s}$
• (c) $71\text{m/s}$
• (d) $72\text{m/s}$

97. A wave has $f=500\text{Hz}$ and $\lambda =0.68\text{m}$. What is the wave speed?

• (a) $340\text{m/s}$
• (b) $350\text{m/s}$
• (c) $360\text{m/s}$
• (d) $370\text{m/s}$

98. A wave $y=0.02\sin (4x-800t)$ (in SI units) propagates. What is the particle velocity at $x=0$, $t=0$?

• (a) $-16\text{m/s}$
• (b) $0\text{m/s}$
• (c) $16\text{m/s}$
• (d) $32\text{m/s}$

99. Two sound waves have $f_1=512\text{Hz}$ and $f_2=520\text{Hz}$. What is the beat frequency?

• (a) $6\text{Hz}$
• (b) $7\text{Hz}$
• (c) $8\text{Hz}$
• (d) $9\text{Hz}$

100. A string ($L=0.4\text{m}$, $v=80\text{m/s}$) is fixed at both ends. What is the fundamental frequency?
     - (a) $99\text{Hz}$
     - (b) $100\text{Hz}$
     - (c) $101\text{Hz}$
     - (d) $102\text{Hz}$

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