Waves—II Problems

This section provides 100 problems to test your understanding of advanced wave phenomena, including sound waves, the Doppler effect, wave intensity, and shock 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. Calculate the speed of sound in air at 27°C ($\gamma = 1.4$, $R = 8.31 , \text{J/mol·K}$, $M = 0.029 , \text{kg/mol}$).

   • (a) $346 , \text{m/s}$
   • (b) $347 , \text{m/s}$
   • (c) $348 , \text{m/s}$
   • (d) $349 , \text{m/s}$

2. A sound wave has displacement $\xi = 0.002 \sin(3 \pi x - 600 \pi t)$ (in SI units). Calculate the maximum particle velocity.

   • (a) $3.75 , \text{m/s}$
   • (b) $3.76 , \text{m/s}$
   • (c) $3.77 , \text{m/s}$
   • (d) $3.78 , \text{m/s}$

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

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

4. A sound wave with $k = 1.5 , \text{rad/m}$, $p_0 = 0.3 , \text{Pa}$ travels in air ($B = 1.4 \times 10^5 , \text{Pa}$). Calculate the displacement amplitude.

   • (a) $1.42 \times 10^{-6} , \text{m}$
   • (b) $1.43 \times 10^{-6} , \text{m}$
   • (c) $1.44 \times 10^{-6} , \text{m}$
   • (d) $1.45 \times 10^{-6} , \text{m}$

5. A car moves at $25 , \text{m/s}$ toward a stationary observer, emitting a horn at $400 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate the observed frequency.

   • (a) $428 , \text{Hz}$
   • (b) $429 , \text{Hz}$
   • (c) $430 , \text{Hz}$
   • (d) $431 , \text{Hz}$

6. An observer moves at $15 , \text{m/s}$ toward a stationary source emitting $800 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate the observed frequency.

   • (a) $834 , \text{Hz}$
   • (b) $835 , \text{Hz}$
   • (c) $836 , \text{Hz}$
   • (d) $837 , \text{Hz}$

7. A rocket moves at $40 , \text{m/s}$ away from an observer, emitting $500 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate the observed frequency.

   • (a) $459 , \text{Hz}$
   • (b) $460 , \text{Hz}$
   • (c) $461 , \text{Hz}$
   • (d) $462 , \text{Hz}$

8. A source and observer move toward each other at $20 , \text{m/s}$ each, with the source emitting $600 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate the observed frequency.

   • (a) $669 , \text{Hz}$
   • (b) $670 , \text{Hz}$
   • (c) $671 , \text{Hz}$
   • (d) $672 , \text{Hz}$

9. A sound wave has $p_0 = 0.4 , \text{Pa}$ in air ($\rho = 1.2 , \text{kg/m}^3$, $v = 340 , \text{m/s}$). Calculate the intensity.

   • (a) $9.78 \times 10^{-5} , \text{W/m}^2$
   • (b) $9.79 \times 10^{-5} , \text{W/m}^2$
   • (c) $9.80 \times 10^{-5} , \text{W/m}^2$
   • (d) $9.81 \times 10^{-5} , \text{W/m}^2$

10. A point source emits $P = 50 , \text{W}$ at $r = 10 , \text{m}$. Calculate the intensity.

    • (a) $0.039 , \text{W/m}^2$
    • (b) $0.040 , \text{W/m}^2$
    • (c) $0.041 , \text{W/m}^2$
    • (d) $0.042 , \text{W/m}^2$

11. A sound intensity is $I = 10^{-4} , \text{W/m}^2$. Calculate the intensity level in dB ($I_0 = 10^{-12} , \text{W/m}^2$).

    • (a) $79 , \text{dB}$
    • (b) $80 , \text{dB}$
    • (c) $81 , \text{dB}$
    • (d) $82 , \text{dB}$

12. A rocket launch produces $I = 10^4 , \text{W/m}^2$ at $r = 30 , \text{m}$. Calculate $I$ at $r = 60 , \text{m}$.

    • (a) $2.4 \times 10^3 , \text{W/m}^2$
    • (b) $2.5 \times 10^3 , \text{W/m}^2$
    • (c) $2.6 \times 10^3 , \text{W/m}^2$
    • (d) $2.7 \times 10^3 , \text{W/m}^2$

13. A jet flies at $510 , \text{m/s}$ ($v_s = 340 , \text{m/s}$). Calculate the Mach number and cone angle.

    • (a) $M = 1.5$, $\theta \approx 41.8^\circ$
    • (b) $M = 1.5$, $\theta \approx 42.0^\circ$
    • (c) $M = 1.6$, $\theta \approx 41.8^\circ$
    • (d) $M = 1.6$, $\theta \approx 42.0^\circ$

14. A rocket at $M = 2.5$ ($v_s = 340 , \text{m/s}$) produces a shock wave. Calculate the speed and cone angle.

    • (a) $v = 850 , \text{m/s}$, $\theta \approx 23.6^\circ$
    • (b) $v = 850 , \text{m/s}$, $\theta \approx 23.7^\circ$
    • (c) $v = 860 , \text{m/s}$, $\theta \approx 23.6^\circ$
    • (d) $v = 860 , \text{m/s}$, $\theta \approx 23.7^\circ$

15. A shock wave with $M = 2$, $\gamma = 1.4$ occurs. Estimate the pressure ratio across the shock.

    • (a) $4.65$
    • (b) $4.66$
    • (c) $4.67$
    • (d) $4.68$

16. A rocket launch at $v = 680 , \text{m/s}$ ($v_s = 340 , \text{m/s}$) produces a shock wave. Calculate $M$ and $\theta$.

    • (a) $M = 2$, $\theta \approx 29.9^\circ$
    • (b) $M = 2$, $\theta \approx 30.0^\circ$
    • (c) $M = 2.1$, $\theta \approx 29.9^\circ$
    • (d) $M = 2.1$, $\theta \approx 30.0^\circ$

17. Calculate the speed of sound in helium at 0°C ($\gamma = 1.67$, $R = 8.31 , \text{J/mol·K}$, $M = 0.004 , \text{kg/mol}$).

    • (a) $970 , \text{m/s}$
    • (b) $971 , \text{m/s}$
    • (c) $972 , \text{m/s}$
    • (d) $973 , \text{m/s}$

18. A sound wave has $\xi = 0.003 \sin(4 \pi x - 800 \pi t)$ (in SI units). Calculate the maximum particle velocity.

    • (a) $7.53 , \text{m/s}$
    • (b) $7.54 , \text{m/s}$
    • (c) $7.55 , \text{m/s}$
    • (d) $7.56 , \text{m/s}$

19. A sound wave in steel ($Y = 2 \times 10^{11} , \text{Pa}$, $\rho = 7800 , \text{kg/m}^3$) propagates. Calculate the speed.

    • (a) $5050 , \text{m/s}$
    • (b) $5060 , \text{m/s}$
    • (c) $5070 , \text{m/s}$
    • (d) $5080 , \text{m/s}$

20. A sound wave with $k = 2 , \text{rad/m}$, $p_0 = 0.6 , \text{Pa}$ travels in air ($B = 1.4 \times 10^5 , \text{Pa}$). Calculate the displacement amplitude.

    • (a) $2.14 \times 10^{-6} , \text{m}$
    • (b) $2.15 \times 10^{-6} , \text{m}$
    • (c) $2.16 \times 10^{-6} , \text{m}$
    • (d) $2.17 \times 10^{-6} , \text{m}$

21. A train moves at $30 , \text{m/s}$ away from a stationary observer, emitting $1000 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate the observed frequency.

    • (a) $914 , \text{Hz}$
    • (b) $915 , \text{Hz}$
    • (c) $916 , \text{Hz}$
    • (d) $917 , \text{Hz}$

22. An observer moves at $10 , \text{m/s}$ away from a stationary source emitting $1200 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate the observed frequency.

    • (a) $1164 , \text{Hz}$
    • (b) $1165 , \text{Hz}$
    • (c) $1166 , \text{Hz}$
    • (d) $1167 , \text{Hz}$

23. A source moves at $50 , \text{m/s}$ toward an observer moving at $20 , \text{m/s}$ toward the source, emitting $700 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate $f'$.

    • (a) $790 , \text{Hz}$
    • (b) $791 , \text{Hz}$
    • (c) $792 , \text{Hz}$
    • (d) $793 , \text{Hz}$

24. A source and observer move away from each other at $15 , \text{m/s}$ each, with the source emitting $900 , \text{Hz}$ ($v = 340 , \text{m/s}$). Calculate $f'$.

    • (a) $819 , \text{Hz}$
    • (b) $820 , \text{Hz}$
    • (c) $821 , \text{Hz}$
    • (d) $822 , \text{Hz}$

25. A sound wave has $p_0 = 0.1 , \text{Pa}$ in air ($\rho = 1.2 , \text{kg/m}^3$, $v = 340 , \text{m/s}$). Calculate the intensity.

    • (a) $1.22 \times 10^{-5} , \text{W/m}^2$
    • (b) $1.23 \times 10^{-5} , \text{W/m}^2$
    • (c) $1.24 \times 10^{-5} , \text{W/m}^2$
    • (d) $1.25 \times 10^{-5} , \text{W/m}^2$

26. A point source emits $P = 200 , \text{W}$ at $r = 8 , \text{m}$. Calculate the intensity.

    • (a) $0.248 , \text{W/m}^2$
    • (b) $0.249 , \text{W/m}^2$
    • (c) $0.250 , \text{W/m}^2$
    • (d) $0.251 , \text{W/m}^2$

27. A sound intensity is $I = 10^{-2} , \text{W/m}^2$. Calculate the intensity level in dB ($I_0 = 10^{-12} , \text{W/m}^2$).

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

28. A rocket launch produces $I = 10^3 , \text{W/m}^2$ at $r = 50 , \text{m}$. Calculate $I$ at $r = 100 , \text{m}$.

    • (a) $249 , \text{W/m}^2$
    • (b) $250 , \text{W/m}^2$
    • (c) $251 , \text{W/m}^2$
    • (d) $252 , \text{W/m}^2$

29. A jet flies at $850 , \text{m/s}$ ($v_s = 340 , \text{m/s}$). Calculate the Mach number and cone angle.

    • (a) $M = 2.5$, $\theta \approx 23.6^\circ$
    • (b) $M = 2.5$, $\theta \approx 23.7^\circ$
    • (c) $M = 2.6$, $\theta \approx 23.6^\circ$
    • (d) $M = 2.6$, $\theta \approx 23.7^\circ$

30. A rocket at $M = 1.8$ ($v_s = 340 , \text{m/s}$) produces a shock wave. Calculate the speed and cone angle.

    • (a) $v = 610 , \text{m/s}$, $\theta \approx 33.7^\circ$
    • (b) $v = 610 , \text{m/s}$, $\theta \approx 33.8^\circ$
    • (c) $v = 612 , \text{m/s}$, $\theta \approx 33.7^\circ$
    • (d) $v = 612 , \text{m/s}$, $\theta \approx 33.8^\circ$

31. A shock wave with $M = 3$, $\gamma = 1.4$ occurs. Estimate the pressure ratio across the shock.

    • (a) $10.48$
    • (b) $10.49$
    • (c) $10.50$
    • (d) $10.51$

32. A rocket launch at $v = 1020 , \text{m/s}$ ($v_s = 340 , \text{m/s}$) produces a shock wave. Calculate $M$ and $\theta$.

    • (a) $M = 3$, $\theta \approx 19.5^\circ$
    • (b) $M = 3$, $\theta \approx 19.6^\circ$
    • (c) $M = 3.1$, $\theta \approx 19.5^\circ$
    • (d) $M = 3.1$, $\theta \approx 19.6^\circ$

33. A sound wave in air at 15°C ($\gamma = 1.4$, $R = 8.31 , \text{J/mol·K}$, $M = 0.029 , \text{kg/mol}$) propagates. Calculate the speed.

    • (a) $339 , \text{m/s}$
    • (b) $340 , \text{m/s}$
    • (c) $341 , \text{m/s}$
    • (d) $342 , \text{m/s}$

34. A sound wave with $p_0 = 0.5 , \text{Pa}$ in air ($\rho = 1.2 , \text{kg/m}^3$, $v = 340 , \text{m/s}$) propagates. Calculate the intensity.

    • (a) $3.06 \times 10^{-4} , \text{W/m}^2$
    • (b) $3.07 \times 10^{-4} , \text{W/m}^2$
    • (c) $3.08 \times 10^{-4} , \text{W/m}^2$
    • (d) $3.09 \times 10^{-4} , \text{W/m}^2$

35. A jet at $M = 2$ ($v_s = 340 , \text{m/s}$) produces a shock wave. Calculate the speed and cone angle.

    • (a) $v = 680 , \text{m/s}$, $\theta \approx 29.9^\circ$
    • (b) $v = 680 , \text{m/s}$, $\theta \approx 30.0^\circ$
    • (c) $v = 690 , \text{m/s}$, $\theta \approx 29.9^\circ$
    • (d) $v = 690 , \text{m/s}$, $\theta \approx 30.0^\circ$

Conceptual Problems

36. What type of wave is a sound wave?

• (a) Transverse
• (b) Longitudinal
• (c) Electromagnetic
• (d) Standing

37. What does the speed of sound in a gas depend on?

• (a) Frequency
• (b) Amplitude
• (c) Temperature, pressure, and density
• (d) Wavelength

38. What does the Doppler effect describe?

• (a) Change in wave speed
• (b) Change in frequency due to relative motion
• (c) Change in amplitude
• (d) Change in wavelength only

39. What happens to the observed frequency when a source moves toward a stationary observer?

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

40. What is the unit of intensity in SI units?

• (a) $\text{W/m}^2$
• (b) $\text{Pa}$
• (c) $\text{dB}$
• (d) $\text{m/s}$

41. What happens to sound intensity with distance from a point source?

• (a) Increases as $r^2$
• (b) Decreases as $1/r^2$
• (c) Remains constant
• (d) Decreases as $1/r$

42. What does a Mach number greater than 1 indicate?

• (a) Subsonic speed
• (b) Supersonic speed
• (c) Sonic speed
• (d) No wave propagation

43. What is the physical significance of $\frac{p_0^2}{2 \rho v}$?

• (a) Wave speed
• (b) Intensity of a sound wave
• (c) Frequency
• (d) Doppler shift

44. What does a sonic boom result from?

• (a) Subsonic motion
• (b) Supersonic motion creating a shock wave
• (c) Interference of waves
• (d) Standing wave formation

45. What is the dimension of intensity?

• (a) $[\text{M} \text{T}^{-3}]$
• (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 Doppler shift indicate?

• (a) No relative motion between source and observer
• (b) Source moving toward observer
• (c) Observer moving toward source
• (d) Supersonic speed

47. What is the significance of $\sin \theta = \frac{1}{M}$ in shock waves?

• (a) Wave speed
• (b) Mach cone angle
• (c) Intensity
• (d) Frequency

48. What happens to the speed of sound if temperature doubles?

• (a) Increases by a factor of $\sqrt{2}$
• (b) Doubles
• (c) Halves
• (d) Remains the same

49. What does a 10 dB increase in sound level indicate?

• (a) Intensity doubles
• (b) Intensity increases by a factor of 10
• (c) Intensity decreases by a factor of 10
• (d) No change in intensity

50. How does the pressure amplitude change with distance from a point source?

• (a) Decreases as $1/r$
• (b) Decreases as $1/r^2$
• (c) Increases as $r$
• (d) Remains constant

Derivation Problems

51. Derive the speed of sound in a gas $v = \sqrt{\frac{\gamma P}{\rho}}$.

52. Derive the pressure-displacement relationship for a sound wave $p = - B \frac{\partial \xi}{\partial x}$.

53. Derive the Doppler effect formula for sound $f' = f \left( \frac{v + v_o}{v - v_s} \right)$.

54. Derive the intensity of a sound wave $I = \frac{p_0^2}{2 \rho v}$.

55. Derive the inverse square law for intensity $I = \frac{P}{4 \pi r^2}$.

56. Derive the decibel scale formula $\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$.

57. Derive the Mach cone angle $\sin \theta = \frac{1}{M}$.

58. Derive the speed of sound dependence on temperature $v \propto \sqrt{T}$.

59. Derive the particle velocity for a sound wave $v_{\text{particle}} = \frac{\partial \xi}{\partial t}$.

60. Derive the pressure ratio across a shock wave $\frac{P_2}{P_1} \approx \frac{2 \gamma M^2}{\gamma + 1}$.

61. Derive the frequency shift for a source moving toward a stationary observer.

62. Derive the amplitude dependence on distance $A \propto \frac{1}{r}$ for a point source.

63. Derive the speed of sound in a solid $v = \sqrt{\frac{Y}{\rho}}$.

64. Derive the energy dissipation relation for a shock wave (entropy increase).

65. Derive the intensity level difference for a 10-fold intensity increase.

NEET-style Conceptual Problems

66. What is the unit of sound wave speed in SI units?

• (a) $\text{m/s}$
• (b) $\text{Hz}$
• (c) $\text{W/m}^2$
• (d) $\text{Pa}$

67. What does a negative Doppler shift indicate?

• (a) Source moving toward observer
• (b) Source moving away from observer
• (c) No relative motion
• (d) Supersonic speed

68. Which phenomenon causes a sonic boom?

• (a) Subsonic motion
• (b) Supersonic motion
• (c) Interference
• (d) Standing waves

69. What happens to sound intensity if distance doubles?

• (a) Increases by a factor of 4
• (b) Decreases by a factor of 4
• (c) Remains the same
• (d) Decreases by a factor of 2

70. What is the dimension of pressure amplitude?

• (a) $[\text{M} \text{L}^{-1} \text{T}^{-2}]$
• (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 adiabatic index $\gamma$ represent in sound speed?

• (a) Density of the medium
• (b) Elasticity of the medium
• (c) Ratio of specific heats
• (d) Frequency of the wave

72. What is the role of relative motion in the Doppler effect?

• (a) Changes wave speed
• (b) Changes observed frequency
• (c) Changes amplitude
• (d) Changes wavelength only

73. What happens to the Mach cone angle as Mach number increases?

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

74. Why does sound intensity decrease with distance?

• (a) Due to interference
• (b) Due to the inverse square law
• (c) Due to Doppler effect
• (d) Due to frequency change

75. What is the unit of intensity level in the decibel scale?

• (a) $\text{W/m}^2$
• (b) $\text{dB}$
• (c) $\text{Pa}$
• (d) $\text{m/s}$

76. What does a constant $v + v_o$ in the Doppler formula indicate?

• (a) No Doppler shift
• (b) Observer speed relative to the medium
• (c) Source speed
• (d) Wave speed

77. Which type of wave produces a shock wave?

• (a) Transverse
• (b) Longitudinal
• (c) Electromagnetic
• (d) Standing

78. What is the direction of particle motion in a sound wave?

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

79. What does a pseudo-force do in a non-inertial frame for the Doppler effect?

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

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

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

81. What is the role of shock waves in rocket launches?

• (a) Increases speed
• (b) Creates pressure jumps, affecting structural design
• (c) Reduces frequency
• (d) Increases intensity

82. What happens to pressure across a shock wave?

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

83. Why does the Doppler effect occur in sound but not in light (non-relativistically)?

• (a) Light speed is constant, sound speed depends on the medium
• (b) Sound waves are transverse
• (c) Light waves have higher frequency
• (d) Sound waves have higher intensity

84. What is the significance of $10^{-12} , \text{W/m}^2$ in the decibel scale?

• (a) Maximum intensity
• (b) Threshold of hearing
• (c) Threshold of pain
• (d) Wave speed

85. What is the unit of the Mach number?

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

86. What does a zero pressure amplitude in a sound wave indicate?

• (a) Maximum intensity
• (b) No sound wave
• (c) Maximum particle velocity
• (d) Doppler shift

87. What is the physical significance of $\sqrt{\frac{\gamma R T}{M}}$?

• (a) Intensity of sound
• (b) Speed of sound in a gas
• (c) Doppler shift
• (d) Mach number

88. Why does a shock wave produce a sonic boom?

• (a) Due to interference
• (b) Due to sudden pressure change from supersonic speed
• (c) Due to Doppler effect
• (d) Due to standing waves

89. What is the dimension of the adiabatic index $\gamma$?

• (a) Dimensionless
• (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 the Doppler effect help in rocket tracking?

• (a) Increases intensity
• (b) Measures velocity via frequency shift
• (c) Reduces wave speed
• (d) Creates shock waves

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

• (a) Increases speed via $v \propto \sqrt{T}$
• (b) Decreases speed
• (c) Affects amplitude
• (d) Affects frequency

92. What does a 20 dB increase in sound level indicate?

• (a) Intensity increases by a factor of 100
• (b) Intensity doubles
• (c) Intensity increases by a factor of 10
• (d) No change

93. What is the physical significance of $\frac{2 \gamma M^2}{\gamma + 1}$?

• (a) Intensity ratio
• (b) Pressure ratio across a shock wave
• (c) Doppler shift
• (d) Wave speed

94. What is the dimension of displacement amplitude in a sound wave?

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

95. Why does sound speed increase in solids compared to gases?

• (a) Higher frequency
• (b) Higher elasticity (Young’s modulus) and density
• (c) Lower density only
• (d) Higher temperature

NEET-style Numerical Problems

96. A sound wave in air at 20°C ($\gamma = 1.4$, $R = 8.31 , \text{J/mol·K}$, $M = 0.029 , \text{kg/mol}$) propagates. What is the speed?

• (a) $342 , \text{m/s}$
• (b) $343 , \text{m/s}$
• (c) $344 , \text{m/s}$
• (d) $345 , \text{m/s}$

97. A source moves at $60 , \text{m/s}$ toward a stationary observer, emitting $300 , \text{Hz}$ ($v = 340 , \text{m/s}$). What is the observed frequency?

• (a) $352 , \text{Hz}$
• (b) $353 , \text{Hz}$
• (c) $354 , \text{Hz}$
• (d) $355 , \text{Hz}$

98. A sound wave has $p_0 = 0.2 , \text{Pa}$ in air ($\rho = 1.2 , \text{kg/m}^3$, $v = 340 , \text{m/s}$). What is the intensity?

• (a) $4.88 \times 10^{-5} , \text{W/m}^2$
• (b) $4.89 \times 10^{-5} , \text{W/m}^2$
• (c) $4.90 \times 10^{-5} , \text{W/m}^2$
• (d) $4.91 \times 10^{-5} , \text{W/m}^2$

99. A jet at $M = 1.2$ ($v_s = 340 , \text{m/s}$) produces a shock wave. What is the speed and cone angle?

• (a) $v = 408 , \text{m/s}$, $\theta \approx 56.4^\circ$
• (b) $v = 408 , \text{m/s}$, $\theta \approx 56.5^\circ$
• (c) $v = 410 , \text{m/s}$, $\theta \approx 56.4^\circ$
• (d) $v = 410 , \text{m/s}$, $\theta \approx 56.5^\circ$

100. A point source emits $P = 100 , \text{W}$ at $r = 4 , \text{m}$. What is the intensity?
     - (a) $0.496 , \text{W/m}^2$
     - (b) $0.497 , \text{W/m}^2$
     - (c) $0.498 , \text{W/m}^2$
     - (d) $0.499 , \text{W/m}^2$

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