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Physics · Textbook solutions

Waves

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 36 questions

Worked Examples

6 q

Solved Examples

Worked · 6
  1. Eg 14.1
    Given below are some examples of wave motion. State in each case if the wave motion is transverse, longitudinal or a combination of both: (a) Motion of a kink in a longitudinal spring produced by displacing one end of the spring sideways. (b) Waves produced in a cylinder containing a liquid by moving its piston back and forth. (c) Waves produced by a motorboat sailing in water. (d) Ultrasonic waves in air produced by a vibrating quartz crystal.
  2. Eg 14.2
    A wave travelling along a string is described by, y(x, t)=0.005sin⁡(80.0 x−3.0 t)y(x,\,t) = 0.005 \sin (80.0\,x - 3.0\,t), in which the numerical constants are in SI units (0.005 m0.005\,\text{m}, 80.0 rad m−180.0\,\text{rad m}^{-1}, and 3.0 rad s−13.0\,\text{rad s}^{-1}). Calculate (a) the amplitude, (b) the wavelength, and (c) the period and frequency of the wave. Also, calculate the displacement yy of the wave at a distance x=30.0 cmx = 30.0\,\text{cm} and time t=20 st = 20\,\text{s} ?
  3. Eg 14.3
    A steel wire 0.72 m0.72\,\text{m} long has a mass of 5.0×10−3 kg5.0 \times 10^{-3}\,\text{kg}. If the wire is under a tension of 60 N60\,\text{N}, what is the speed of transverse waves on the wire ?
  4. Eg 14.4
    Estimate the speed of sound in air at standard temperature and pressure. The mass of 11 mole of air is 29.0×10−3 kg29.0 \times 10^{-3}\,\text{kg}.
  5. Eg 14.5
    A pipe, 30.0 cm30.0\,\text{cm} long, is open at both ends. Which harmonic mode of the pipe resonates a 1.1 kHz1.1\,\text{kHz} source? Will resonance with the same source be observed if one end of the pipe is closed ? Take the speed of sound in air as 330 m s−1330\,\text{m s}^{-1}.
  6. Eg 14.6
    Two sitar strings A and B playing the note 'Dha' are slightly out of tune and produce beats of frequency 5 Hz5\,\text{Hz}. The tension of the string B is slightly increased and the beat frequency is found to decrease to 3 Hz3\,\text{Hz}. What is the original frequency of B if the frequency of A is 427 Hz427\,\text{Hz} ?

Exercises

30 q
  1. Ex 14.1
    A string of mass 2.50 kg2.50\,\text{kg} is under a tension of 200 N200\,\text{N}. The length of the stretched string is 20.0 m20.0\,\text{m}. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
  2. Ex 14.2
    A stone dropped from the top of a tower of height 300 m300\,\text{m} splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m s−1340\,\text{m s}^{-1} ? (g=9.8 m s−2g = 9.8\,\text{m s}^{-2})
  3. Ex 14.3
    A steel wire has a length of 12.0 m12.0\,\text{m} and a mass of 2.10 kg2.10\,\text{kg}. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 ∘C=343 m s−120\,^{\circ}\text{C} = 343\,\text{m s}^{-1}.
  4. Ex 14.4(a)
    Use the formula v=γPρv = \sqrt{\dfrac{\gamma P}{\rho}} to explain why the speed of sound in air (a) is independent of pressure,
  5. Ex 14.4(b)
    Use the formula v=γPρv = \sqrt{\dfrac{\gamma P}{\rho}} to explain why the speed of sound in air (b) increases with temperature,
  6. Ex 14.4(c)
    Use the formula v=γPρv = \sqrt{\dfrac{\gamma P}{\rho}} to explain why the speed of sound in air (c) increases with humidity.
  7. Ex 14.5(a)
    You have learnt that a travelling wave in one dimension is represented by a function y=f(x, t)y = f(x,\,t) where xx and tt must appear in the combination x−v tx - v\,t or x+v tx + v\,t, i.e. y=f(x±v t)y = f(x \pm v\,t). Is the converse true? Examine if the following functions for yy can possibly represent a travelling wave : (a) (x−vt)2(x - vt)^{2}
  8. Ex 14.5(b)
    You have learnt that a travelling wave in one dimension is represented by a function y=f(x, t)y = f(x,\,t) where xx and tt must appear in the combination x−v tx - v\,t or x+v tx + v\,t, i.e. y=f(x±v t)y = f(x \pm v\,t). Is the converse true? Examine if the following functions for yy can possibly represent a travelling wave : (b) log⁡[(x+vt)/x0]\log\left[(x + vt)/x_{0}\right]
  9. Ex 14.5(c)
    You have learnt that a travelling wave in one dimension is represented by a function y=f(x, t)y = f(x,\,t) where xx and tt must appear in the combination x−v tx - v\,t or x+v tx + v\,t, i.e. y=f(x±v t)y = f(x \pm v\,t). Is the converse true? Examine if the following functions for yy can possibly represent a travelling wave : (c) 1/(x+vt)1/(x + vt)
  10. Ex 14.6
    A bat emits ultrasonic sound of frequency 1000 kHz1000\,\text{kHz} in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m s−1340\,\text{m s}^{-1} and in water 1486 m s−11486\,\text{m s}^{-1}.
  11. Ex 14.7
    A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s−11.7\,\text{km s}^{-1} ? The operating frequency of the scanner is 4.2 MHz4.2\,\text{MHz}.
  12. Ex 14.8
    A transverse harmonic wave on a string is described by y(x, t)=3.0sin⁡(36 t+0.018 x+π/4)y(x,\,t) = 3.0 \sin (36\,t + 0.018\,x + \pi/4) where xx and yy are in cm and tt in s. The positive direction of xx is from left to right. (a) Is this a travelling wave or a stationary wave ? If it is travelling, what are the speed and direction of its propagation ? (b) What are its amplitude and frequency ? (c) What is the initial phase at the origin ? (d) What is the least distance between two successive crests in the wave ?
  13. Ex 14.9
    For the wave described in Exercise 14.8, plot the displacement (yy) versus (tt) graphs for x=0x = 0, 22 and 4 cm4\,\text{cm}. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase ?
  14. Ex 14.10
    For the travelling harmonic wave y(x, t)=2.0cos⁡2π (10t−0.0080 x+0.35)y(x,\,t) = 2.0 \cos 2\pi\,(10t - 0.0080\,x + 0.35) where xx and yy are in cm and tt in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (a) 4 m4\,\text{m}, (b) 0.5 m0.5\,\text{m}, (c) λ/2\lambda/2, (d) 3λ/43\lambda/4
  15. Ex 14.11
    The transverse displacement of a string (clamped at its both ends) is given by y(x, t)=0.06sin⁡(2π3x)cos⁡(120 πt)y(x,\,t) = 0.06 \sin\left(\dfrac{2\pi}{3}x\right)\cos (120\,\pi t) where xx and yy are in m and tt in s. The length of the string is 1.5 m1.5\,\text{m} and its mass is 3.0×10−2 kg3.0 \times 10^{-2}\,\text{kg}. Answer the following : (a) Does the function represent a travelling wave or a stationary wave? (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ? (c) Determine the tension in the string.
  16. Ex 14.12
    (i) For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m0.375\,\text{m} away from one end?
  17. Ex 14.13(a)
    Given below are some functions of xx and tt to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (a) y=2cos⁡(3x)sin⁡(10t)y = 2\cos (3x)\sin (10t)
  18. Ex 14.13(b)
    Given below are some functions of xx and tt to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (b) y=2x−vty = 2\sqrt{x - vt}
  19. Ex 14.13(c)
    Given below are some functions of xx and tt to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (c) y=3sin⁡(5x−0.5t)+4cos⁡(5x−0.5t)y = 3\sin (5x - 0.5t) + 4\cos (5x - 0.5t)
  20. Ex 14.13(d)
    Given below are some functions of xx and tt to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (d) y=cos⁡xsin⁡t+cos⁡2xsin⁡2ty = \cos x \sin t + \cos 2x \sin 2t
  21. Ex 14.14
    A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz45\,\text{Hz}. The mass of the wire is 3.5×10−2 kg3.5 \times 10^{-2}\,\text{kg} and its linear mass density is 4.0×10−2 kg m−14.0 \times 10^{-2}\,\text{kg m}^{-1}. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?
  22. Ex 14.15
    A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz340\,\text{Hz}) when the tube length is 25.5 cm25.5\,\text{cm} or 79.3 cm79.3\,\text{cm}. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
  23. Ex 14.16
    A steel rod 100 cm100\,\text{cm} long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz2.53\,\text{kHz}. What is the speed of sound in steel?
  24. Ex 14.17
    A pipe 20 cm20\,\text{cm} long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz430\,\text{Hz} source ? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m s−1340\,\text{m s}^{-1}).
  25. Ex 14.18
    Two sitar strings A and B playing the note 'Ga' are slightly out of tune and produce beats of frequency 6 Hz6\,\text{Hz}. The tension in the string A is slightly reduced and the beat frequency is found to reduce to 3 Hz3\,\text{Hz}. If the original frequency of A is 324 Hz324\,\text{Hz}, what is the frequency of B?
  26. Ex 14.19(a)
    Explain why (or how): (a) in a sound wave, a displacement node is a pressure antinode and vice versa,
  27. Ex 14.19(b)
    Explain why (or how): (b) bats can ascertain distances, directions, nature, and sizes of the obstacles without any "eyes",
  28. Ex 14.19(c)
    Explain why (or how): (c) a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,
  29. Ex 14.19(d)
    Explain why (or how): (d) solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and
  30. Ex 14.19(e)
    Explain why (or how): (e) the shape of a pulse gets distorted during propagation in a dispersive medium.