Physics · Textbook solutions

Superposition of Waves

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 45 questions

6. Superposition of Waves — worked examples

13 q

Solved Examples

Worked · 13
  1. Solved Ex.6.1
    The displacements of two sinusoidal waves propagating through a string are given by the following equations y1=4sin⁡(20x−30t)y_1 = 4 \sin (20x - 30t) y2=4sin⁡(25x−40t)y_2 = 4 \sin (25x - 40t) where xx and yy are in centimeter and tt is in second. a) Calculate the phase difference between these two waves at the points x=5x = 5 cm and t=2t = 2 s. b) When these two waves interfere, what are the maximum and minimum values of the intensity?
  2. Solved Ex.6.2
    A progressive wave travels on a stretched string. A particle on this string takes 4.0 ms to move from its mean position to one of its extreme positions. The distance between two consecutive points on the string which are at their mean positions (at a certain time instant) is 2.0 cm. Find the frequency, wavelength and speed of the wave.
  3. Solved Ex.6.3
    Find the distance between two successive nodes in a stationary wave on a string vibrating with frequency 64 Hz. The velocity of progressive wave that resulted in the stationary wave is 48 m s−148\ \mathrm{m\ s}^{-1}.
  4. Solved Ex.6.4
    An air column is of length 17 cm long. Calculate the frequency of 5th overtone if the air column is (a) closed at one end and (b) open at both ends. (Velocity of sound in air =340 ms−1= 340\ \mathrm{ms}^{-1}).
  5. Solved Ex.6.5
    A closed pipe and an open pipe have the same length. Show that no mode of the closed pipe has the same wavelength as any mode of the open pipe.
  6. Solved Ex.6.6
    A string is fixed at both ends. What is the ratio of the frequency of the first harmonic to that of the second harmonic?
  7. Solved Ex.6.7
    The velocity of a transverse wave on a string of length 0.5 m is 225 m/s. (a) What is the fundamental frequency of a standing wave on this string if both ends are kept fixed? (b) While this string is vibrating in the fundamental harmonic, what is the wavelength of sound produced in air if the velocity of sound in air is 330 m/s?
  8. Solved Ex.6.8
    A string 105 cm long is fixed at one end. Transverse vibrations of frequency 15 Hz are imposed at the free end. A stationary wave, produced in the string, consists of 3 loops. Calculate the speed of progressive waves which have produced the stationary wave in the string.
  9. Solved Ex.6.9
    A sonometer wire of length 50 cm is stretched by keeping weights equivalent of 3.5 kg. The fundamental frequency of vibration is 125 Hz. Determine the linear density of the wire.
  10. Solved Ex.6.10
    Two wires of the same material and the same cross section are stretched on a sonometer in succession. Length of one wire is 60 cm and that of the other is 30 cm. An unknown load is applied to the first wire and second wire is loaded with 1.5 kg. If both the wires vibrate with the same fundamental frequencies, calculate the unknown load.
  11. Solved Ex.6.11
    A wire has linear density 4.0×10−34.0 \times 10^{-3} kg/m. It is stretched between two rigid supports with a tension of 360 N. The wire resonates at a frequency of 420 Hz and 490 Hz in two successive modes. Find the length of the wire.
  12. Solved Ex.6.12
    Two sound waves having wavelengths 81 cm and 82.5 cm produce 8 beats per second. Calculate the speed of sound in air.
  13. Solved Ex.6.13
    Two tuning forks having frequencies 320 Hz and 340 Hz are sounded together to produce sound waves. The velocity of sound in air is 326.4 m s−1326.4\ \mathrm{m\ s}^{-1}. Find the difference in wavelength of these waves.

Exercises

32 q

Choose the correct option

Practice · 5
  1. Choose the correct option.
    Ex Q.1 (i)
    When an air column in a pipe closed at one end vibrates such that three nodes are formed in it, the frequency of its vibrations is …….times the fundamental frequency.
    1. A.
      2
    2. B.
      3
    3. C.
      4
    4. D.
      5
  2. Ex Q.1 (ii)
    If two open organ pipes of length 50 cm and 51 cm sounded together produce 7 beats per second, the speed of sound is.
    1. A.
      307 m/s
    2. B.
      327 m/s
    3. C.
      350 m/s
    4. D.
      357 m/s
  3. Ex Q.1 (iii)
    The tension in a piano wire is increased by 25%. Its frequency becomes ….. times the original frequency.
    1. A.
      0.8
    2. B.
      1.12
    3. C.
      1.25
    4. D.
      1.56
  4. Ex Q.1 (iv)
    Which of the following equations represents a wave travelling along the yy-axis?
    1. A.
      x=Asin⁡(ky−ωt)x = A \sin (ky - \omega t)
    2. B.
      y=Asin⁡(kx−ωt)y = A \sin (kx - \omega t)
    3. C.
      y=Asin⁡(ky)cos⁡(ωt)y = A \sin (ky) \cos(\omega t)
    4. D.
      y=Acos⁡(ky)sin⁡(ωt)y = A \cos (ky) \sin(\omega t)
  5. Ex Q.1 (v)
    A standing wave is produced on a string fixed at one end with the other end free. The length of the string
    1. A.
      must be an odd integral multiple of λ/4\lambda/4.
    2. B.
      must be an odd integral multiple of λ/2\lambda/2.
    3. C.
      must be an odd integral multiple of λ\lambda.
    4. D.
      must be an even integral multiple of λ\lambda.

Answer in brief

Practice · 5
  1. Answer in brief.
    Ex Q.2 (i)
    A wave is represented by an equation y=Asin⁡(Bx+Ct)y = A \sin (Bx + Ct). Given that the constants A, B and C are positive, can you tell in which direction the wave is moving?
  2. Ex Q.2 (ii)
    A string is fixed at the two ends and is vibrating in its fundamental mode. It is known that the two ends will be at rest. Apart from these, is there any position on the string which can be touched so as not to disturb the motion of the string? What will be the answer to this question if the string is vibrating in its first and second overtones?
  3. Ex Q.2 (iii)
    What are harmonics and overtones?
  4. Ex Q.2 (iv)
    For a stationary wave set up in a string having both ends fixed, what is the ratio of the fundamental frequency to the second harmonic?
  5. Ex Q.2 (v)
    The amplitude of a wave is represented by y=0.2sin⁡4π[t0.08−x0.8]y = 0.2 \sin 4\pi \left[ \dfrac{t}{0.08} - \dfrac{x}{0.8} \right] in SI units. Find (a) wavelength, (b) frequency and (c) amplitude of the wave.

Solve the following

Practice · 22
  1. Ex Q.3
    State the characteristics of progressive waves.
  2. Ex Q.4
    State the characteristics of stationary waves.
  3. Ex Q.5
    Derive an expression for equation of stationary wave on a stretched string.
  4. Ex Q.6
    Find the amplitude of the resultant wave produced due to interference of two waves given as y1=A1sin⁡ωty_1 = A_1 \sin \omega t y2=A2sin⁡(ωt+φ)y_2 = A_2 \sin (\omega t + \varphi)
  5. Ex Q.7
    State the laws of vibrating strings and explain how they can be verified using a sonometer.
  6. Ex Q.8
    Show that only odd harmonics are present in the vibrations of air column in a pipe closed at one end.
  7. Ex Q.9
    Prove that all harmonics are present in the vibrations of the air column in a pipe open at both ends.
  8. Ex Q.10
    A wave of frequency 500 Hz is travelling with a speed of 350 m/s. (a) What is the phase difference between two displacements at a certain point at times 1.0 ms apart? (b) what will be the smallest distance between two points which are 45∘45^\circ out of phase at an instant of time?
  9. Ex Q.11
    A sound wave in a certain fluid medium is reflected at an obstacle to form a standing wave. The distance between two successive nodes is 3.75 cm. If the velocity of sound is 1500 m/s, find the frequency.
  10. Ex Q.12
    Two sources of sound are separated by a distance 4 m. They both emit sound with the same amplitude and frequency (330 Hz), but they are 180∘180^\circ out of phase. At what points between the two sources, will the sound intensity be maximum? (Take velocity of sound to be 330 m/s)
  11. Ex Q.13
    Two sound waves travel at a speed of 330 m/s. If their frequencies are also identical and are equal to 540 Hz, what will be the phase difference between the waves at points 3.5 m from one source and 3 m from the other if the sources are in phase?
  12. Ex Q.14
    Two wires of the same material and same cross section are stretched on a sonometer. One wire is loaded with 1.5 kg and another is loaded with 6 kg. The vibrating length of first wire is 60 cm and its fundamental frequency of vibration is the same as that of the second wire. Calculate vibrating length of the other wire.
  13. Ex Q.15
    A pipe closed at one end can produce overtones at frequencies 640 Hz, 896 Hz and 1152 Hz. Calculate the fundamental frequency.
  14. Ex Q.16
    A standing wave is produced in a tube open at both ends. The fundamental frequency is 300 Hz. What is the length of tube in the fundamental mode? (speed of the sound =340 m s−1= 340\ \mathrm{m\ s}^{-1}).
  15. Ex Q.17
    Find the fundamental, first overtone and second overtone frequencies of a pipe, open at both the ends, of length 25 cm if the speed of sound in air is 330 m/s.
  16. Ex Q.18
    A pipe open at both the ends has a fundamental frequency of 600 Hz. The first overtone of a pipe closed at one end has the same frequency as the first overtone of the open pipe. How long are the two pipes? (Take velocity of sound to be 330 m/s)
  17. Ex Q.19
    A string 1 m long is fixed at one end. Transverse vibrations of frequency 15 Hz are imposed at the free end. Due to this, a stationary wave with four complete loops, is produced on the string. Find the speed of the progressive wave which produces the stationary wave. [Hint: Remember that the free end is an antinode.]
  18. Ex Q.20
    A violin string vibrates with fundamental frequency of 440 Hz. What are the frequencies of first and second overtones?
  19. Ex Q.21
    A set of 8 tuning forks is arranged in a series of increasing order of frequencies. Each fork gives 4 beats per second with the next one and the frequency of last fork is twice that of the first. Calculate the frequencies of the first and the last fork.
  20. Ex Q.22
    A sonometer wire is stretched by tension of 40 N. It vibrates in unison with a tuning fork of frequency 384 Hz. How many numbers of beats get produced in two seconds if the tension in the wire is decreased by 1.24 N?
  21. Ex Q.23
    A sonometer wire of length 0.5 m is stretched by a weight of 5 kg. The fundamental frequency of vibration is 100 Hz. Calculate linear density of wire.
  22. Ex Q.24
    The string of a guitar is 80 cm long and has a fundamental frequency of 112 Hz. If a guitarist wishes to produce a frequency of 160 Hz, where should the person press the string?