Principle of Superposition - UNSOLVED PRACTICE SET

Q7. State the principle of superposition of waves. Under what conditions is it valid?

Q8. Two waves of the same frequency and amplitude but differing in phase by π rad interfere. What is the amplitude of the resultant wave? Explain.

Q9. Two waves y₁ = A sin(kx – ωt) and y₂ = A sin(kx – ωt + φ) superpose. Write the expression for the resultant wave using the principle of superposition.

Q10. Explain the difference between constructive interference and destructive interference with one example of each.

Q11. Three waves of equal amplitude A and same frequency meet at a point with phases 0, 2π/3, and 4π/3. What is the resultant amplitude? Show your calculation.

Q12. Why does the principle of superposition not apply to very intense light waves or very large amplitude water waves?

Q13. State and explain the principle of superposition of waves. Using this principle, derive the expression for the resultant wave when two sinusoidal waves of the same frequency and amplitude but different phases interfere:

(i) Show that the resultant is also a sinusoidal wave

(ii) Derive the expression for resultant amplitude

(iii) Discuss the conditions for constructive and destructive interference

(iv) Explain the phenomenon of beats as an application of superposition

Q14. Two waves traveling in the same direction are described by:

y₁ = 0.03 sin(2πx – 400πt)

y₂ = 0.03 sin(2πx – 400πt + π/3)

(i) Using the principle of superposition, find the resultant wave.

(ii) Calculate the amplitude of the resultant wave.

(iii) Calculate the phase of the resultant wave relative to y₁.

(iv) What would be the resultant amplitude if the phase difference were π?

Q15. A student sets up two speakers in the school auditorium connected to the same audio source. She walks across the room and notices that the sound is loud at some places and faint at others.

(i) Explain this observation using the principle of superposition.

(ii) What is this phenomenon called?

(iii) How would the pattern change if the frequency of the sound were doubled?

(iv) What would happen if the two speakers were connected to different audio sources?

Q16. Two waves traveling in the same direction are given by:

y₁ = 0.04 sin(5x – 200t)

y₂ = 0.04 sin(5x – 200t + π/4)

(i) Write the equation of the resultant wave.

(ii) Calculate the amplitude of the resultant wave.

(iii) Calculate the phase angle of the resultant wave.

(iv) Calculate the ratio of maximum intensity to minimum intensity if these waves were to interfere with a phase difference varying from 0 to 2π.

(v) At what phase difference would the intensity be half the maximum intensity?

Q17. Three waves of the same frequency meet at a point:

y₁ = 0.02 sin(ωt)

y₂ = 0.02 sin(ωt + π/3)

y₃ = 0.02 sin(ωt + 2π/3)

(i) Find the resultant wave using the principle of superposition.

(ii) Calculate the amplitude of the resultant wave.

(iii) Calculate the phase of the resultant wave.

(iv) What would be the resultant amplitude if y₃ had phase 4π/3 instead of 2π/3?

(v) Generalize: what is the condition for N waves of equal amplitude and equally spaced phases to produce zero resultant amplitude?

Q18. In a school science exhibition, a student creates a "wave interference" demonstration using two identical ripple tank vibrators placed 10 cm apart, oscillating at 20 Hz. The speed of water waves is 0.4 m/s.

(i) Calculate the wavelength of the water waves.

(ii) Explain why there are regions of calm water and regions of large amplitude.

(iii) Calculate the distance between two consecutive regions of maximum amplitude along the line joining the two sources.

(iv) If the student increases the frequency to 30 Hz, how does the interference pattern change?

(v) The student places a barrier between the two sources. Explain why the interference pattern disappears.