Superposition and stationary waves

Sample questions

Original practice questions written in the style of the exam.

1. Question 14 marks

   A string of vibrating length $0.80\,\text{m}$ is fixed at both ends. Its fundamental (first harmonic) frequency is $120\,\text{Hz}$. The mass per unit length of the string is $4.0 \times 10^{-3}\,\text{kg m}^{-1}$.

   (a) Calculate the speed of waves on the string.

   (b) Calculate the tension in the string.

   (c) State the frequency of the third harmonic.

   Mark scheme

   • Fundamental: $\lambda = 2L = 1.60\,\text{m}$ (1)
   • $v = f\lambda = 120 \times 1.60 = 192\,\text{m s}^{-1}$ (1)
2. Question 26 marks

   A string is stretched between a vibration generator and a fixed pulley. When the frequency of the generator is adjusted, a stationary wave with three loops is seen on the string.

   Explain how the stationary wave is formed, and describe how it differs from a progressive wave.

   Mark scheme

   Indicative content:

   • Waves travel along the string and are reflected at the fixed end (pulley) (1)
   • The incident and reflected waves have the same frequency and (similar) amplitude and travel in opposite directions (1)
   • They superpose / interfere (1)
   • Nodes form where the waves are always in antiphase (destructive interference), giving zero amplitude; antinodes form where they are in phase, giving maximum amplitude (1)
   • A stationary wave does not transfer energy along the string, whereas a progressive wave does (1)
   • In a stationary wave all points between adjacent nodes oscillate in phase with different amplitudes; in a progressive wave all points have the same amplitude but phase varies continuously with position (1)

   Award 1 mark per relevant point, max 6.