4 The rope shown at an instant is carrying a wave travelling towards right, created by a source vibrating at…

4 The rope shown at an instant is carrying a wave travelling towards right, created by a source vibrating at a frequency n. [figure] Choose the correct statement(s). Schematic of a rope at an instant with a travelling wave (to the right). Horizontal axis with points labeled a and b on left hump, vertical dashed line at point c (center), and points d and e on right hump; two successive crests shown.
  1. (a) The speed of the wave is 4n × ab
  2. (b) The phase difference between b and e is $\tfrac{3\pi}{2}$
  3. (c) Both (a) and (b) are correct
  4. (d) Neither (a) nor (b) are incorrect

Solution

Given the geometry $ab=\dfrac{\lambda}{4}$ so a path between points b and e is $\dfrac{3\lambda}{4}$. Phase difference $\Delta\phi=\dfrac{2\pi}{\lambda}\times{\text{path difference}}=\dfrac{2\pi}{\lambda}\times\dfrac{3\lambda}{4}=\dfrac{3\pi}{2}$. (Using $v=n\lambda$ and $ab=\lambda/4$ as given leads to the stated phase difference.)

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