A wave travelling on a string at a speed of 10 m s−1 causes each particle of the string to oscillate with a time period of 20 ms. (a) What is the wavelength of the wave? (b) If the displacement of a particle of 1⋅5 mm at a certain instant, what will be the displacement of a particle 10 cm away from it at the same instant?
Solution
Given,
Wave speed (v) = 10 ms−1
Time period (T) = 20 ms
\[= 20 \times {10}^{- 3} = 2 \times {10}^{- 2} s\]
(a) Wavelength of the wave:
\[\lambda = \nu t = 10 \times 2 \times {10}^{- 2} \]
\[ = 0 . 02 m = 20 cm\]
(b) Displacement of the particle at a certain instant:
\[y = a\sin\left( \omega t - kx \right)\]
\[ \Rightarrow 1 . 5 = a\sin\left( \omega t - kx \right)\]
Phase difference of the particle at a distance x = 10 cm:
\[\phi = \frac{2\pi x}{\lambda} = 2\pi \times \frac{10}{20} = \pi\]
\[The displacement is given by\]
\[ y' = a\sin\left( \omega t - kx + \pi \right)\]
\[ = a\sin\left( \omega t - kx \right) = 1 . 5 mm\]
\[ \therefore Displacement = 1 . 5 \text{ mm }\]
