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:

\[\phi = \frac{2\pi x}{\lambda} = 2\pi \times \frac{10}{20} = \pi\]

\[The displacement is given by\]

\[ y' = a\sin\left( \omega t - kx + \pi \right)\]

\[ = a\sin\left( \omega t - kx \right) = 1 . 5 mm\]

\[ \therefore Displacement = 1 . 5 \text{ mm }\]