Question

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?

Answer 1

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

\[The  displacement  is  given  by\] 

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

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

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