Question

A wave is described by the equation \[y = \left( 1 \cdot 0  mm \right)  \sin  \pi\left( \frac{x}{2 \cdot 0  cm} - \frac{t}{0 \cdot 01  s} \right) .\] 
(a) Find the time period and the wavelength? (b) Write the equation for the velocity of the particles. Find the speed of the particle at x = 1⋅0 cm at time t = 0⋅01 s. (c) What are the speeds of the particles at x = 3⋅0 cm, 5⋅0 cm and 7⋅0 cm at t = 0⋅01 s?
(d) What are the speeds of the particles at x = 1⋅0 cm at t = 0⋅011, 0⋅012, and 0⋅013 s?

Answer 1

The wave equation is represented by \[y = \left( 1 \cdot 0  mm \right)  \sin  \pi\left( \frac{x}{2 \cdot 0  cm} - \frac{t}{0 \cdot 01  s} \right)\]
Let:
Time period = T
Wavelength = λ

\[\left( a \right)  T = 2 \times 0 . 01 = 0 . 02  s = 20  ms\] 

\[ \lambda = 2 \times 2 = 4  cm\]
(b) Equation for the velocity of the particle:

\[=  - 0 . 50  \cos  2\pi\left\{ \left( \frac{x}{4} \right) - \left( \frac{t}{0 . 02} \right) \right\} \times \frac{1}{0 . 02}\] 

\[ \Rightarrow \nu =  - 0 . 50  \cos  2\pi  \left\{ \left( \frac{x}{4} \right) - \left( \frac{t}{0 . 02} \right) \right\}\] 

\[At  x = 1  \text{ and }  t = 0 . 01  s,   \nu =  - 0 . 50  \cos  2\pi  \left\{ \frac{1}{4} - \frac{1}{2} \right\} = 0 .\]

(c) (i) Speed of the particle:
\[At  x = 3  cm  \text{ and }  t = 0 . 01  s,\] 
\[\nu =  - 0 . 50\cos2\pi\left\{ \frac{3}{4} - \frac{1}{2} \right\} = 0\] 
(ii) \[\text{ At } x = 5  cm  \text{ and }  t = 0 . 01  s,\] 

\[\nu = 0  \] 

\[\left( iii \right)  At    x = 7  cm  \text{ and } t = 0 . 1  s,   \nu = 0 . \] 

\[\] 

\[\left( iv \right)  At  x = 1  cm  \text{ and }  t = 0 . 011  s, \] 

\[\nu = 50  \cos  2\pi\left\{ \left( \frac{1}{4} \right) - \left( \frac{0 . 011}{0 . 02} \right) \right\}\] 

\[     =  - 50  \cos  \left( \frac{3\pi}{5} \right) =  - 9 . 7  cm/s\]
(By changing the value of t, the other two can be calculated.)