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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?
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Solution
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.)
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