First slide

Equation of wave

Question

The displacement y of a particle in a medium can be expressed as $y = 10^{- 6} \sin (100 t + 20 x + \frac{x}{4})$ , where
t is in second and x in metre. The speed of the wave is

Moderate

A

2000 ${ms}^{- 1}$
B

5 ${ms}^{- 1}$
C

20 ${ms}^{- 1}$
D

5 $π$ ${ms}^{- 1}$

Solution

The displacement is given by
$y = 10^{- 6} \sin (100 t + 20 x + \frac{π}{4})$
Comparing with standard equation $y = A \sin (ω t + k x + ϕ)$
we get $ω = 100, k = 20$
where, $ω$ is angular frequency and k is angular wave number.
Now, $k = \frac{2 π}{λ} = \frac{2 π v}{λ v} = \frac{ω}{v} = 20$
$\Rightarrow v = \frac{ω}{20} = \frac{100}{20} = 5 {ms}^{- 1}$
So, the speed of the wave is 5 ms^-1.

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