First slide

Characterstics of wave

Question

When a wave travels in a medium, displacement of a particle is given by $y = a \sin 2 π (b t - c x)$ where
a, b and c are constants. The maximum particle velocity will be twice the wave velocity, if

Moderate

A

b = ac
B

$b = \frac{1}{a c}$
C

$c = π a$
D

$c = \frac{1}{π a}$

Solution

Given, $y = a \sin 2 π (b t - c x)$
On comparing this equation with general equation
$y = r \sin (\frac{2 π t}{T} - \frac{2 π}{λ} x), we get$
$\frac{2 π}{T} = ω = 2 π b, r = a$
$λ = \frac{1}{c} and T = \frac{1}{b}$
Maximum particle velocity, $ω r = 2 π b a$
Wave velocity, $v = \frac{λ}{T} = \frac{b}{c}$
Given, maximum particle velocity = 2 x wave velocity
$2 π b a = 2 \times \frac{b}{c} \Rightarrow c = \frac{1}{π a}$

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