Question

A longitudinal wave is represented by x = 10 sin 2π `("nt" - x/lambda)` cm. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to ______.

पर्याय

•  2π

• 5π

• π

• `(5pi)/2`

Answer 1

A longitudinal wave is represented by x = 10 sin 2π `("nt" - x/lambda)` cm. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to 5π.

Explanation:

y = 10 sin 2π `("nt" - x/lambda)`

y = 10 sin `(2pi"nt" - (2pi)/lambdax)`

On comparing it with standard equation of wave

⇒ ω 2pn and k = `(2pi)/lambda`

Given : Maximum particle velocity = 4 × wave velocity

In wave propagation, medium particle oscillates and maximum velocity of oscillating particle is equal to Aω 

Where, A ⇒ Amplitude

ω ⇒ Angular frequency

⇒ Wave velocity is given by V = `ω/"k"` where, k is angular wave number and k = `(2pi)/lambda`

V_wave = `(2pin)/(2pilambda)` = nλ

`"V"_("particle")` = 4 V_wave 

Aω = 4nλ

A × 2πn = 4nλ

A × 2 πn = 4nλ

λ = `(2pi"A")/4 = (pi"A")/2`

As,  A = 10

⇒ λ = `(pixx10)/2` 

= 5π