Question

Illustration 8.A progressive wave having amplitude 5 m and wavelength 3 m. If the maximum average velocity of particlhalf time period is 5 m/s and wave is moving in the positive x-direction then find which may be the comequation(s) of the wave? (where x in meter)(1) 5sin ( 94 - 21 x)(2) 4 sin nyt 29 x) + 3 cos (21 x)(3) 5sin ( )() scola) -4.sin( a , 2 )SolutionAns. (2,​

Answer 1

The equation is $5\sin(\dfrac{\pi}{2}t-\dfrac{2\pi}{3}x)$

Explanation:

Given that,

Amplitude = 5 m

Wavelength = 3 m

Average velocity = 5 m/s

We need to calculate the maximum displacement

Maximum displacement in half time period is

$y=2a=2\times5$

$y=10$

We need to calculate the wave number

Using formula of wave number

$k=\dfrac{2\pi}{\lambda}$

Put the value into the formula

$k=\dfrac{2\pi}{3}$

We need to calculate the time period

Using formula of period

$v=\dfrac{y}{t}$

$\dfrac{T}{2}=\dfrac{y}{v}$

Put the value into the formula

$\dfrac{T}{2}=\dfrac{10}{5}$

$T=4\ sec$

We need to calculate the angular frequency

Using formula of angular frequency

$\omega=\dfrac{2\pi}{T}$

Put the value into the formula

$\omega=\dfrac{2\pi}{4}$

$\omega=\dfrac{\pi}{2}$

The wave is moving in the positive x-direction then

The general equation is

$y=A\sin(\omega t-kx)$

Put the value in the equation

$y=5\sin(\dfrac{\pi}{2}t-\dfrac{2\pi}{3}x)$

Hence, The equation is $5\sin(\dfrac{\pi}{2}t-\dfrac{2\pi}{3}x)$

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Topic : wave equation

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