A sinusoidal wave longitudinal or transverse is propagating through a medium in the direction of - ve x-axis- The parameters of the waves are A- - and k- The particle at x -4 executes the motion y t - A sin- t- Possible equation of the wave is A- y x - t - A sin - t - kx - - 2-B- y x - t - A sin - t - kx - - 2-C- y x - t - A sin - t - kx - - 2-D- y x - t - A sin - t - kx - - 2-

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Revisiting Standing Waves

A sinusoidal ...

Question

A sinusoidal wave (longitudinal or transverse) is propagating through a medium in the direction of $-$ ve $x -$ axis. The parameters of the waves are $A, ω$ and $k$ . The particle at $x = \frac{λ}{4}$ executes the motion $y (t) = A sin ω t$ . Possible equation of the wave is

y (x, t) = A sin [ω t - k x + (π / 2)]

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y (x, t) = A sin [ω t + k x + (π / 2)]

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y (x, t) = A sin [ω t - k x - (π / 2)]

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y (x, t) = A sin [ω t + k x - (π / 2)]

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Solution

The correct option is D $y (x, t) = A sin [ω t + k x - (π / 2)]$
Let equation of wave as it is moving along $-$ ve $x -$ axis is
$y = A sin (k x + ω t + α)$
where $k = \frac{2 π}{λ}$
But, $y (λ / 4, t) = A sin ω t$
Hence at $x = \frac{λ}{4}$
$k x + α = 0 \Rightarrow α = - k x = - \frac{2 π}{λ} * \frac{λ}{4} = - π / 2$

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