A wave disturbance in a medium is described b | vedclass

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Question

(A, B, D) The given equation is $y(x, t) = 0.02 \cos(50 \pi t + \frac{\pi}{2}) \cos(10 \pi x)$.Nodes occur where the spatial part $\cos(10 \pi x) = 0$

Vedclass · Accepted Answer

The wavelength of the wave is $0.2 \ m$

Vedclass · Answer

(A, B, D) The given equation is $y(x, t) = 0.02 \cos(50 \pi t + \frac{\pi}{2}) \cos(10 \pi x)$.Nodes occur where the spatial part $\cos(10 \pi x) = 0$.$10 \pi x = (n + \frac{1}{2}) \pi \implies x = \frac{n + 0.5}{10} = 0.05, 0.15, 0.25, \dots \ m$. Thus, a node occurs at $x = 0.15 \ m$.Antinodes occur where $|\cos(10 \pi x)| = 1$.$10 \pi x = n \pi \implies x = \frac{n}{10} = 0, 0.1, 0.2, 0.3, \dots \ m$. Thus, an antinode occurs at $x = 0.3 \ m$.Comparing with the standard stationary wave equation $y = A \cos(\omega t + \phi) \cos(kx)$, we get $k = 10 \pi$ and $\omega = 50 \pi$.Wavelength $\lambda = \frac{2 \pi}{k} = \frac{2 \pi}{10 \pi} = 0.2 \ m$.Wave speed $v = \frac{\omega}{k} = \frac{50 \pi}{10 \pi} = 5 \ m/s$.

$A$ wave disturbance in a medium is described by $y(x, t) = 0.02 \cos(50 \pi t + \frac{\pi}{2}) \cos(10 \pi x)$, where $x$ and $y$ are in meters and $t$ is in seconds. Which statement$(s)$ is/are correct?

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