Question

A wave disturbance in a medium is described by $y(x,t)=0.02\cos \left(10\pi x\right)\cos (50\pi t+\frac{\pi }{2})$, where $x$ and $y$ are in $\text{meter}$ and $t$ in $\text{seconds}$. Identify the incorrect option.

• A. Node occurs at $x=0.15\text{m}$
• B. Antinode occurs at $x=0.25\text{m}$
• C. Wavelength of the constituent waves is $0.2\text{m}$
• D. Speed of the constituent waves is $5\text{m/s}$

Answer 1

The correct option is B Antinode occurs at $x=0.25\text{m}$

For the stationary wave represented by, $y=0.02\text{cos}\left(10\pi x\right)\cos (50\pi t+\frac{\pi }{2})$
Considering a snapshot of string at some given instant of time, i.e., $t=$ fixed
Nodes will be formed where displacement of particle is zero.
It will happen when,
$\cos \left(10\pi x\right)=0$
$\Rightarrow 10\pi x=\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},......$
$\Rightarrow x=0.05,0.15,0.25,....$
So, nodes occurs at $x=0.15\text{m}$ and $x=0.25\text{m}$.

On comparing given equation with standard equation of standing wave,
$y=2A\cos \left(kx\right)\cos (\omega t+\varphi )$,
Wavenumber, $k=10\pi {\text{m}}^{-1}$
Angular frequency, $\omega =50\pi {\text{s}}^{-1}$
So, Velocity of wave,
$v=\frac{\omega }{k}=\frac{50\pi }{10\pi }=5\text{m/s}$
Also, wavelength,
$\lambda =\frac{2\pi }{k}=\frac{2\pi }{10\pi }=0.2\text{m}$