Question

Which of the following represents a standing wave?

$\text{y}=\text{A}\sin (\omega \text{t}-\text{kx})$
$\text{y}=\text{A}{\text{e}}^{-\text{bx}}\sin (\omega \text{t}-\text{kx}+\alpha )$
$\text{y}=\text{A}\sin (\text{kx})\cos (\omega \text{t})$
$\text{y}=(\text{ax}+\text{b})\sin (\omega \text{t}-\text{kx})$

Answer 1

Hint:Michael Faraday was the first to detect standing waves in 1831. Standing waves on the surface of a liquid in a vibrating container were discovered by Faraday. Around 1860, Franz Melde created the name "standing wave" and proved the phenomena with vibrating strings in his classic experiment.

Complete step-by-step solution:
A standing wave, also known as a stationary wave, is a wave that oscillates in time but does not move in space due to its peak amplitude profile. The wave oscillations' peak amplitude is constant with time at any place in space, and the oscillations at various locations throughout the wave are in phase. Nodes are the areas where the absolute value of the amplitude is lowest, while antinodes are the sites where the absolute value of the amplitude is highest. This phenomena can occur in a stationary medium as a result of interference between two waves going in opposing directions, or it can occur when the medium moves in the opposite direction of the wave. Standing waves are most commonly caused by resonance, which occurs within a resonator as a result of interference between waves reflected back and forth at the resonator's resonant frequency. There is no net energy propagation for waves of equal amplitude flowing in opposite directions.
Consider a string that is infinitely long along the x-axis and may be extended transversely in the y direction.
The displacement of the string in the y direction as a function of position x and time t for a harmonic wave going to the right along the string is
$y_{\text{R}}(x,t)=y_{\text{max}}\sin \left({\displaystyle \frac{2\pi x}{\lambda }}-\omega t\right)$
For an identical harmonic wave going to the left, the displacement in the y-direction is
$$y_{\text{L}}(x,t)=y_{\text{max}}\sin \left({\displaystyle \frac{2\pi x}{\lambda }}+\omega t\right),$$
The overall displacement of the string is equal to the sum of $y_R$and$y_L$ for identical right- and left-traveling waves on the same string.
$$y(x,t)=y_{\text{R}}+y_{\text{L}}=y_{\text{max}}\sin \left({\displaystyle \frac{2\pi x}{\lambda }}-\omega t\right)+y_{\text{max}}\sin \left({\displaystyle \frac{2\pi x}{\lambda }}+\omega t\right).$$
Now Using the trigonometric sum-to-product identity $\sin a+\sin b=2\sin \left({\displaystyle \frac{a+b}{2}}\right)\cos \left({\displaystyle \frac{a-b}{2}}\right)$
$$y(x,t)=2y_{\text{max}}\sin \left({\displaystyle \frac{2\pi x}{\lambda }}\right)\cos (\omega t).$$
i.e, $\text{y}=\text{A}\sin (\text{kx})\cos (\omega \text{t})$
Hence $\text{y}=\text{A}\sin (\text{kx})\cos (\omega \text{t})$ option C is correct.

Note:In two- and three-dimensional resonators, standing waves can also exist. Standing waves on two-dimensional membranes, such as drumheads, result in nodal lines, which are lines on the surface that have no movement and divide areas vibrating in opposite phase, as shown in the videos above. Chladni figures are nodal line patterns with several nodes. There are nodal surfaces in three-dimensional resonators like musical instrument sound boxes and microwave cavity resonators. To demonstrate how to expand the concept to higher dimensions, this section presents a two-dimensional standing wave example with a rectangular border.