Question

Equation of a stationary wave is given by:
A. $y = A\sin (\omega t - kx)$
B. $y = 2A\sin kx.\cos \omega t$
C. $y = A\cos 2\pi (kx - t/T)$
D. $y = A\cos (2\pi t/I)$

Answer 1

In this question, first we will know the basic definition of a stationary wave. Using the basic definition, we can answer this question. Further, we will study the basics of a wave and also, we will see the superposition of two or more waves, for our better understanding.

Complete answer:
As we know a standing wave can also be termed as a stationary wave. A stationary wave is a wave which oscillates in time but its peak amplitude value does not move in space. So, this peak amplitude of the wave oscillations at any point in space is constant with time. Also, the oscillations of the stationary wave at different points throughout the wave are in phase.
When we see the given options, in the equation:
$y = 2A\sin kx.\cos \omega t$ , here the amplitude $2A\sin kx$ is changing with x. So, this is the equation of stationary waves.
Whereas, in rest of the given equations amplitude is A which is fixed. So, all other equations are equations of travelling waves.
Therefore, the correct option is B) i.e., $y = 2A\sin kx.\cos \omega t$

Additional information:
As we know, a wave involves the transfer of energy without the transfer of the matter. So, we can say that a wave can be defined as a disturbance that travels through a medium, transporting energy from one point to another point without transfer of matter.
Further, the frequency is defined as the number of waves that pass a fixed point in unit time. It can also be defined as the number of cycles or vibrations undergone during one unit of time. The S.I unit of frequency is Hertz or Hz and the unit of wavelength is meter or m. Furthermore we also know the S.I unit of time which is given by second or s.
Also, we know that two waves are said to be coherent if they are moving with the same frequency and have constant phase difference.
When the two or more waves are propagating, the summation or adding or subtraction of all these waves travelling in a particular medium, will give us the superposition of waves. If the direction or amplitude of these waves are opposite then the superposition of these waves are calculated by subtracting the waves. If these two or more waves are travelling in the same direction or they have the same amplitude the resultant is given by adding up the two or more waves.
Also, the phase of a wave gives us the location of a point within a wave cycle of a repetitive waveform.

Note:
We should remember that, depending on the direction of propagation. The phase of the wave can be positive or negative. Also, a sine wave starts from zero point, whereas the cosine wave starts from one. A wave which has the same amplitude but opposite orientation will cancel out each other and thereby give zero output.