Question

The plots of $\xi ,\partial \xi /\partial t$ and $\partial \xi /\partial x,x$:

Answer 1

Hint: In this question, we will use the formula of displacement of a wave. This will help us to get the required result and thereby, we will be able to plot the graphs. Further, we will discuss the basics of a wave and also, propagation of two or more waves, for our better understanding.

Formula used:
 $y = A\cos (\omega t - kx)$

Complete step-by-step answer:
Here, we have the equation:
$\xi = a\cos (\omega t - kx)$
At t=0, the above equation becomes:
$\xi = a\cos (kx)$
Now, by differentiating this equation with respect to t, we get:
$\dfrac{{\partial \xi }}{{\partial t}} = - a\omega \sin (\omega t - kx)$
 At t=0, the above equation becomes:
$\dfrac{{\partial \xi }}{{\partial t}} = a\omega \sin (kx)$
Similarly, by differentiating the equation with respect to x, we get:
$\dfrac{{\partial \xi }}{{\partial x}} = ak\sin (\omega t - kx)$
At t=0, the above equation becomes:
$\dfrac{{\partial \xi }}{{\partial t}} = - ak\sin (kx)$

Therefore, this shows that all the plots will be similar but all the plots will have different amplitude.

Additional Information: Waves involve the transfer of energy without the transfer of the matter. So, it can be said that waves can be described as a disturbance that travels through a medium, transporting energy from one location to another location 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.
Two waves are said to be coherent if they are moving with the same frequency and have constant phase difference.
The summation or adding or subtraction of all the waves travelling in a particular medium, gives us the superposition of waves. If the direction or amplitude of the waves are opposite then the superposition of waves are calculated by subtracting the waves, whereas if the two waves are travelling in the same direction or have the same amplitude the resultant is given by adding up the two or more waves.
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.
Phase of a wave specifies the location of a point within a wave cycle of a repetitive waveform. Generally, the phase differences between two or more sound waves are important. When two sound waves combine, like- the difference between the phases of the two waves is important in determining the resulting waveform.

Note: The phase of the wave can be positive or negative depending on its direction of propagation. A sine wave starts from zero, 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.