Question

The equation of a progressive wave where $$t$$ is the time in second $x$ is the distance in meter is $y=A\cos 240\left(t-{\displaystyle \frac{x}{12}}\right)$. The phase difference (in SI unit) between two position $0.5m$ apart is
A.40
B.20
C.10
D.5

Answer 1

Hint: We will use the definition of phase difference in this question. The lateral difference between two or more waveforms along a common axis and the same frequency sinusoidal waveforms is known as phase difference. The phase differential equation would therefore be: $y=A\cos \left(\omega t-Kx\right)$,where $k=$propagation wave vector, $\omega =$angular frequency, $t=$time, $x=$position vector, $A=$maximum amplitude.
Formula used:
$\mathrm{\Delta }\phi ={\displaystyle \frac{2\pi }{\lambda }}\times \mathrm{\Delta }x$, where $\mathrm{\Delta }x=$path difference, and $K={\displaystyle \frac{2\pi }{\lambda }}$.

Complete answer:
According to the question the equation for a progressive wave is $y=A\cos 240\left(t-{\displaystyle \frac{x}{12}}\right)$, where $$t$$ is the time in second $x$ is the distance in meter.
So, we can also write the above equation as follows,
$\Rightarrow y=A\cos \left(240t-240\times {\displaystyle \frac{x}{12}}\right)$
$\Rightarrow y=A\cos \left(240t-20x\right)$-------equation (1)
Now if we see the standard equation for the progressive wave, which is as
$y=A\cos \left(\omega t-Kx\right)$---------equation (2)
On comparing the equation (1) and equation (2), we see that
$\omega =240$and $K=20$
Now we have to find phase difference, formula for the phase difference is,
$\mathrm{\Delta }\phi ={\displaystyle \frac{2\pi }{\lambda }}\times \mathrm{\Delta }x$------equation (3), where $\mathrm{\Delta }x=$path difference.
Here it is given in the question, the path difference = $\mathrm{\Delta }x=0.5$ and we know that the $K={\displaystyle \frac{2\pi }{\lambda }}=20$, So putting the values in the equation (3), we get
$\Rightarrow \mathrm{\Delta }\phi =K\times \mathrm{\Delta }x$
$\Rightarrow \mathrm{\Delta }\phi =K\times \mathrm{\Delta }x$
$\Rightarrow \mathrm{\Delta }\phi =20\times 0.5$
$\Rightarrow \mathrm{\Delta }\phi =10$
Hence the phase difference (in SI unit) between two positions $0.5m$ apart is $10$.

So, option (C) is the correct answer.

Note:
In these types of questions it is best to consider the basic concepts such as progressive waves, i.e. a wave that moves in the same direction continuously in a medium without the change in its amplitude. Let's take one example on a string of a progressive wave. So, here we'll define the displacement relationship of any element on the string as a function of time, and the vibration of the string elements along the length at a given time.