Question

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

Answer 1

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:
$\Delta \varphi = \dfrac{{2\pi }}{\lambda } \times \Delta x$, where $\Delta x =$path difference, and $K = \dfrac{{2\pi }}{\lambda }$.

Complete answer:
According to the question the equation for a progressive wave is $y = A\cos 240\left( {t - \dfrac{x}{{12}}} \right)$, where $t$ is the time in second $x$ is the distance in meter.
So, we can also write the above equation as follows,
$\Rightarrow y = A\cos \left( {240t - 240 \times \dfrac{x}{{12}}} \right)$
$\Rightarrow y = A\cos \left( {240t - 20x} \right)$-------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,
$\Delta \varphi = \dfrac{{2\pi }}{\lambda } \times \Delta x$------equation (3), where $\Delta x =$path difference.
Here it is given in the question, the path difference = $\Delta x = 0.5$ and we know that the $K = \dfrac{{2\pi }}{\lambda } = 20$, So putting the values in the equation (3), we get
$\Rightarrow \Delta \varphi = K \times \Delta x$
$\Rightarrow \Delta \varphi = K \times \Delta x$
$\Rightarrow \Delta \varphi = 20 \times 0.5$
$\Rightarrow \Delta \varphi = 10$
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.