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Answer
i. Consider two simple harmonic progressive waves of equal amplitudes (a) and wavelength $(\lambda)$ propagating on a long uniform string in opposite directions.
ii. The equation of wave travelling along the $X$-axis in the positive direction is given by,
$y_1=a \sin \left[2 \pi\left(n t-\frac{x}{\lambda}\right)\right]$
The equation of wave travelling along the $X$-axis in the negative direction is given by,
$y_2=a \sin \left[2 \pi\left(n t+\frac{x}{\lambda}\right)\right]$
iii. When these waves interfere, the resultant displacement of particles of the string is given by the principle of superposition of waves as $y$
$ = y _1+ y _2$
$\therefore y = a \sin \left[2 \pi\left(n t-\frac{x}{\lambda}\right)\right]+a \sin \left[2 \pi\left(n t+\frac{x}{\lambda}\right)\right] $
iv. By using trigonometry formula,
$ \sin C +\sin D =2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$
$y =2 a \sin (2 \pi nt ) \cos \frac{2 \pi x}{\lambda}$
$y =2 a \cos \frac{2 \pi x}{\lambda} \sin (2 \pi nt ) \ldots .(1) $
v. Substituting $2 a \cos \frac{2 \pi x}{\lambda}= A$ in equation (1),
$ y=A \sin (2 \pi n t)$
$\therefore y=A \sin \omega t \ldots .(\therefore \omega=2 \pi n) $
This is the equation of a stationary wave which gives resultant displacement due to two simple harmonic progressive waves.
vi. Nodes: The points of a medium, which vibrate with minimum amplitude are called nodes.
Condition for node: Amplitude is minimum, i.e., $A=0$.
$\therefore 2 a \cos \frac{2 \pi x}{\lambda}=0$
$\therefore \cos \frac{2 \pi x}{\lambda}=0$
$\therefore \frac{2 \pi x}{\lambda}=\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{5 \pi}{2}, \ldots \ldots \ldots$
$\therefore x =\frac{\lambda}{4}, \frac{3 \lambda}{4}, \frac{5 \lambda}{4}, \ldots \ldots$
i.e., $x=(2 p-1) \frac{\lambda}{4}$ where $p=1,2,3, \ldots$
Distance between two successive nodes:
$ \frac{3 \lambda}{4}-\frac{\lambda}{4}=\frac{\lambda}{2}$
$\frac{5 \lambda}{4}-\frac{3 \lambda}{4}=\frac{\lambda}{2} $
$\therefore$ Distance between two successive nodes is $\frac{\lambda}{2}$
vii. Antinodes: The points of a medium, which vibrate with maximum amplitude are called antinodes. Condition for antinode:
Amplitude is maximum, i.e., $A = \pm 2 a$
$\therefore 2 a \cos \frac{2 \pi x}{\lambda}= \pm 2 a$
$\therefore \cos \frac{2 \pi x}{\lambda}= \pm 1$
$\therefore \frac{2 \pi x}{\lambda}=0, \pi, 2 \pi, 3 \pi, \ldots$.
$\therefore x =0, \frac{\lambda}{2}, \lambda, \frac{3 \lambda}{2}, \ldots \ldots \ldots$
i.e., $x =\frac{\lambda p}{2}$ where $p =0,1,2,3, \ldots$
$\therefore$ Distance between two successive antinodes:
$\frac{\lambda}{2}-0=\frac{\lambda}{2}$
$\lambda-\frac{\lambda}{2}=\frac{\lambda}{2}$
$\therefore$ Distance between two successive antinodes is $\frac{\lambda}{2}$.
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