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Q.
A Wave is represented by an equation; $Y=A\,cos\,kx\,sin\omega t,$ then
Waves
Solution:
Consider two travelling waves 1 and 2 .
Let the displacements at time $t$ and position $x$ be $y_{1}$ and $y_{2}$.
$y_{1}=a \sin (t-k x)$ (say right-left)
$y_{2}=a \sin (t+k x)$ (say left- right)
Therefore:
$y_{1}+y_{2}=a \sin (w t-k x)+a \sin (t+k x)=2 a \sin (t) \cdot \cos (k x)=A \sin (t)$
Note that this expression is composed of two terms:
(a) $\sin ( t )$ - this shows a varying amplitude with time at a particular place.
(b) $\cos ( kx )$ - this shows a varying amplitude with position at a particular time.
When $x=0, \frac{I}{2} \ldots A$ is a maximum and we have an antinode;
When $x =\frac{ I }{4}, \frac{3 l }{4}, \frac{5 l }{4} \ldots A$ is a minimum and we have a node.
Notice that the maximum value of $A$ is $2 a$.