Question

A Wave is represented by an equation; $Y=Acoskxsin\omega t,$ then

• A. it is a progressive wave with amplitude A
• B. it is a progressive wave with amplitude A cos kx
• C. it is a stationary wave with amplitude A
• D. it is a stationary wave with amplitude A cos kx

Answer 1

Answer: (D)

Consider two travelling waves 1 and 2 .
Let the displacements at time $t$ and position $x$ be $y_1$ and $y_2$.
$y_1=a\sin (t-kx)$ (say right-left)
$y_2=a\sin (t+kx)$ (say left- right)
Therefore:
$y_1+y_2=a\sin (wt-kx)+a\sin (t+kx)=2a\sin (t)\cdot \cos (kx)=A\sin (t)$
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 $x=0,\frac{I}{2}\dots A$ is a maximum and we have an antinode;
When $x=\frac{I}{4},\frac{3l}{4},\frac{5l}{4}\dots A$ is a minimum and we have a node.
Notice that the maximum value of $A$ is $2a$.