Question

1.
Given below is function of $x\text{and}t$to represent the displacement of an elastic wave.
$\left(a\right)y=5\cos \left(4x\right)\sin \left(20t\right)$ Does this equation represent. Give reason for your answer.


$A$: a travelling wave along $-x$ direction

$B$: a stationary wave

$C$: beats

$D$: a travelling wave along $+x$ direction


2.
Given below is function of $x\text{and}t$ to represent the displacement of an elastic wave.
$\left(b\right)y=4\sin (5x-t/2)+3\cos (5x+t/2)$ Does this equation represent. Give reason for your answer.


$A$: a travelling wave along $-x$ direction

$B$: a stationary wave

$C$: beats

$D$: a travelling wave along $+x$ direction

3.
Given below is function of $x\text{and}t$ to represent the displacement of an elastic wave.
$\left(c\right)y=10\cos \left[\right(252-250\left)\pi t\right]\cos (252+250)\pi t]$
Does this equation represent. Give reason for your answer.


$A$: a travelling wave along $-x$ direction

$B$: a stationary wave

$C$: beats

$D$: a travelling wave along $+x$ direction

4.
Given below is function of $x\text{and}t$ to represent the displacement of an elastic wave.
$\left(d\right)y=100\cos \left(100\pi t\right)+0.5x$
Does this equation represent. Give reason for your answer.


$A$: a travelling wave along $-x$ direction

$B$: a stationary wave

$C$: beats

$D$: a travelling wave along $+x$ direction

Answer 1

$A$:

Stationary wave: When two progressive waves of same amplitude and wavelength travelling along a straight line in opposite directions superimpose on each other, stationary waves are formed.
Given, displacement equation,
$y=5\cos \left(4x\right)\sin \left(20t\right)$
It is the combination of two progressive wave (contain $\sin ,\cos$,terms). Therefore, it represents a stationary wave.

Final answer: b) Stationary wave

$B$:

Given, displacement equation,

$y=4\sin (5x-t/2)+3\cos (5x+t/2)$

$y=5(\frac{4}{5}\sin (5x-\frac{t}{2})+(5x-\frac{t}{2}))$

$y=5\left(\sin \right({53}^{\circ })\sin (5x-\frac{t}{2})+\cos ({53}^{\circ }\left)\cos (5x-\frac{t}{2})\right)$

$y=5\cos (5x-\frac{t}{2}-{53}^{\circ })=5\cos (\frac{t}{2}-5x+{53}^{\circ })$

As in the displacement equation $k<0$, so it represents travelling wave along $x-\text{direction}$.

Final answer: d) a travelling wave along $+x$ direction


$C$:

Beats: The pulsation caused by the combination of two waves of slightly different frequencies.
Given, displacement equation,

$y=10\cos \left[\right(252-250\left)\pi t\right]\cos \left[\right(252+250\left)\pi t\right]$

As the given equation involves sum and difference of two nearby frequencies (i.e., 252,250), so it represent beats formation.
Final answer: c) Beats


$D$:

Given, displacement equation,
$y=100\cos \left(100\pi t\right)+0.5x$
In the given equation $k>0$, so it represents travelling wave along negative $x-\text{direction}$.

Final answer: a) a travelling wave along $-x\text{direction}$.