Question
A wave in a string has an amplitude of $$2\,cm.$$ The wave travels in the positive direction of $$x$$-axis with a speed of $$128\,m{s^{ - 1}}$$ and it is noted that 5 complete waves fit in $$4\,m$$ length of the string. The equation describing the wave is
A.
$$y = \left( {0.02} \right)m\sin \left( {7.85\,x + 1005\,t} \right)$$
B.
$$y = \left( {0.02} \right)m\sin \left( {15.7\,x - 2010\,t} \right)$$
C.
$$y = \left( {0.02} \right)m\sin \left( {15.7\,x + 2010\,t} \right)$$
D.
$$y = \left( {0.02} \right)m\sin \left( {7.85\,x - 1005\,t} \right)$$
Answer :
$$y = \left( {0.02} \right)m\sin \left( {7.85\,x - 1005\,t} \right)$$
Solution :
Given, amplitude of wave, $$A = 2\,cm$$
direction $$= +ve\,\,x$$ direction
Velocity of wave
$$v = 128\,m{s^{ - 1}}$$
and length of string, $$5\lambda = 4$$
$$\eqalign{
& {\text{We know that,}}\,\, \cr
& k = \frac{{2\pi }}{\lambda } = \frac{{2\pi \times 5}}{4} = 7.85 \cr
& {\text{and}}\,\,v = \frac{\omega }{k} = 128\,m{s^{ - 1}} \cr
& \left[ {\omega = {\text{Angular frequency}}} \right] \cr
& \Rightarrow \omega = v \times k = 128 \times 7.85 = 1005 \cr} $$
As, the wave travelling towards $$+ x$$ -axis is given by
$$\eqalign{
& y = A\sin \left( {kx - \omega t} \right) \cr
& {\text{So,}}\,y = 2\sin \left( {7.85\,x - 1005\,t} \right) \cr
& y = \left( {0.02} \right)m\sin \left( {7.85\,x - 1005\,t} \right) \cr} $$