The vibrations of a string of length 60 cm fixed at both ends are represented by the equation. `y=4 sin [(pi x)/(15)] cos(96 pi t)` where x and y are in cm and t in sec.
(a) What is the maximum displacement al r = 5 cm?
(b) What are the nodes located along the string ?
(c) What is the velocity of the particle at `x=7.5 and t=0.25 s ?`
(d) Write down the equations of component waves whose superposition gives the above wave.

Explanation

For `x=5 cm, y=4 sin (5pi//15) cos(96 pi t) or y=2 sqrt(3) cos (96 pit)`
So y will be maximum when `cos(96 pit)=1.,e (y_(max))_(x=5)=2 sqrt(3)cm`.,
(b) At nodes amplitude of wave is zero
`4 sin [(pix)/(15)]=0 or (pix)/(15)=0, pi, 2pi, 3pi....`
So `x=0,15,30,45,60` cm [ as length of string =60 cm]
(c) As `y= 4 sin (pix//15) cos (96 pit)`
`(dy)/(dt)=-4 sin [(pix)/(15)][ sin (96 pit)xx(96 pi)`
So the velocity of the particle at x = 7.5 cm and `t=0.25 s, v_(pa)=-384pi sin(7.5 pi//15) sin (96 pi xx0.25)`
`v_(pa)=-384pi xx 1 xx 0=0`
(d) `y=y_(1)+y_(2) "with" y_(1)= 2in [96pit+(pix)/(15)]`
`y_(2)=-2isn [96 pi t-(pix)/(15)]`