The vibrations of a string of length `60cm` fixed at both ends are represented by the equation----------------------------
`y = 4 sin ((pix)/(15)) cos (96 pit)`
Where `x` and `y` are in `cm` and `t` in seconds.
(i) What is the maximum displacement of a point at `x = 5cm`?
(ii) Where are the nodes located along the string?
(iii) What is the velocity of the particle at `x = 7.5 cm` at `t = 0.25 sec`.?
(iv) Write down the equations of the component waves whose superpositions gives the above wave.

Explanation

The given equation is
`y = 4"sin" ((pix)/(15))cos(96pit)`
This can be written as
`y = 2 xx 2 "sin" ((2pix)/(300)) "cos" (2pi(1440)t)/(30)`
Comparing equation - (6.208) with standard equation of stationary wave, we get
This shows that A = 2 cm `lambda` = 30 cm and v = 1440cm/s.
(i) The maxium displacement is given by
`y_(max) = 4"sin"(2pix)/(30)`
(ii) For a point at x = 5 cm,
`y_(max) = 4"sin" (2pi5)/(30) = (4sqrt(3))/(2)2sqrt(3)cm`
(iii) As `lambda = 30cm` hence the nodes are located along the string at places
0,15cm, 30cm, 45cm, 60cm
The velocity of the particle is given by
`(dy)/(dt) = - 4"sin" ((pix)/(15))sin(96pit)96pi`
x = 7.5 cm and t = 0.25 sec, we have `(dy)/(dt) = 4"sin"((pi.7.5)/(15))sin(96pi xx 0.25)96 pi` = 0
The equations of component waves are
`y_(1) = A "sin"(2pi)/(lambda)(x - vt) = 2"sin"(2pi)/(30)(x - 1440t)`
`y = 2sin 2pi ((x)/(30)-48t)`
`y_(2) = A"sin" (2pi)/(lambda)(x + vt) = 2"sin"(2pi)/(30)(x div 1440t)`
`y_(2) = 2sin2pi((x)/(30) + 48t)`
Equations-(6.209) and (6.210) represent the equations of component waves superposition of which results the given stationary wave