One end of a long string of linear mass dnesity `8.0xx10^(-3)kgm^(-1)` is connected to an electrically driven tuning fork of frequency 256 Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude. At `t=0` the left end (fork end) of the string `x=0` has zero transverse displacement `(y=0)` and is moving along positive y-direction. The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as function of x and t that describest the wave on the string.

Explanation

The equation of a travelling wave propagating along the positive y-direction is given by the displacement equation:
`y(x,t)=a sin (wt-kx)...(i)`
Linear mass density, ` mu=8.0xx10^(-3) kg m^(-1)`
Frequency of the tuning fork, v=256 HZ
Amplitude of the wave, a = 5.0 cm = 0.05 m … (ii)
Mass of the pan, m = 90 kg
Tension in the string, `T = mg = 90 xx 9.8 = 882 N`
The velocity of the transverse wave v, is given by the relation:
`v=sqrt((T)/(mu))`
`=sqrt((882)/(8.0xx10^(-3)))=332 m//s`
Angular frequency `omega=2piv`
`" " =2xx3.14xx256`
`" " =1608.5=1.6xx10^(3)"rad"//s ..... (iii)`
Wavelength `lambda=(v)/(v)=(332)/(256)m`
`:.` Propagation constant , `k=(2pi)/(lambda)`
`" " =(2xx3.14)/((332)/(256))=4.84 m^(-1)..... (iv)`
Substituting the values from equations (ii), (iii), and (iv) in equation (i), we get the displacement equation:
`y(x,t)=0.05 sin (1.6xx10^(3)t-4.84xx)m`