A transverse harmonic wave on a string is described by `y(x,t)=3.0sin(36t+0.018x+pi//4)` where x, y are in cm, t in second. The positive direction of x is from left to right . (i) Is this a travelling or stationary wave ? If travellying, what are the speed and direction of its propagation ? (ii) what are its amplitude and frequency ? (iii) what is the inital phase at the origing ? (iv) What is the least distance between two successive creests in the wave?

Explanation

(i) The given equation is `y(x,t)=3.0sin(36t+0.018x +pi//4)`
Compare it with the standard equation, of a harmonic wave:
`y(x,t)=rsin((2pit)/(T)-(2pix)/(lambda)+phi_(0))`
We find that, `r=3.0 cm` `(2pi)/(T)=36, T=(2pi)/(36)=(pi)/(18) second`
`-(2pi)/(T)=0.018, lambda=-(2pi)/(0.018)=-(2000pi)/(18)cm`
`v=(lambda)/(T)=-(2000pi)/(18xxpi)xx18=-2000cm//s=-20m//s`
Negative sign indicates that wave is travelling from right to left, with a velocity of 20`m//s`. It cannot be a stationay wave as terms cotnaining x and t are not independent.
(ii) Amplitude, `r=3.0cm,` frequency `v=(1)/(T)=(18)/(pi)=(18xx7)/(22)=5.7s^(_1)`
(ii) Initial phase at the origin , `phi_(0)=pi//4rad`
(iv) Distance between centres of two consecutive troughs `=lambda=(2000pi)/(18)cm=(20)/(18)xx(22)/(7)m =3.49m`