A transverse harmonic wave on a string is described by `y(x,t) = 3.0 sin (36 t + 0.018x + pi//4)` where x and y are in cm and t in s. The positive direction of x is from left to right
(a) Is this a travelling wave or a stationary wave?
If it is travelling what are the speed and direction of its propagation?
(b) What are its amplitude and frequency?
(c ) What is the initial phase at the origin ?
(d) What is the least distance between two successive crests in the wave?

Explanation

Comparing `y = 3 sin (36 t +0.018 x + (pi)/(4))`
with `y = a sin (omega t + kx + phi)`
we get `a = 3 cm, omega 36 rad//s, k =0.018 rad//cm, `
`phi = (pi)/(4) rad`
(a) Given equation represents one dimensional travelling harmonic wave propagationg along -X axis as its form matches with
`y =a sin (omega t+ kx + phi)`
`v = (omega)/(k) = (36)/(0.018) =2000 (cm)/(s)`
`= 20 (m)/(s)`
(b) Amplitude of given wave is a =3 cm
Also, `omega = 36 rad//s`
`implies f = (36)/(2pi) = (36)/(2xx3.14) =5.732 Hz`
(c) Initial phase at origin is `theta = phi`
`(because theta = omega t + kx + phi and ` here `t =0 and x =0)`
` therefore theta = phi = (pi)/(4) rad`
(d) Least distance between two consecutive
crests is `lamda = (2pi)/(k) " "(because k = (2pi)/(lamda))`
`therefore lamda = (2 xx 3.14)/(0.018) =348.9cm`