A transverse harmonic wave on a string is described by
`y ( x,t) = 3.0 sin ( 36t +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

Compare the given equation with that of plane progressive wave of amplitude r, travelling with a velocity V from from right to left.
`y ( x,t) = r sin [ (2pi)/( lambda) ( vt + x ) + phi_(0)]` ....(1)
We find that
(a) The given equation represents a transverse harmonic wave travelling from right to left. It is not a stationary wave.
(b) The given equation can be written as
`y (x,t ) = 3.0 sin [ 0.018 ((36)/( 0.018)t+x)+(pi)/(4)]` equation coefficient of t in the two (1) & (2) we get,
`V=( 36)/( 0.018) = 2000 cm//sec`
Obviously, `r = 3.0 cm`
Also, ` ( 2pi)/(lambda) = 0.018`
`lambda= ( 2pi)/( 0.018) cm`
Frequency , `v= ( upsilon)/( lambda) = ( 2000)/( 2pi) xx 0.018`
`= 5.7351`
(c )Initial phase,` phi_(0)= ( pi)/(4)` radian
For different values of t,we calculate y using eq(i). These values are tabulated below `:`
`{:(t,0,(T)/(8),(2T)/(8),(3T)/(8),(4T)/(8),(5T)/(8),(6T)/(8),(7T)/(8),T),(x,(3)/(sqrt(2)),3,(3)/(sqrt(2)),0,(-3)/(sqrt(2)),-3,(-3)/(sqrt(2)),0,(3)/(sqrt(2))):}`
On plotting y versus t graph , we obtain a sinusoidal curve as shown in figure.


Similarly graphs are obtained for x= 2cm & x = 4cm . The oscillary motion in travelling wave differs from one point to another only in terms of phase. Amplitude and frequency of oscillatory motion remain the same in all the three areas.
(d) Least distance between two successive crests of the wave
Wave length , ` lambda= ( 2pi)/( 0.018 cm ) = 349 cm`