A transverse harmonic wave on a strin is decribed 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 starting point ?
What is the least distance between two successive crests in the wave ?

Explanation

(a) Yes, Speed = 20 m/s, Direction = Right to left
(b) 3 cm, 5.73 Hz
(c) `(pi)/(4)`
(d) 3.49 m
Explanation:
(a) The equation of a progressive wave travelling from right to left is given by the displacement function:
`y (x, t) = a "sin" (omegat + kx + pi)" " … (i)`
The given equation is:
`y(x,t)=3.0 "sin"(36t+0.018x+(pi)/(4))" "...(iii)`
On comparing both the equations, we find that equation (ii) represents a travelling wave, propagating from right to left.
Now, using equations (i) and (ii), we can write:
`omega=36 rad//s` and `k=0.018 m^(-1)`
We know that
`v=(omega)/(2pi)` and `lambda=(2pi)/(k)`
Also
`v=vlambda`
`therefore v=((omega)/(2pi))xx((2pi)/(k))=(omega)/(k)`
`=(36)/(0.018)=2000 cm//s =20 m//s`
Hence, the speed of the given travelling wave is 20 m/s.
(b) Amplitude of the given wave, a = 3 cm
Frequency of the given wave:
`v=(omega)/(2pi)=(36)/(2xx3.14)=5.73 Hz`
(c) On comparing equations (i) and (ii), we find that the initial phase angle, `phi=(pi)/(4)`
(d) The distance between two successive crests or troughs is equal to the wavelength of the wave.
Wavelength is given by the relation:
`k=(2pi)/(lambda)`
`therefore lambda=(2pi)/(k)=(2xx3.14)/(0.018)=348.89 cm =3.49 m`