The wave function for a travelling wave on a taut string is
`y(x,t)=(0.350 m)sin(10pit-3pix+pi//4)`. (SI units)
(a) what is the speed and direction of travel of the wave ?
(b) what if the vertically position of an element of the string at `t=0, x=0.100 m?`
(c ) what is the wavelength and frequency of the wave?
(d) what is the maximum transverse speed of an element of the string?

Explanation

Let us compare the given equation with `y=A sin(omegat-kx+phi)`.
we find that `k=3pi rad//m` and `omega=10pi rad//s`
(a) The speed and direction of the wave are both specified by the vector wave velocity:
`vec(v)=f lambda hat(i)=(omega)/(k)hat(i)=(10pi rad//s)/(3pi rad//m)hat(i)=3.33hat(i) m//s`
(b) substituting `t=0` and `x=0.100 m`, we have `y=(0.350 m) sin (-0.300pi+0.250pi)`
`=(0.350 m) sin (-0.157)`
`=(0.350 m)(-0.156)=-0.0548 m=-5.48 cm`
note that when you take the sine of a quantity with no units, the quantity is not in degrees, but in radians.
(c ) The wavelength is
`lambda=(2pi rad)/(k)=(2pi rad)/(3pi rad//m)=0.667 m`
and the frequency is
`f(omega)/(2pi rad) (10pi rad//s)/(2pi rad)=5.00 Hz`
(d) The particle speed is `v_(y)=dely//delt= (0.350 m)(10pi rad)cos(10pi t-3pix+pi//4)`
The maximum occurs when the cosine term is `1:`
`v_(y.max) =(10pi rad//s)(0.350 m//s)=11.0 m//s`