A travelling harmonic wave on a string is described by yx, t=7.5sin0.0050x+12t+π4.

(a) what are the displacement and velocity of oscillation of a point at x = 1 cm, and t = 1 sec? Is this velocity equal to the velocity of wave propagation?

(b) Locate the points of the string which have the same transverse displacements and velocity as that of point at x = 1 cm and t = 2 s, 5 s and 11 s.

The given harmonic wave is:

yx, t=7.5 sin0.0050x12t+π4
For x = 1 cm and t = 1s,
y1,1=7.5sin0.0050+12+π4
=7.5sin12.0050+π4
=7.5sinθ
where, θ=12.0050+π4=12.0050+3.144=12.79 rad
=1803.14×12.79=732.81
 y1,1=7.5sin(732.81)=7.5sin(90×8+12.81)=7.5sin12.81=7.5×0.2217=1.66291.663cm
The velocity of the oscillation:
v=ddtyx,t=ddt7.5sin0.0050x+12t+π4=7.5×12cos0.0050x+12t+π4
At x=1 cm and t=1sec:
v(1, 1)=90cos(12.005+π4)=90cos(732.81)=90cos(90×8+12.81)=90cos(12.81)=90×0.975=87.75cm/s
Now, comparing the given equation with the equation of a propagating wave:
yx, t=a sinkx+ωt+ϕ
So, 
ω=12 rad/s
k=0.0050 m-1
 v=120.0050=2400cm/s
Hence, the velocity of the wave oscillation at x = 1 cm and t = 1 s 
is not equal to the velocity of the wave propagation.
b
Propagation constant is given by:
k=2πλ
 λ=2πk=2×3.140.0050=1256 cm=12.56 m
All the points at distances , i.e. ±12.56 m, ±25.12 m, ... where n=±1,±2,... and so on
will have the same displacement as that of he point at x=1 cm at t=2 s, 5 s, and 11 s.