Here y(x,t)=7.5‌sin(0.0050x+12t+

π

4

)
=7.5‌sin[0.0050(

12t

0.0050

+x)+

π

4

] ...(i)
(a) Comparing it with general equation of travelling wave
y=A‌sin[

2π

λ

(vt+x)+

π

4

], we get
v= velocity of wave propagation
=

12

0.0050

=

12×104

50

=12×200‌cm‌s−1=24ms−1
At x=1 cm, t=1s , the displacement is given by
y(1,1)=7.5‌sin(0.005×1+12×1+

π

4

)
=7.5‌sin(12.005+

3.14

4

)
=7.5‌sin‌732.83°
=7.5‌sin(720°+12.83°)
=7.5‌sin(12.83°)
=7.5×0.2215=1.67‌cm
Velocity of oscillation of the point isgiven by
v=

dy

dt

=7.5×12‌cos(0.0050x+12t+

π

4

)
=90.0‌cos(0.0050x+12t+

π

4

)
At x = 1 cm, t = 1 s,
v = 90 cos (0.05 × 1 + 12 × 1 + 0.785)
= 90 cos 732.83°
= 90 cos 12.83° = 90 × 0.9751 cm‌s–1
= 87.76 cm‌s–1 ≈ 88 cm‌s–1
But velocity of wave propagation is 24 cm‌s–1
Clearly the velocity of oscillation of point is not equal to the velocity of wave propagation.
∴ No, this velocity is not equal to velocity of wavepropagation which is 24 ms–1 in magnitude.
(b) The given equations is
y(x,t)=7.5‌sin(0.0050x+12t+

π

4

)
Comparing it with equation of a progressive wave,
y=Asin(ωt+kx+ϕ), we get
k=0.005‌rad‌cm–1
∴λ=

2π

k

=

2×3.142

0.005

=12.57m
In a wave all point have same transverse displacement which are separated by λ,2λ,3λ etc. Thus points having separation 12.57 m, 25.14 m, 37.71 m etc. will have same displacement and velocity as x=1 cm point. Thus all points given by nλ when n=±1,±2,±3,±4.... have displacement i.e., distances 12.57 m, 25.14 m .... from x=1 cm.