Here $y(x,t) = 7.5 \sin\left(0.0050x + 12t + \frac{\pi}{4}\right)$ $= 7.5 \sin \left[ 0.0050 \left( \frac{12t}{0.0050} + x \right) + \frac{\pi}{4} \right]$ ...(i) (a) Comparing it with general equation of travelling wave $y = A \sin \left[ \frac{2\pi}{\lambda}(vt + x) + \frac{\pi}{4} \right],$ we get $v =$ velocity of wave propagation $= \frac{12}{0.0050} = \frac{12 \times 10^{4}}{50}$ $= 12 \times 200 \, \text{cm} \, \text{s}^{-1} = 24 \, \text{m} \, \text{s}^{-1}$ At $x = 1$ cm, $t = 1 \, \text{s}$ , the displacement is given by $y(1,1) = 7.5 \sin \left(0.005 \times 1 + 12 \times 1 + \frac{\pi}{4}\right)$ $= 7.5 \sin \left(12.005 + \frac{3.14}{4}\right)$ $= 7.5 \sin 732.83^{\circ}$ $= 7.5 \sin (720^{\circ} + 12.83^{\circ})$ $= 7.5 \sin (12.83^{\circ})$ $= 7.5 \times 0.2215 = 1.67 \, \text{cm}$ Velocity of oscillation of the point isgiven by $v = \frac{dy}{dt}$ $= 7.5 \times 12 \cos \left(0.0050x + 12t + \frac{\pi}{4}\right)$ $= 90.0 \cos \left(0.0050x + 12t + \frac{\pi}{4}\right)$ 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 $\text{cm} \, \text{s}^{-1}$= 87.76 $\text{cm} \, \text{s}^{-1}$ ≈ 88 $\text{cm} \, \text{s}^{-1}$ But velocity of wave propagation is 24 $\text{cm} \, \text{s}^{-1}$ 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 $\text{m} \, \text{s}^{-1}$ in magnitude. (b) The given equations is $y(x,t) = 7.5 \sin\left(0.0050x + 12t + \frac{\pi}{4}\right)$ Comparing it with equation of a progressive wave, $y = A \sin(\omega t + kx + \phi)$, we get $k = 0.005 \, \text{rad} \, \text{cm}^{-1}$$\therefore \lambda = \frac{2\pi}{k} = \frac{2 \times 3.142}{0.005} = 12.57 \, \text{m}$ In a wave all point have same transverse displacement which are separated by $\lambda, 2\lambda, 3\lambda$ 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 $n\lambda$ when $n = \pm 1, \pm 2, \pm 3, \pm 4 \ldots$ have displacement i.e., distances 12.57 m, 25.14 m .... from $x = 1$ cm.