A travelling harmonic wave on a string is described by .

(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:

$\begin{array}{rl}\mathrm{y}\left(1,1\right)& =7.5\mathrm{sin}\left(0.0050+12+\frac{\mathrm{\pi }}{4}\right)\end{array}$
$=7.5\mathrm{sin}\left(12.0050+\frac{\mathrm{\pi }}{4}\right)$
$=7.5\mathrm{sin\theta }$

$=\frac{180}{3.14}×12.79=732{.81}^{\circ }$

$\begin{array}{rl}\mathrm{v}=\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{y}\left(\mathrm{x},\mathrm{t}\right)& =\frac{\mathrm{d}}{\mathrm{dt}}\left[7.5\mathrm{sin}\left(0.0050\mathrm{x}+12\mathrm{t}+\frac{\mathrm{\pi }}{4}\right)\right]\\ & =7.5×12\mathrm{cos}\left(0.0050\mathrm{x}+12\mathrm{t}+\frac{\mathrm{\pi }}{4}\right)\end{array}$

$\left(\mathrm{b}\right)$

$\mathrm{k}=\frac{2\mathrm{\pi }}{\mathrm{\lambda }}$