The given harmonic wave is:

$\mathrm{y}\left(\mathrm{x}, \mathrm{t}\right)=7.5 \sin \left(0.0050\mathrm{x}12\mathrm{t}+\frac{\mathrm{\pi }}{4}\right)$
$\mathrm{For} \mathrm{x} = 1 \mathrm{cm} \mathrm{and} \mathrm{t} = 1\mathrm{s},$
$\begin{array}{rl}\mathrm{y}\left(1,1\right)& =7.5\sin \left(0.0050+12+\frac{\mathrm{\pi }}{4}\right)\end{array}$
$=7.5\sin \left(12.0050+\frac{\mathrm{\pi }}{4}\right)$
$=7.5\mathrm{sin\theta }$
$\mathrm{where}, \mathrm{\theta }=12.0050+\frac{\mathrm{\pi }}{4}=12.0050+\frac{3.14}{4}=12.79 \mathrm{rad}$
$=\frac{180}{3.14}\times 12.79=732{.81}^{\circ }$
$\begin{array}{rl}\therefore \mathrm{y}\left(1,1\right)& =7.5\sin \left(732{.81}^{\circ }\right)\\ & =7.5\sin (90\times 8+12{.81}^{\circ })=7.5\sin 12{.81}^{\circ }\\ & =7.5\times 0.2217\\ & =1.6629\approx 1.663\mathrm{cm}\end{array}$
$\mathrm{The} \mathrm{velocity} \mathrm{of} \mathrm{the} \mathrm{oscillation}:$
$\begin{array}{rl}\mathrm{v}=\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{y}\left(\mathrm{x},\mathrm{t}\right)& =\frac{\mathrm{d}}{\mathrm{dt}}\left[7.5\sin \left(0.0050\mathrm{x}+12\mathrm{t}+\frac{\mathrm{\pi }}{4}\right)\right]\\ & =7.5\times 12\cos \left(0.0050\mathrm{x}+12\mathrm{t}+\frac{\mathrm{\pi }}{4}\right)\end{array}$
$\mathrm{At} \mathrm{x}=1 \mathrm{cm} \mathrm{and} \mathrm{t}=1\sec :$
$\begin{array}{rl}\mathrm{v}(1, 1)& =90\cos (12.005+\frac{\mathrm{\pi }}{4})\\ & =90\cos \left(732{.81}^{\circ }\right)=90\cos (90\times 8+12{.81}^{\circ })\\ & =90\cos \left(12{.81}^{\circ }\right)\\ & =90\times 0.975=87.75\mathrm{cm}/\mathrm{s}\end{array}$
$\mathrm{Now}, \mathrm{comparing} \mathrm{the} \mathrm{given} \mathrm{equation} \mathrm{with} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{a} \mathrm{propagating} \mathrm{wave}:$
$\mathrm{y}\left(\mathrm{x}, \mathrm{t}\right)=\mathrm{a} \sin \left(\mathrm{kx}+\mathrm{\omega t}+\mathrm{\varphi }\right)$
$\mathrm{So},$
$\mathrm{\omega }=12 \mathrm{rad}/\mathrm{s}$
$\mathrm{k}=0.0050 {\mathrm{m}}^{-1}$
$\therefore \mathrm{v}=\frac{12}{0.0050}=2400\mathrm{cm}/\mathrm{s}$
$\mathrm{Hence}, \mathrm{the} \mathrm{velocity} \mathrm{of} \mathrm{the} \mathrm{wave} \mathrm{oscillation} \mathrm{at} \mathrm{x} = 1 \mathrm{cm} \mathrm{and} \mathrm{t} = 1 \mathrm{s}$
$\mathrm{is} \mathrm{not} \mathrm{equal} \mathrm{to} \mathrm{the} \mathrm{velocity} \mathrm{of} \mathrm{the} \mathrm{wave} \mathrm{propagation}.$
$\left(\mathrm{b}\right)$
$\mathrm{Propagation} \mathrm{constant} \mathrm{is} \mathrm{given} \mathrm{by}:$
$\mathrm{k}=\frac{2\mathrm{\pi }}{\mathrm{\lambda }}$
$\therefore \mathrm{\lambda }=\frac{2\mathrm{\pi }}{\mathrm{k}}=\frac{2\times 3.14}{0.0050}=1256 \mathrm{cm}=12.56 \mathrm{m}$
$\mathrm{All} \mathrm{the} \mathrm{points} \mathrm{at} \mathrm{distances} \mathrm{n\lambda }, \mathrm{i}.\mathrm{e}. \pm 12.56 \mathrm{m}, \pm 25.12 \mathrm{m}, ... \mathrm{where} \mathrm{n}=\pm 1,\pm 2,... \mathrm{and} \mathrm{so} \mathrm{on}$
$\mathrm{will} \mathrm{have} \mathrm{the} \mathrm{same} \mathrm{displacement} \mathrm{as} \mathrm{that} \mathrm{of} \mathrm{he} \mathrm{point} \mathrm{at} \mathrm{x}=1 \mathrm{cm} \mathrm{at} \mathrm{t}=2 \mathrm{s}, 5 \mathrm{s}, \mathrm{and} 11 \mathrm{s}.$
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