The equation of wave is given by

$\mathrm{Y}=10^{-2} \sin 2 \pi(160 t-0.5 x+\pi / 4)$

where $x$ and $Y$ are in $\mathrm{m}$ and $\mathrm{t}$ in $s$. The speed of the wave is ________ $\mathrm{km} ~\mathrm{h}^{-1}$.

<p>Given the wave equation:</p> <p>$Y = 10^{-2} \sin 2 \pi(160t - 0.5x + \pi/4)$</p> <p>Comparing this equation with the general form:</p> <p>$Y = A \sin(2\pi(ft - kx + \phi))$</p> <p>We can identify the wave number $k = 0.5\,\mathrm{m}^{-1}$ and the frequency $f = 160\,\mathrm{Hz}$. The wave speed $v$ can be found using the relationship between wave number, wave speed, and frequency:</p> <p>$v = \frac{\omega}{k} = \frac{2\pi f}{2\pi k}$</p> <p>Now, we can calculate the wave speed:</p> <p>$v = \frac{2\pi \times 160}{2\pi \times 0.5} = \frac{160}{0.5}\,\mathrm{m/s}$</p> <p>$v = 320\,\mathrm{m/s}$</p> <p>Now, we need to convert the wave speed from meters per second to kilometers per hour:</p> <p>$$v = 320 \frac{\mathrm{m}}{\mathrm{s}} \times \frac{1\,\mathrm{km}}{1000\,\mathrm{m}} \times \frac{3600\,\mathrm{s}}{1\,\mathrm{h}}$$</p> <p>$v = 320 \times \frac{1}{1000} \times 3600\,\mathrm{km/h}$</p> <p>$v = 1152\,\mathrm{km/h}$</p> <p>So, the speed of the wave is $1152\,\mathrm{km/h}$.</p>