Question 89: Waves

Question

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 ________ $$\mathrm{km} ~\mathrm{h}^{-1}$$.

Accepted Answer

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

Waves

The equation of wave is given by $\mathrm{Y}=10^{-2} \sin 2 \pi(160 t-0.5 x+\pi / 4) where x and Y are in \mathrm{m} and \mathrm{t} in s. The speed of the wave is ________ \mathrm{km} ~\mathrm{h}^{-1}$.

Choose the correct answer: