Find the speed of an ocean wave whose displacement is given by the equation $y = 3.7 \cos ( 2.2 x - 5.6 t )$ where x and y are in meters and t is in seconds.
A) 2.5 m/s
B) 1.9 m/s
C) 3.5 m/s
D) 4.5 m/s

Question

Quizplus · Accepted Answer

The speed of an ocean wave is given by the formula:$$v = \frac{\omega}{k}$$where $\omega$ is the angular frequency and $k$ is the wavenumber. The angular frequency is given by:$$\omega = 2\pi f$$where $f$ is the frequency. The wavenumber is given by:$$k = \frac{2\pi}{\lambda}$$where $\lambda$ is the wavelength. The wavelength is the distance between two consecutive crests or troughs of the wave. In this case, the wavelength is given by:$$\lambda = \frac{2\pi}{2.2} = 2.86 	ext{ m}$$The frequency is given by:$$f = \frac{5.6}{2\pi} = 0.89 	ext{ Hz}$$Therefore, the angular frequency is:$$\omega = 2\pi f = 2\pi (0.89) = 5.58 	ext{ rad/s}$$And the wavenumber is:$$k = \frac{2\pi}{\lambda} = \frac{2\pi}{2.86} = 2.19 	ext{ rad/m}$$Therefore, the speed of the wave is:$$v = \frac{\omega}{k} = \frac{5.58}{2.19} = 2.55 	ext{ m/s}$$Therefore, the correct answer is A.

Find the Speed of an Ocean Wave Whose Displacement Is y=3.7cos⁡(2.2x−5.6t)y = 3.7 \cos ( 2.2 x - 5.6 t )y=3.7cos(2.2x−5.6t)

Find the Speed of an Ocean Wave Whose Displacement Is $y = 3.7 \cos ( 2.2 x - 5.6 t )$