A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the $x$-direction in 0.3 sec . The crest P is at $x=0$ at $\mathrm{t}=0 \mathrm{sec}$ and maximum displacement of the wave is 2 cm . Which equation correctly represents this wave?

<p>To find the equation of the sinusoidal wave, we need to determine several properties of the wave:</p> <p><strong>Wave Velocity (v)</strong>:</p> <p><p>Given that the wave travels a distance of 1.2 cm in 0.3 seconds, the velocity $ v $ can be computed as:</p> <p>$ v = \frac{\text{distance}}{\text{time}} = \frac{1.2 \, \text{cm}}{0.3 \, \text{s}} = 4 \, \text{cm/s} $</p></p> <p><strong>Wave Number (k)</strong>:</p> <p><p>The wave number $ k $ is calculated using the wavelength $ \lambda = 7.5 \, \text{cm} $:</p> <p>$ k = \frac{2\pi}{\lambda} = \frac{2\pi}{7.5} = \frac{4\pi}{15} \approx 0.83 $</p></p> <p><strong>Angular Frequency ($\omega$)</strong>:</p> <p><p>Angular frequency $ \omega $ is related to the wave number and velocity by the equation $ v = \frac{\omega}{k} $, thus:</p> <p>$ \omega = vk = 4 \times \frac{4\pi}{15} = \frac{16\pi}{15} \approx 3.35 $</p></p> <p><strong>Wave Equation</strong>:</p> <p><p>The general form for a sinusoidal wave traveling in the positive x-direction is:</p> <p>$ y = A \cos(kx - \omega t) $</p></p> <p><p>Given the maximum displacement (amplitude $ A $) of the wave is 2 cm, the equation becomes:</p> <p>$ y = 2 \cos(0.83x - 3.35t) \, \text{cm} $</p></p> <p>This equation accurately describes the sinusoidal wave given the provided parameters.</p>