Question 133: Waves

Question

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?

Accepted Answer

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

Answer

$y=2 \cos (0.83 x-3.35 t) \mathrm{cm}$

Answer

$y=2 \sin (0.83 x-3.5 \mathrm{t}) \mathrm{cm}$

Answer

$y=2 \cos (0.13 x-0.5 t) \mathrm{cm}$

Answer

$y=2 \cos (3.35 x-0.83 \mathrm{t}) \mathrm{cm}$

Waves

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?

Choose the correct answer: