Question

Write an equation for a sine wave with amplitude \(0.120 \mathrm{m}\) wavelength \(0.300 \mathrm{m},\) and wave speed \(6.40 \mathrm{m} / \mathrm{s}\) traveling in the \(-x\) -direction.

Short answer

Answer: \(y(x, t) = 0.120\sin(20.9x + 134t)\)

Step-by-step solution

Identify the important parameters given

The given parameters for the sine wave are: - Amplitude (A) = \(0.120\) m - Wavelength (\(\lambda\)) = \(0.300\) m - Wave speed (v) = \(6.40\) m/s - Direction: \(-x\)

Calculate the wave number

The wave number (k) can be calculated using the formula: $$ k=\frac{2\pi}{\lambda} $$ Plug in the value of \(\lambda\): $$ k=\frac{2\pi}{0.300} $$ Calculate the wave number: $$ k=6.283 \approxeq 20.9 \mathrm{rad} / \mathrm{m} $$

Calculate the angular frequency

The angular frequency (ω) can be calculated using the formula: $$ \omega=\frac{2\pi v}{\lambda} $$ Plug in the values of v and \(\lambda\): $$ \omega=\frac{2\pi (6.40)}{0.300} $$ Calculate the angular frequency: $$ \omega=134 \mathrm{rad} / \mathrm{s} $$

Write the equation of the sine wave

Since the wave travels in the \(-x\) direction, the equation of the sine wave is: $$y(x, t) = A\sin(kx + \omega t)$$ Plug in the values for A, k, and ω: $$ y(x, t) = 0.120\sin(20.9x + 134t) $$ The equation of the sine wave with the given parameters is: $$ y(x, t) = 0.120\sin(20.9x + 134t) $$