Question

A wave along a string has the following equation $y = 0.05 \sin(28t - 2.0x)$ m (where $t$ is in seconds and $x$ is in meters). What are the amplitude, frequency and wavelength of the wave?

• A. amplitude $= 0.05$ m, frequency $= 4.456$ Hz and wavelength $= 3.518$ m
• B. amplitude $= 0.05$ m, frequency $= 28$ Hz and wavelength $= 2.0$ m
• C. amplitude $= 5.0$ m, frequency $= 4.456$ Hz and wavelength $= 3.518$ m
• D. amplitude $= 0.05$ m, frequency $= 2.0$ Hz and wavelength $= 28$ m
• E. amplitude $= 0.05$ m, frequency $= 3.456$ Hz and wavelength $= 4.518$

Hint:

The coefficient of $t$ is always $\omega$ and the coefficient of $x$ is always $k$[ 56]. Use these to jump straight into finding $f$ and $\lambda$

Answer 1

The Correct Option is A

Concept: The general equation for a travelling wave is $y = A \sin(\omega t - kx)$, where $A$ is amplitude, $\omega$ is angular frequency ($\omega = 2\pi f$), and $k$ is the wave number ($k = 2\pi / \lambda$)

Step 1: {Extract $A$, $\omega$, and $k$ from the given equation.}
By comparing $y = 0.05 \sin(28t - 2.0x)$ with the general form:
• $A = 0.05$ m
• $\omega = 28$ rad/s
• $k = 2.0$ rad/m

Step 2: {Calculate the frequency ($f$).}
$$\omega = 2\pi f \implies f = \frac{\omega}{2\pi}$$ $$f = \frac{28}{2\pi} = \frac{28}{2 \times 3.14159} = \frac{14}{3.14159} \approx 4.456 \text{ Hz}$$

Step 3: {Calculate the wavelength ($\lambda$).}
$$k = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{k}$$ $$\lambda = \frac{2\pi}{2.0} = \pi \approx 3.14159 \text{ m}$$ Looking at the options, $3.518$ is the closest match provided for a calculated wavelength of $\pi$ (likely due to a slightly different approximation or typo in the original test).