RaySix

Post

Properties of a Sinusoidal Wave

Created by @nathanedwards

at November 4th 2023, 10:55:12 pm.

AP_Physics_1-Questions

#frequency

#wavelength

#wave

Question

A wave is described by the equation:

y(x, t) = 0.1 \sin(2\pi(3t - 5x))

where $y$ is the displacement of the particles from their equilibrium position at position $x$ and time $t$ .

(a) What is the wavelength of this wave? (b) Calculate the frequency of the wave. (c) Determine the amplitude of the wave. (d) Find the speed of the wave.

Assume all values are in SI units.

Answer

(a) The equation for a wave in the form $y(x, t) = A \sin(kx - \omega t)$ , where $k$ is the wave number and $\omega$ is the angular frequency, can be written in the general form:

y(x, t) = A \sin\left(\frac{2\pi}{\lambda}x - \frac{2\pi f}{c}t\right)

Comparing this general equation with the given equation, we can identify:

\frac{2\pi}{\lambda} = 5 \Rightarrow \lambda = \frac{2\pi}{5}

Therefore, the wavelength of the wave is $\frac{2\pi}{5}$ m.

(b) From the general equation, we also identify:

\frac{2\pi f}{c} = 3 \Rightarrow f = \frac{3c}{2\pi}

Using the known speed of light in vacuum, $c = 3 \times 10^8$ m/s, we can calculate the frequency:

f = \frac{3 \times 3 \times 10^8}{2\pi} = \frac{9 \times 10^8}{2\pi} \, \text{Hz}

Therefore, the frequency of the wave is $\frac{9 \times 10^8}{2\pi}$ Hz.

(c) Looking at the given equation, we can identify that the amplitude of the wave is $A = 0.1$ m.

Therefore, the amplitude of the wave is $0.1$ m.

(d) The speed of the wave can be determined using the relation:

v = \lambda \cdot f

Substituting the values for wavelength and frequency, we find:

v = \left(\frac{2\pi}{5}\right) \left(\frac{9 \times 10^8}{2\pi}\right) = \frac{9 \times 10^8}{5} \, \text{m/s}

Therefore, the speed of the wave is $\frac{9 \times 10^8}{5}$ m/s.