Question:

A wave in a medium is described by the equation:

where $y$ is the displacement of a particle along the $y$ axis at a position $x$ and a time $t$. Assume that $x$ and $y$ are in meters, and $t$ is in seconds.

a) Determine the wave speed.

b) Find the frequency of the wave.

c) Calculate the wavelength of the wave.

Answer:

a) The wave speed ($v$) is given by the equation:

where $\lambda$ is the wavelength and $T$ is the period. Since the equation for the wave is $y(x, t) = 3 \sin(2 \pi (0.5x - t))$, we can see that the expression inside the sine function represents the argument of the wave. By comparing this expression with the general equation for a wave $y(x, t) = A \sin(kx - \omega t)$, we can conclude that the wave number $k$ is equal to $2\pi \times 0.5 = \pi$ and the angular frequency $\omega$ is equal to $2\pi$.

Therefore, the period $T$ is given by:

Substituting the values of $\lambda$ and $T$ into the equation for the wave speed, we get:

Hence, the wave speed is equal to the wavelength.

b) The frequency ($f$) can be determined using the equation:

where $T$ is the period. From part (a), we found that the period is $T = 1 \, \text{s}$. Therefore, the frequency is:

Thus, the frequency of the wave is 1 Hz.

c) The wavelength ($\lambda$) is related to the wave number ($k$) by the equation:

From part (a), we found that the wave number is $k = \pi$. Substituting this value into the equation, we get:

Thus, the wavelength of the wave is 2 meters.