The provided image contains information about a wave function and its characteristics. Let's analyze it step by step.

Wave Function:

The function given is: $y(x, t) = 3 \sin(4\pi x - 6\pi t + \pi)$ where:

• $x$ represents the position in meters.
• $t$ represents the time in seconds.
• $y(x, t)$ is the displacement of the wave in meters.

Key Components of the Wave:

1. Amplitude:
   The amplitude is the coefficient in front of the sine function, which is 3 meters. This means the wave oscillates between $y = 3$ and $y = -3$.

2. Angular Wave Number ($k$):
   From the equation, the term $4\pi$ corresponds to $k$, the angular wave number.
   $k = 4\pi \implies \lambda = \frac{2\pi}{k} = \frac{2\pi}{4\pi} = \frac{1}{2} \, \text{meters}$ So, the wavelength $\lambda$ is $0.5$ meters.

3. Angular Frequency ($\omega$):
   The term $6\pi$ corresponds to $\omega$, the angular frequency.
   $\omega = 6\pi \implies T = \frac{2\pi}{\omega} = \frac{2\pi}{6\pi} = \frac{1}{3} \, \text{seconds}$ The period $T$ is $1/3$ seconds.

4. Phase Shift:
   The term $\pi$ inside the sine function represents the phase shift. A phase shift of $\pi$ corresponds to a shift by half the wavelength.

5. Wave Direction:
   Since the equation contains $4\pi x - 6\pi t$, the wave moves in the positive $x$-direction (because of the negative sign with $\omega t$).

Summary:

• Amplitude: 3 meters
• Wavelength ($\lambda$): 0.5 meters
• Period (T): 1/3 seconds
• Direction: Positive $x$-direction
• Phase Shift: $\pi$ (half a wavelength)

Do you need further details or have any questions?
Here are five related questions to explore further:

1. How do you determine the wave velocity from the given function?
2. What is the difference between phase velocity and group velocity?
3. How would the wave function change if the amplitude doubled?
4. What happens to the wave if the phase shift changes from $\pi$ to $\pi/2$?
5. How would the graph look if the wave moved in the negative $x$-direction?

Tip: The wave speed can be computed as $v = \frac{\lambda}{T}$, using the wavelength and period.