D

The equation given is
$$ \vec{\phi}(x,t) = \hat{j} \sin(\frac{2\pi}{\lambda} vt) \cos(\frac{2\pi}{\lambda} x) $$
Step 1: Identify the components of the wave equation. The sine function in time \( vt \) and the cosine function in space \( x \) indicate that the wave is a function of time and position separately.
Step 2: Examine the waveforms. The sine term depends on time while the cosine term depends on position. This is a characteristic of a stationary wave, which has fixed nodes and antinodes.
Step 3: Determine the type of wave. Since the wave is dependent on a sine function for time and cosine for space, it does not propagate through space like a progressive wave. It represents vibrations in a fixed medium. Therefore, it is a stationary wave.
Step 4: Check the direction. The vector \( \hat{j} \) indicates that the wave oscillates in the vertical direction (y-direction), suggesting it is transverse.
Therefore, this wave is a transverse stationary wave.