1. **Problem statement:** A plane harmonic wave propagates along the negative x-axis with angular frequency $\omega$ and wave speed $u$. At time $t=\frac{T}{4}$, the wave shape is given, and we need to find the correct wave expression from the options. 2. **Wave propagation direction and general form:** For a wave traveling in the negative x direction, the wave function is generally written as $$y = A \cos \left[ \omega \left(t + \frac{x}{u} \right) + \phi \right]$$ where $\phi$ is the phase constant. 3. **Given time and wave shape:** At $t=\frac{T}{4}$, the wave shape corresponds to a cosine function shifted by some phase. The period $T$ relates to angular frequency by $T = \frac{2\pi}{\omega}$. 4. **Evaluate the phase at $t=\frac{T}{4}$:** $$\omega t = \omega \cdot \frac{T}{4} = \omega \cdot \frac{2\pi}{\omega} \cdot \frac{1}{4} = \frac{\pi}{2}$$ 5. **Check the phase shift options:** The wave crosses zero at $x=0$ at $t=\frac{T}{4}$, and the crest is to the left, indicating a phase shift of $-\frac{\pi}{2}$ for the wave traveling in the negative x direction. 6. **Select the correct expression:** The wave traveling in the negative x direction with phase shift $-\frac{\pi}{2}$ is $$y = A \cos \left[ \omega \left(t + \frac{x}{u} \right) - \frac{\pi}{2} \right]$$ which corresponds to option (C). **Final answer:** $$y = A \cos \left[ \omega \left(t + \frac{x}{u} \right) - \frac{\pi}{2} \right]$$