Physics

This is a standing wave problem—the product form tells you that position and time are separated, so the string oscillates up and down at each point rather than traveling. Extract the amplitude at each location from the spatial part $\sin \frac{\pi x}{3}$, then use the temporal part $\cos 40\pi t$ to find how fast and in what pattern that amplitude changes. The key is recognizing that the wavelength and frequency are hidden in the coefficients of $x$ and $t$.

This is a standing wave problem—the product form tells you that position and time are separated, so the string oscillates up and down at each point rather than traveling. Extract the amplitude at each location from the spatial part $\sin \frac{\pi x}{3}$, then use the temporal part $\cos 40\pi t$ to find how fast and in what pattern that amplitude changes. The key is recognizing that the wavelength and frequency are hidden in the coefficients of $x$ and $t$.