Does there always exist a sum of sines whose value is always greater than any value k>0? I may be way over my league here, but as I was in a swimming pool, watching the waves rise and fall, I noticed that they went above and below a certain line in the tiles of the pool's walls. It gave rise to a question, which I hope is formulated correctly.

For any $k > 0$, does there exist a sum of sine waves $g(t)$ whose value $\geq k$ for all $t$, where the absolute value of the amplitudes of the sines is $\lt k$? If this is possible, is it possible with a finite number of terms?

At first, I asked if such a function existed for any k, and the answer was trivial as, for example, $sin(t) \ge -1$ everywhere. I think this also means that a final form could be $g(t)=sin(t)+k+C$ where $C \ge 1$, if the constant $k+C$ can be approximated by sine functions.
I think that, mathematically, it looks like this $$\exists ? g(t) = \sum_{j=1}^L A_jsin(\omega_j t + \varphi_j)\ge k; \omega_j = 2\pi f_j, k>0, \lvert A_j \rvert<k$$ where L is the number of necessary sines.
For the visualization, what I mean to be asking is if some combination of waves on the pool could make the water remain above a line on the pool's wall (which is above the resting water line) at an arbitrary point on the line, for an arbitrary amount of time.
Thank you for your time and forgive me if the answer is trivial or any point is unclear.
 A: If the only requirement is that it remain above the line for an arbitrary amount of time, this is in fact trivial (unfortunately, because I enjoyed reading your question and thinking about it). Suppose you wanted a sine wave to stay above $k$ for an arbitrary large open interval $(x_1, x_2).$ Then take $l=x_2-x_1$. This is the length of that interval. $sin(x)$ is greater than or equal to $\frac 1 2$ for $\frac \pi 6 \geq x\geq\frac {5\pi} 6$. This period has length $\frac 2 3 \pi$. If we multiply the input by $\frac{3l}{2\pi}$ (i.e. $sin(\frac{3l}{2\pi}x)$, the function is greater than or equal to $\frac 12$ between $\frac\pi4$ and $\frac{5\pi}4$, which has length $l$. Adding up $2k$ of these waves will keep the function above $k$ for a length $l$.
Unfortunately, this is impossible if you need it to be $\textit{infinitely}$ long :( Whatever the period of each sine wave is, if you multiply all of them together, you get the period of the wave formed by their sum (or a multiple of it). At half the period of a wave, it is always equal to $0$, as waves are rotationally symmetric around that point.
