Solution: The function $ S(t) = \left| \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right) \right| $ has period $ T $, where both components repeat. The fundamental period is the least common multiple of the periods of $ \sin\left(\frac{3\pi}{7} t\right) $ and $ \cos\left(\frac{4\pi}{7} t\right) $.

["Understanding the Period of the Function $ S(t) = \left| \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right) \right| $", "When analyzing periodic functions, especially those involving trigonometric expressions, determining the fundamental period is essential for understanding the full behavior of the function. In this article, we explore the periodicity of the function:", "$$\nS(t) = \left| \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right) \right|\n$$", "This function combines a sine and a cosine term with different angular frequencies, and its absolute value introduces symmetry that simplifies period analysis. The core idea is that the fundamental period $ T $ is the least common multiple (LCM) of the individual periods of the two components.", "---", "### Step 1: Determine the Periods of Individual Components", "The standard period of $ \sin(\omega t) $ or $ \cos(\omega t) $ is $ \frac{2\pi}{\omega} $.", "For $ \sin\left(\frac{3\pi}{7} t\right) $:\n$$\n\omega_1 = \frac{3\pi}{7} \Rightarrow T_1 = \frac{2\pi}{\frac{3\pi}{7}} = \frac{2\pi \cdot 7}{3\pi} = \frac{14}{3}\n$$", "For $ \cos\left(\frac{4\pi}{7} t\right) $:\n$$\n\omega_2 = \frac{4\pi}{7} \Rightarrow T_2 = \frac{2\pi}{\frac{4\pi}{7}} = \frac{2\pi \cdot 7}{4\pi} = \frac{14}{4} = \frac{7}{2}\n$$", "So, one component completes a full cycle every $ \frac{14}{3} $, and the other every $ \frac{7}{2} $.", "---", "### Step 2: Find the Least Common Multiple (LCM) of Periods", "We seek the smallest $ T > 0 $ such that both components repeat, i.e., $ T $ satisfies:\n$$\nT = k \cdot \frac{14}{3} = m \cdot \frac{7}{2}, \quad \ ext{for integers } k, m\n$$", "Multiply both sides of the equality $ k \cdot \frac{14}{3} = m \cdot \frac{7}{2} $:\n$$\n\frac{14k}{3} = \frac{7m}{2} \Rightarrow \frac{14k}{3} \cdot \frac{2}{7} = m \Rightarrow m = \frac{28k}{21} = \frac{4k}{3}\n$$", "For $ m $ to be an integer, $ \frac{4k}{3} \in \mathbb{Z} $, so $ k $ must be a multiple of 3. Let $ k = 3 $:\n$$\nT = 3 \cdot \frac{14}{3} = 14\n$$", "Check with $ m = \frac{4 \cdot 3}{3} = 4 $:\n$$\nT = 4 \cdot \frac{7}{2} = 14\n$$", "Thus, the fundamental period is $ T = 14 $.", "---", "### Step 3: Why the Absolute Value Simplifies Period Analysis", "The function $ S(t) $ involves the absolute value of the sum:\n$$\nS(t) = \left| f(t) \right|, \quad \ ext{where } f(t) = \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right)\n$$", "Since $ |\sin x| $ and $ |\cos x| $ have periods half that of their original functions (when $ \omega > 0 $), but in this case, the absolute value does not reduce the fundamental period—it preserves it. However, because $ f(t) $ is a sum of periodic functions with incommensurate fundamental periods, taking the absolute value does not usually yield a smaller period unless symmetry forces it.", "Crucially, the sum $ f(t) $ still repeats every $ T = 14 $, and the absolute value does not introduce a shorter repetition cycle due to phase and frequency alignment at this period. Thus, the fundamental period of $ S(t) $ remains the LCM of $ \frac{14}{3} $ and $ \frac{7}{2} $, which is $ 14 $.", "---", "### Step 4: Practical Implications of the Period", "Knowing that $ S(t) $ has period $ 14 $ allows:\n- Predicting behavior across all real $ t $ with exact repetition.\n- Simplifying Fourier analysis, signal processing, or solving differential equations involving this function.\n- Visualizing waveform symmetry and constructing periodic plots without redundancy.", "---", "### Conclusion", "The function\n$$\nS(t) = \left| \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right) \right|\n$$\nhas a fundamental period of $ \boxed{14} $, determined by the least common multiple of the individual component periods $ \frac{14}{3} $ and $ \frac{7}{2} $. The absolute value preserves this periodicity, maintaining the full cycle length without reduction. Understanding this periodicity is fundamental for deeper analysis in applied mathematics, engineering, and computational modeling."]