Show that the period of the fundamental mode of a stretched strings is equal to double the time the component waves forming stationary waves in the string take in traversing the distance between the fixed ends.

Explanation

To show that the period of the fundamental mode of a stretched string is equal to double the time taken by the component waves forming stationary waves in the string to traverse the distance between the fixed ends, we can follow these steps:

### Step 1: Understand the setup
Consider a stretched string fixed at both ends. The length of the string is denoted as \( L \). The fundamental mode of vibration of the string is the simplest form of standing wave that can be formed, which has one antinode in the middle and a node at each end.

### Step 2: Determine the wave speed
The speed \( v \) of a wave on a stretched string is given by the formula:
\[
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