Two identical sinusoidal waves travel in opposite direction in a wire `15 m` long and produce a standing wave in the wire . If the speed of the wave is `12 ms^(-1)` and there are `6` nodes in the standing wave . Find the frequency .

Explanation

To find the frequency of the standing wave produced by two identical sinusoidal waves traveling in opposite directions in a wire, we can follow these steps:

### Step 1: Understand the relationship between nodes and wavelength
In a standing wave, the distance between two consecutive nodes is half the wavelength (λ/2). If there are 6 nodes, we can determine the number of half-wavelengths in the wire.

### Step 2: Calculate the number of half-wavelengths
The number of segments between nodes is one less than the number of nodes. Therefore, for 6 nodes, there are 5 segments:
- Number of half-wavelengths = 5

### Step 3: Relate the total length of the wire to the wavelength
The total length (L) of the wire is given as 15 m. Since there are 5 half-wavelengths in this length, we can express the total length as:
\[ L = \frac{5\lambda}{2} \]
This means:
\[ 15 = \frac{5\lambda}{2} \]

### Step 4: Solve for the wavelength (λ)
To find the wavelength, rearrange the equation:
\[ \lambda = \frac{2L}{5} \]
Substituting the value of L:
\[ \lambda = \frac{2 \times 15}{5} = \frac{30}{5} = 6 \text{ m} \]

### Step 5: Use the wave speed to find the frequency
The speed (v) of the wave is given as 12 m/s. The relationship between speed (v), frequency (f), and wavelength (λ) is given by:
\[ v = f \cdot \lambda \]
Rearranging this gives:
\[ f = \frac{v}{\lambda} \]

### Step 6: Substitute the values to find frequency
Now, substituting the known values:
\[ f = \frac{12 \text{ m/s}}{6 \text{ m}} = 2 \text{ Hz} \]

### Conclusion
The frequency of the standing wave is **2 Hz**.

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