In a standing wave pattern in a vibrating air column, nodes are formed at a distance of 4.0 cm. If the speed of sound in air is` 328 ms^-1` what is the frequency of the source ?

Explanation

To find the frequency of the source in a standing wave pattern in a vibrating air column, 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 equal to half the wavelength (λ/2).

### Step 2: Calculate the wavelength (λ)
Given that the distance between nodes is 4.0 cm, we can express this in meters:
\[ \text{Distance between nodes} = 4.0 \, \text{cm} = 0.04 \, \text{m} \]
Since this distance is equal to λ/2:
\[ \frac{\lambda}{2} = 0.04 \, \text{m} \]
To find λ, we multiply both sides by 2:
\[ \lambda = 2 \times 0.04 \, \text{m} = 0.08 \, \text{m} \]

### Step 3: Use the speed of sound to find frequency
The speed of sound (v) is given as 328 m/s. The relationship between speed (v), frequency (f), and wavelength (λ) is given by the formula:
\[ v = f \cdot \lambda \]
Rearranging this formula to solve for frequency gives:
\[ f = \frac{v}{\lambda} \]

### Step 4: Substitute the known values
Now substitute the values of v and λ into the frequency equation:
\[ f = \frac{328 \, \text{m/s}}{0.08 \, \text{m}} \]

### Step 5: Calculate the frequency
Perform the calculation:
\[ f = \frac{328}{0.08} = 4100 \, \text{Hz} \]

### Step 6: Convert to kilohertz
To convert the frequency from hertz to kilohertz:
\[ f = 4100 \, \text{Hz} = 4.1 \, \text{kHz} \]

### Final Answer
The frequency of the source is **4.1 kHz**.

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