A source of frequency 'f' is stationary and an obsercer starts moving towards it at t=0 with constant small acceleration. Then the variation of observed frequency 'f' registered by the observer with time is best represented as

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Explanation

To solve the problem, we need to analyze how the observed frequency changes as the observer moves towards a stationary sound source with constant small acceleration. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Doppler Effect
The Doppler effect describes how the frequency of a wave changes for an observer moving relative to the source of the wave. In this case, the source is stationary, and the observer is moving towards it.

### Step 2: Define the Variables
Let:
- \( f \) = frequency of the stationary source
- \( v \) = speed of sound in the medium
- \( a \) = constant acceleration of the observer
- \( t \) = time
- \( v_0 \) = velocity of the observer at time \( t \)

### Step 3: Determine the Velocity of the Observer
Since the observer starts from rest and accelerates with a constant acceleration \( a \), the velocity of the observer at time \( t \) can be expressed as:
\[ v_0 = a \cdot t \]

### Step 4: Apply the Doppler Effect Formula
The observed frequency \( f_1 \) when the observer is moving towards the source can be given by the formula:
\[ f_1 = f \cdot \frac{v + v_0}{v} \]
Substituting \( v_0 \) from Step 3:
\[ f_1 = f \cdot \frac{v + a \cdot t}{v} \]

### Step 5: Simplify the Expression
This can be simplified to:
\[ f_1 = f + \frac{a \cdot f}{v} \cdot t \]
This shows that the observed frequency \( f_1 \) is a linear function of time \( t \).

### Step 6: Identify the Relationship
The equation \( f_1 = f + \frac{a \cdot f}{v} \cdot t \) indicates that:
- The frequency increases linearly with time.
- The slope of the line is \( \frac{a \cdot f}{v} \), which is positive since \( a \), \( f \), and \( v \) are all positive.

### Step 7: Graphical Representation
The relationship between observed frequency \( f_1 \) and time \( t \) can be represented graphically as a straight line with:
- A positive slope (indicating increasing frequency)
- A positive intercept (the frequency at \( t = 0 \) is \( f \))

### Conclusion
Thus, the variation of the observed frequency \( f_1 \) registered by the observer with time is best represented as a straight line with a positive slope and a positive intercept.