There is a destructive interference between the two waves of wavelength `lambda` coming from two different paths at a point. To get maximum sound or constructive interference at that point, the path of one wave is to be increased by

Answer

To solve the problem of achieving constructive interference from a state of destructive interference between two waves, we can follow these steps:

### Step 1: Understand the Condition for Destructive Interference
Destructive interference occurs when the path difference between two waves is an odd multiple of half the wavelength. Mathematically, this can be expressed as:
\[ \Delta x = \left( n + \frac{1}{2} \right) \lambda \quad \text{where } n = 0, 1, 2, \ldots \]
For the simplest case, let's consider \( n = 0 \):
\[ \Delta x = \frac{\lambda}{2} \]

### Step 2: Determine the Condition for Constructive Interference
Constructive interference occurs when the path difference is an integer multiple of the wavelength. This can be expressed as:
\[ \Delta x = m \lambda \quad \text{where } m = 0, 1, 2, \ldots \]

### Step 3: Relate the Two Conditions
To transition from destructive interference to constructive interference, we need to adjust the path difference. Starting from the condition for destructive interference:
\[ \Delta x = \frac{\lambda}{2} \]
we want to find the new path difference that satisfies the condition for constructive interference.

### Step 4: Calculate the Required Path Difference for Constructive Interference
To achieve constructive interference, we can set the path difference to the next possible integer multiple of the wavelength. The simplest case after \(\frac{\lambda}{2}\) is:
\[ \Delta x = \lambda \]

### Step 5: Determine the Increase in Path Length
To find out how much we need to increase the path of one wave, we calculate the difference between the new path difference for constructive interference and the existing path difference for destructive interference:
\[ \text{Increase in path length} = \Delta x_{\text{constructive}} - \Delta x_{\text{destructive}} = \lambda - \frac{\lambda}{2} = \frac{\lambda}{2} \]

### Conclusion
To achieve maximum sound or constructive interference at the point where there was previously destructive interference, the path of one wave must be increased by:
\[ \frac{\lambda}{2} \]