The equation of two interfering waves are `y_(1) =b_(1)cos omegat` and `y_(2) =bcos(omegat + phi)` respectively. Destructive interference will take place at the point of observation for the following value of `phi`.

Options

A. `0^(@)`
B. `360^(@)`
C. `180^(@)`
D. `720^(@)`

Answer

To determine the value of \(\phi\) for which destructive interference occurs between the two interfering waves given by:

\[ y_1 = b_1 \cos(\omega t) \]
\[ y_2 = b \cos(\omega t + \phi) \]

we follow these steps:

### Step 1: Understand the Condition for Destructive Interference
Destructive interference occurs when the phase difference between two waves is an odd multiple of \(\pi\) radians (or 180 degrees). This can be expressed mathematically as:

\[ \Delta \phi = (2n + 1)\pi \quad \text{where } n \text{ is an integer} \]

### Step 2: Identify the Phase Difference
The phase difference \(\Delta \phi\) between the two waves can be expressed as:

\[ \Delta \phi = \phi - 0 = \phi \]

### Step 3: Set Up the Equation for Destructive Interference
For destructive interference, we set the phase difference equal to \(\pi\) (the simplest case when \(n = 0\)):

\[ \phi = (2n + 1)\pi \quad \Rightarrow \quad \phi = \pi \quad \text{(for } n = 0\text{)} \]

### Step 4: Conclusion
Thus, the value of \(\phi\) for which destructive interference occurs is:

\[ \phi = \pi \text{ radians} \quad \text{or } 180^\circ \]

### Final Answer
Destructive interference will take place at the point of observation for the following value of \(\phi\):

\[ \phi = \pi \text{ radians} \text{ (or } 180^\circ\text{)} \]

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