In a stationary wave respresented by `y=2acos(kx)sin(omegat)` the intensity at a certain point is maximum when

Options

A. `cos(kx)` is maximum
B. `cos(kx)` is minimum
C. `sin(omega t)` is maximum
D. `sin (omega t)` is minimum

Answer

To determine when the intensity of the stationary wave represented by the equation \( y = 2A \cos(kx) \sin(\omega t) \) is at its maximum, we can follow these steps:

### Step 1: Understand the relationship between intensity and amplitude
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \). Therefore, we can express this relationship as:
\[ I \propto A^2 \]

### Step 2: Identify the amplitude in the given wave equation
In the given equation \( y = 2A \cos(kx) \sin(\omega t) \), the amplitude of the wave is:
\[ A = 2A \cos(kx) \]
This indicates that the amplitude varies with position \( x \) due to the \( \cos(kx) \) term.

### Step 3: Determine when the amplitude is maximum
For the intensity to be maximum, the amplitude must also be maximum. The term \( \cos(kx) \) reaches its maximum value of 1 when:
\[ \cos(kx) = 1 \]
This occurs at specific points given by:
\[ kx = 2n\pi \quad (n \in \mathbb{Z}) \]
where \( n \) is any integer.

### Step 4: Conclusion about the maximum intensity
Thus, the intensity is maximum when:
\[ \cos(kx) = 1 \]
This means that the intensity will be maximum at points where \( kx \) is an even multiple of \( \pi \).

### Final Answer
The intensity at a certain point is maximum when \( \cos(kx) = 1 \).

---