A sinusoidal wave `y_(1) = a sin ( omega t - kx) ` is reflected from a rigid support and superpose with the incident wave `y_(1)` . Assume the rigid support to be at `x = 0`.

Options

A. Stationary waves are obtained with antinodes at the rigid support.
B. Stationary waves are obtained with nodes at the rigid support.
C. Stationary waves are obtained with intensity varying periodically with distance.
D. Stationary waves are obtained with intensity varying periodically with time.

Explanation

To solve the problem, we need to analyze the behavior of the sinusoidal wave when it is reflected from a rigid support. Let's break down the solution step by step.

### Step 1: Understand the Incident Wave
The incident wave is given by the equation:
\[ y_1 = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( t \) is time,
- \( x \) is the position.

### Step 2: Reflection from a Rigid Support
When a wave is reflected from a rigid support, it undergoes a phase change of \( \pi \) (or 180 degrees). This means that the reflected wave will have the same amplitude but will change its sign.

### Step 3: Write the Equation of the Reflected Wave
The reflected wave can be expressed as:
\[ y_r = A \sin(\omega t + kx) \]
This is because the wave is now traveling in the opposite direction (to the left) after reflection.

### Step 4: Superposition of the Incident and Reflected Waves
The total displacement \( y \) at any point \( x \) and time \( t \) is the sum of the incident wave and the reflected wave:
\[ y = y_1 + y_r \]
Substituting the equations:
\[ y = A \sin(\omega t - kx) + A \sin(\omega t + kx) \]

### Step 5: Use the Principle of Superposition
Using the trigonometric identity for the sum of sine functions:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = \omega t - kx \) and \( B = \omega t + kx \):
\[ y = 2A \sin\left(\omega t\right) \cos\left(kx\right) \]

### Step 6: Identify Nodes and Antinodes
From the equation \( y = 2A \sin(\omega t) \cos(kx) \):
- The nodes occur where \( \cos(kx) = 0 \), which happens at \( kx = \frac{\pi}{2} + n\pi \) (where \( n \) is an integer).
- The antinodes occur where \( \cos(kx) = 1 \), which happens at \( kx = n\pi \).

### Step 7: Conclusion
- At the rigid support (at \( x = 0 \)), there is a node.
- The stationary wave pattern is formed, with nodes at the rigid support and antinodes at positions \( \frac{\lambda}{4}, \frac{3\lambda}{4}, \) etc.

### Final Answer
The correct statements are:
1. Stationary waves are obtained with nodes at the rigid support.
2. The intensity varies periodically with distance.