A travelling wave is partly reflected and partly transmitted from a rigid boudary. Let `a_(i),a_(r)and a_(t)` be the amplitude of incident wave, reflected wave and transmitted wave and `I_(i),I_(r)and I_(t)` be the corresponding intensities. Then choose the correct alternatives

Options

A. `(l_(i))/(l_(r))=((a_(i))/(a_(r)))^(2)`
B. `(l_(i))/(l_(t))=((a_(i))/(a_(r)))^(2)`
C. `(l_(r))/(l_(t))=((a_(r))/(a_(t)))^(2)`
D. All the above

Answer

To solve the problem regarding the relationship between the amplitudes and intensities of incident, reflected, and transmitted waves at a rigid boundary, 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 \( a \). This can be expressed mathematically as:
\[ I \propto a^2 \]

### Step 2: Write the Ratios of Intensities
For the incident wave, reflected wave, and transmitted wave, we can write:
\[ \frac{I_i}{I_r} = \frac{a_i^2}{a_r^2} \]
where \( I_i \) is the intensity of the incident wave, \( I_r \) is the intensity of the reflected wave, \( a_i \) is the amplitude of the incident wave, and \( a_r \) is the amplitude of the reflected wave.

### Step 3: Consider the Rigid Boundary Condition
At a rigid boundary, the reflected wave will have an amplitude that is related to the incident wave. The amplitude of the reflected wave can be expressed as:
\[ a_r = -k \cdot a_i \]
where \( k \) is a constant that depends on the properties of the boundary (for a rigid boundary, \( k = 1 \)). Thus, we can say:
\[ a_r = a_i \]

### Step 4: Substitute the Amplitudes into the Intensity Ratio
Substituting \( a_r \) into the intensity ratio gives:
\[ \frac{I_i}{I_r} = \frac{a_i^2}{a_i^2} = 1 \]
This indicates that the intensity of the reflected wave is equal to the intensity of the incident wave.

### Step 5: Analyze the Transmitted Wave
Since the wave is partly transmitted, we can denote the amplitude of the transmitted wave as \( a_t \) and the intensity as \( I_t \). The relationship for the transmitted wave can be similarly expressed as:
\[ \frac{I_i}{I_t} = \frac{a_i^2}{a_t^2} \]

### Conclusion
From the above steps, we can conclude that:
1. The intensity of the reflected wave is equal to the intensity of the incident wave.
2. The relationship between the intensities and amplitudes can be expressed as:
\[ \frac{I_i}{I_r} = \frac{a_i^2}{a_r^2} \]
and
\[ \frac{I_i}{I_t} = \frac{a_i^2}{a_t^2} \]

### Final Answer
The correct alternative is:
- \( \frac{I_i}{I_r} = \left(\frac{a_i}{a_r}\right)^2 \)