The AM wave contans three frequencies viz:

Options

A. `(f_(c)),(f_(c)+f_(s))/(2),(f_(c)-f_(s))/(2)`
B. `2f_(c),2(f_(c)+f_(s)),2(f_(c)-f_(s))`
C. `f_(c),(f_(c)+f_(s)),(f_(c)-f_(s))`
D. `f_(c),f_(c),f_(c)`

Answer

To solve the problem of identifying the frequency components in an Amplitude Modulated (AM) wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the AM Wave Equation**:
The general equation for an AM wave can be expressed as:
\[ E_{AM} = E_C + E_M \cos(\omega_M t) \cos(\omega_C t) \]
where \(E_C\) is the carrier wave amplitude, \(E_M\) is the message signal amplitude, \(\omega_C\) is the angular frequency of the carrier wave, and \(\omega_M\) is the angular frequency of the message signal.

2. **Factor Out the Carrier Wave**:
We can rewrite the equation by factoring out the carrier wave:
\[ E_{AM} = E_C \left(1 + \frac{E_M}{E_C} \cos(\omega_M t)\right) \]
Here, \(\frac{E_M}{E_C}\) is known as the modulation index \(M\).

3. **Expand the Equation**:
We can express the AM wave as:
\[ E_{AM} = E_C \cos(\omega_C t) + \frac{M E_C}{2} \cos(\omega_C + \omega_M)t + \frac{M E_C}{2} \cos(\omega_C - \omega_M)t \]
This uses the trigonometric identity for the product of cosines.

4. **Identify Frequency Components**:
From the expanded equation, we can identify the frequency components:
- The carrier frequency \(f_C\) corresponds to the term \(E_C \cos(\omega_C t)\).
- The upper side frequency \(f_C + f_M\) corresponds to the term \(\frac{M E_C}{2} \cos(\omega_C + \omega_M)t\).
- The lower side frequency \(f_C - f_M\) corresponds to the term \(\frac{M E_C}{2} \cos(\omega_C - \omega_M)t\).

5. **Final Frequency Components**:
Thus, the three frequencies present in the AM wave are:
- \(f_C\) (Carrier Frequency)
- \(f_C + f_M\) (Upper Side Frequency)
- \(f_C - f_M\) (Lower Side Frequency)

### Conclusion:
The AM wave contains three frequency components: the carrier frequency, the upper side frequency, and the lower side frequency.

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