A message signal of freuency `omega_(m)` is superposed on a carrier wave of frequency `omega_(c)` to get an amplititude modulated wave (AM). The frequency of the AM wave will be

Options

A. `omega_(m)`
B. `omega_(c )`
C. `(omega_(c )+omega_(m))/(2)`
D. `(omega_(c )-omega_(m))/(2)`

Answer

To determine the frequency of the amplitude modulated (AM) wave when a message signal of frequency \( \omega_m \) is superposed on a carrier wave of frequency \( \omega_c \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Components**:
- The message signal (modulating wave) is represented as \( y_m = A_m \sin(\omega_m t) \), where \( A_m \) is the amplitude and \( \omega_m \) is the angular frequency of the message signal.
- The carrier wave is represented as \( y_c = A_c \sin(\omega_c t) \), where \( A_c \) is the amplitude and \( \omega_c \) is the angular frequency of the carrier wave.

2. **Formulate the AM Wave**:
- The amplitude modulated wave is formed by combining the message signal with the carrier wave. The general form of the AM wave can be expressed as:
\[ y_{AM} = A_c \sin(\omega_c t) + \frac{\mu A_c}{2} \cos((\omega_c - \omega_m)t) - \frac{\mu A_c}{2} \cos((\omega_c + \omega_m)t) \]
- Here, \( \mu \) is the modulation index.

3. **Identify the Frequency Components**:
- In the resultant AM wave, the first term \( A_c \sin(\omega_c t) \) indicates that the primary frequency of the AM wave is determined by the carrier frequency \( \omega_c \).
- The other terms \( \cos((\omega_c - \omega_m)t) \) and \( \cos((\omega_c + \omega_m)t) \) introduce sidebands at frequencies \( \omega_c - \omega_m \) and \( \omega_c + \omega_m \), but they do not affect the fundamental frequency of the AM wave.

4. **Conclusion**:
- Therefore, the frequency of the amplitude modulated wave is given by the frequency of the carrier wave, which is \( \omega_c \).

### Final Answer:
The frequency of the AM wave is \( \omega_c \).

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