The AM wave contains three frequencies,viz:

Options

A. `(f_(c))/(2),(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 question regarding the frequencies contained in an Amplitude Modulated (AM) wave, we can follow these steps:

### Step 1: Understand the AM Wave Equation
The instantaneous value of an AM wave can be expressed mathematically as:
\[ V(t) = V_c \sin(\omega_c t) + \frac{m V_c}{2} \cos((\omega_c - \omega_m)t) + \frac{m V_c}{2} \cos((\omega_c + \omega_m)t) \]
Where:
- \( V_c \) is the amplitude of the carrier wave.
- \( \omega_c \) is the angular frequency of the carrier wave.
- \( \omega_m \) is the angular frequency of the modulating wave.
- \( m \) is the modulation index.

### Step 2: Identify the Frequencies
From the equation, we can see that there are three components:
1. The carrier wave: \( V_c \sin(\omega_c t) \)
- Frequency: \( f_c = \frac{\omega_c}{2\pi} \)

2. The first sideband (lower frequency): \( \frac{m V_c}{2} \cos((\omega_c - \omega_m)t) \)
- Frequency: \( f_c - f_m = \frac{\omega_c - \omega_m}{2\pi} \)

3. The second sideband (higher frequency): \( \frac{m V_c}{2} \cos((\omega_c + \omega_m)t) \)
- Frequency: \( f_c + f_m = \frac{\omega_c + \omega_m}{2\pi} \)

### Step 3: Summarize the Frequencies
Thus, the three frequencies present in the AM wave are:
1. \( f_c \) (Carrier frequency)
2. \( f_c - f_m \) (Lower sideband frequency)
3. \( f_c + f_m \) (Upper sideband frequency)

### Step 4: Verify the Options
Now, we can check the options provided in the question to see which one matches our derived frequencies.

### Final Answer
The three frequencies contained in the AM wave are:
- \( f_c \)
- \( f_c - f_m \)
- \( f_c + f_m \)