Identify the mathematical expression for amplitude modulated wave:

Options

A. `A_(c)sin(omega_(c )+k_(1)v_(m)(t)t+phi`
B. `A_(c)sin(omega_(c)t+phi+k_(2)v_(m)(t))`
C. `(A_(c)+k_(2)v_(m)(t))sin(omega_(c)t+phi)`
D. `A_(c)v_(m)(t)sin(omega_(c)t+phi)`

Answer

To identify the mathematical expression for an amplitude modulated (AM) wave, we can follow these steps:

### Step 1: Understand the Components of AM Wave
An amplitude modulated wave consists of two main components: the carrier wave and the message signal. The carrier wave is typically represented as \( C(t) \) and the message signal as \( M(t) \).

### Step 2: Write the Carrier Wave Equation
The carrier wave can be expressed mathematically as:
\[ C(t) = A_c \sin(\omega_c t) \]
where:
- \( A_c \) is the amplitude of the carrier wave,
- \( \omega_c \) is the angular frequency of the carrier wave.

### Step 3: Write the Message Signal Equation
The message signal can be expressed as:
\[ M(t) = A_m \sin(\omega_m t) \]
where:
- \( A_m \) is the amplitude of the message signal,
- \( \omega_m \) is the angular frequency of the message signal.

### Step 4: Combine the Carrier and Message Signal
The amplitude modulated wave \( C_M(t) \) can be represented as the sum of the carrier wave and the modulated message signal. The general form is:
\[ C_M(t) = [A_c + k \cdot A_m \cdot M(t)] \cdot \sin(\phi) \]
where \( k \) is the modulation index.

### Step 5: Substitute the Expressions
Substituting the expressions for \( C(t) \) and \( M(t) \) into the equation, we get:
\[ C_M(t) = \left[A_c + k \cdot A_m \cdot \sin(\omega_m t)\right] \cdot \sin(\omega_c t) \]

### Step 6: Final Expression
The final expression for the amplitude modulated wave can be simplified to:
\[ C_M(t) = A_c \sin(\omega_c t) + \frac{A_m}{2} \sin(\left(\omega_c + \omega_m\right)t) - \frac{A_m}{2} \sin(\left(\omega_c - \omega_m\right)t) \]
This represents the AM wave, showing how the carrier wave is modulated by the message signal.

### Conclusion
Thus, the mathematical expression for an amplitude modulated wave is:
\[ C_M(t) = A_c \sin(\omega_c t) + A_m \sin(\omega_m t) \]