The AM wave of equivalent to the summation of

Options

A. two sinusoidal waves
B. three sinusoidal waves
C. four sinusoidal waves
D. none of these

Answer

To determine how many sinusoidal waves an amplitude modulated (AM) wave is equivalent to, we can analyze the components of an AM wave step by step.

### Step-by-Step Solution:

1. **Understanding AM Wave Components**:
An AM wave consists of a carrier wave and a modulating signal. The modulating signal can be represented as:
\[ m(t) = A_m \sin(\omega_m t) \]
where \(A_m\) is the amplitude of the modulating signal and \(\omega_m\) is its angular frequency.

2. **Carrier Wave Representation**:
The carrier wave can be expressed as:
\[ c(t) = A_c \sin(\omega_c t) \]
where \(A_c\) is the amplitude of the carrier wave and \(\omega_c\) is its angular frequency.

3. **Combining the Waves**:
The AM wave can be represented as:
\[ AM(t) = (A_c + m(t)) \cdot \sin(\omega_c t) \]
Substituting \(m(t)\) into the equation gives:
\[ AM(t) = (A_c + A_m \sin(\omega_m t)) \cdot \sin(\omega_c t) \]

4. **Expanding the Equation**:
Expanding this expression leads to:
\[ AM(t) = A_c \sin(\omega_c t) + A_m \sin(\omega_m t) \sin(\omega_c t) \]

5. **Using Trigonometric Identities**:
We can use the trigonometric identity:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]
to rewrite the product \(A_m \sin(\omega_m t) \sin(\omega_c t)\):
\[ A_m \sin(\omega_m t) \sin(\omega_c t) = \frac{A_m}{2} \left( \cos((\omega_c - \omega_m)t) - \cos((\omega_c + \omega_m)t) \right) \]

6. **Final Expression**:
Thus, the complete expression for the AM wave becomes:
\[ AM(t) = A_c \sin(\omega_c t) + \frac{A_m}{2} \cos((\omega_c - \omega_m)t) - \frac{A_m}{2} \cos((\omega_c + \omega_m)t) \]
This shows that the AM wave consists of:
- One carrier wave
- Two additional sidebands (resulting from the modulation)

7. **Counting the Sinusoidal Waves**:
Therefore, the total number of sinusoidal waves in an AM wave is:
- 1 (carrier wave)
- 2 (sidebands)

This results in a total of **3 sinusoidal waves**.

### Conclusion:
The AM wave is equivalent to the summation of **three sinusoidal waves**.