Understanding the Concept:
In a standard Amplitude Modulated (AM) signal, the total transmitted power consists of the power in the carrier wave plus the power contained in the two symmetrical sidebands: the Upper Sideband (USB) and the Lower Sideband (LSB). The mathematical equation for total AM power is given by:
\[
P_t = P_c + P_{\text{USB}} + P_{\text{LSB}} = P_c \left(1 + \frac{\mu^2}{2}\right)
\]
where $P_c$ is the carrier power and $\mu$ is the modulation index.
Step 1: Formulating the Power in the Sidebands
The total power allocated to both sidebands combined ($P_{\text{sb}}$) is the difference between the total power and the carrier power:
\[
P_{\text{sb}} = P_t - P_c = P_c \cdot \frac{\mu^2}{2}
\]
Since a standard AM signal splits its sideband power equally between the Upper Sideband (USB) and the Lower Sideband (LSB), the power inside each individual sideband is exactly half of the total sideband power:
\[
P_{\text{each sideband}} = \frac{P_{\text{sb}}}{2} = \frac{1}{2} \left( P_c \cdot \frac{\mu^2}{2} \right) = P_c \cdot \frac{\mu^2}{4}
\]
Step 2: Extracting Values from the Problem Statement
From the problem description, we have:
• Carrier wave power: $P_c = 176 \, \text{W}$
• Modulation percentage: $60\%$
Convert the modulation percentage to a decimal value to find the modulation index ($\mu$):
\[
\mu = \frac{60}{100} = 0.6
\]
Step 3: Calculating Power for Each Sideband
Substitute $P_c = 176$ and $\mu = 0.6$ into our individual sideband power formula:
\[
P_{\text{each sideband}} = 176 \times \frac{(0.6)^2}{4}
\]
First, calculate the square of the modulation index:
\[
(0.6)^2 = 0.36
\]
Now substitute this back into the equation:
\[
P_{\text{each sideband}} = 176 \times \frac{0.36}{4}
\]
Simplify the fraction $\frac{0.36}{4}$:
\[
\frac{0.36}{4} = 0.09
\]
Now multiply by the carrier power:
\[
P_{\text{each sideband}} = 176 \times 0.09
\]
Let us calculate this final product step by step:
\[
176 \times 9 = 1584 \implies 176 \times 0.09 = 15.84 \, \text{W}
\]
Therefore, the power contained within each individual sideband is exactly 15.84W, matching option (D).