In amplitude modulation, the sinusoidal carrier frequency used is denoted by $$\omega_{c}$$ and the signal frequency is denoted by $$\omega_{m}$$. The bandwidth $$\Delta\omega_{m}$$ of the signal is such that $$\Delta\omega_{m} \ll \omega_{c}$$. Which of the following frequencies is not contained in the modulated wave?

In amplitude modulation we first take a high-frequency carrier wave. Let its instantaneous value be written as

$$c(t)=A_c \cos(\omega_c t),$$

where $$A_c$$ is the carrier amplitude and $$\omega_c$$ is the angular frequency of the carrier. Now a comparatively low-frequency message or signal wave

$$m(t)=A_m \cos(\omega_m t)$$

with angular frequency $$\omega_m$$ (and bandwidth $$\Delta\omega_m$$ such that $$\Delta\omega_m \ll \omega_c$$) is superposed on the carrier. In the simplest textbook treatment the modulated (AM) wave is written as

$$s(t)=\bigl[\,1+k\,m(t)\bigr]\;A_c\cos(\omega_c t),$$

where $$k$$ is called the modulation index. Substituting $$m(t)=A_m\cos(\omega_m t)$$ we obtain

$$s(t)=A_c\cos(\omega_c t)+kA_cA_m\cos(\omega_m t)\cos(\omega_c t).$$

We now expand the product term. Before doing that we explicitly state the trigonometric identity that will be used:

$$\cos\alpha\,\cos\beta=\frac{1}{2}\Bigl[\cos(\alpha+\beta)+\cos(\alpha-\beta)\Bigr].$$

Applying this identity with $$\alpha=\omega_m t$$ and $$\beta=\omega_c t$$ gives

$$\cos(\omega_m t)\cos(\omega_c t)=\frac{1}{2}\Bigl[\cos\!\bigl((\omega_c+\omega_m)t\bigr)+\cos\!\bigl((\omega_c-\omega_m)t\bigr)\Bigr].$$

Placing this result back into the expression for $$s(t)$$ we get

$$\begin{aligned} s(t)&=A_c\cos(\omega_c t)+kA_cA_m\;\frac{1}{2}\Bigl[\cos\!\bigl((\omega_c+\omega_m)t\bigr)+\cos\!\bigl((\omega_c-\omega_m)t\bigr)\Bigr]\\[4pt] &=A_c\cos(\omega_c t)+\frac{kA_cA_m}{2}\cos\!\bigl((\omega_c+\omega_m)t\bigr)+\frac{kA_cA_m}{2}\cos\!\bigl((\omega_c-\omega_m)t\bigr). \end{aligned}$$

From this final expression we can read off all the distinct angular frequencies that physically appear in the modulated wave:

1. The original carrier frequency $$\omega_c$$.
2. The upper sideband frequency $$(\omega_c+\omega_m)$$.
3. The lower sideband frequency $$(\omega_c-\omega_m)$$.

Notice carefully that the lone message frequency $$\omega_m$$ is not present as a separate component in the AM spectrum; it shows up only in the two sidebands combined with the carrier frequency.

Therefore, among the given options, the frequency that is not contained in the amplitude-modulated wave is precisely $$\omega_m$$, corresponding to Option B.

Hence, the correct answer is Option B.