If a message signal of frequency $$f_m$$ is amplitude modulated with a carrier signal of frequency $$f_c$$ and radiated through an antenna, the wavelength of the corresponding signal in air is

In amplitude modulation (AM), a message signal of frequency $$f_m$$ modulates a carrier signal of frequency $$f_c$$. The resulting AM signal contains three frequency components: the carrier frequency $$f_c$$, and two sidebands at frequencies $$f_c + f_m$$ and $$f_c - f_m$$.

When this AM signal is radiated through an antenna, the signal that is actually transmitted through the air is the carrier wave with the modulated amplitude. The wavelength of the radiated signal corresponds to the carrier frequency, since the carrier is the dominant component that determines the electromagnetic wave's wavelength.

The wavelength of an electromagnetic wave in air is given by $$\lambda = \frac{c}{f}$$, where $$c$$ is the speed of light.

For the carrier signal, the wavelength is $$\lambda = \frac{c}{f_c}$$.

Therefore, the wavelength of the corresponding signal in air is $$\frac{c}{f_c}$$.