Trouble in understanding AM modulation Amplitude modulation is in fact, superimposing the low frequency transmission signal into a high frequency carrier signal, right?
So, if the transmission signal can be represented as $c(t)=A_c \sin (\omega_ct )$ and the carrier wave can be represented as $c(t)=A_m \sin (\omega_mt)$, then upon superimposing and simplifying the equation using trigonometric identities we will have
$$
c_m(t)=A_c \sin (\omega_ct )+\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t-\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t,
$$
but this equation has the carrier wave and two sinusoidal waves with frequencies $\omega_c-\omega_m$ and $\omega_c+\omega_m$. So where is the signal which is transmitted? What we've got is a corrupted signal with uneven frequencies i.e. $\omega_c-\omega_m$ and $\omega_c+\omega_m$
Correct me if I'm wrong at any point.
 A: You are correct in your assertion that the AM procedure, when applied to a monochromatic signal, will generate a radio emission that, spectrally, has a spike at the carrier and then two sidebands on either side, separated from the carrier frequency $\omega_c$ by the modulation frequency $\omega_m$.
More specifically:


*

*The strength of the signal is encoded in the depth of the modulation and therefore on the strength of the sidebands when compared to the carrier.

*The frequency of the signal is encoded in the separation of the sidebands with respect to the carrier.

*The phase of the signal is encoded in the phase of the sidebands with respect to the carrier.


This means that the amplitude-modulation procedure has successfully encoded all the relevant information of your signal into a higher-frequency radio beam that can be easily transmitted. It is then the job of the detector (i.e. your consumer radio) to decode that information into audio signals that can be played by a speaker; how the decoder does that is up to the device, but the important thing is that the full information is there to be decoded.
The decoding procedure itself can be done by forgetting about this sideband business and just looking at the signal in the time domain with an envelope detector, or you can explicitly use this carrier-sideband structure with a locally-generated carrier that you then use to extract information from the sidebands in a synchronous detector.
A: You are right that by superimposing a constant modulation on the carrier wave you cannot transmit a signal because you get a constant single frequency carrier wave plus two constant sideband frequencies. For the transmission of a signal, you need a modulation changing in time (frequency and/or amplitude) like in speech transmission in AM radio or in Morse signals transmission in amateur shortwave radio. 
A: The low frequency transmission signal means slowly changing $A_c(t)$, not fixed $A_c$ with fixed $\omega$ which does not carry any information.
