Amplitude Modulation of a wave

[Prashasti]

Homework Statement

I am utterly confused. When I was reading my textbook , I found something unacceptable.


While deriving an expression for a modulated wave,
It's been given that
"A sinusoidal carrier wave can be represented as c (t) = A_c sin
(ωt + Φ)
where c (t) is the signal strength of the carrier wave.
Let m (t) = A_m sinω_mt represent the message or the modulating signal.
The modulated signal c_m (t) can be written as
c_m= (A_c+A_msin ω_mt) sin ω_ct

I wonder how's it possible! Shouldn't it be c_m (t) = A_csin ω_ct + A_m sinω_mt ?

But then I made an adhoc assumption - which was not satisfactory - but I thought it could be justified from a more rigorous application of mathematics. So, I continued reading

On the next page, I found something in contrast to my "assumption".
In the topic "Production of amplitude modulated wave" -
According to my textbook "Here the modulating signal A_m
sinω_mt is added to the carrier signal A_csinωt to produce the signal x (t). This signal x (t) = A_m sinω_mt + A_c sin ω_ct is passed through a square law device."

Now this equation for x (t) is different from the one which was used (in the textbook) earlier.
What even is happening?

 

[ehild]

If the sum of two signals of different frequencies ω₁ and ω₂ is the input of a linear network, the output would be again the sum of two signals with the same frequencies.
Using a nonlinear device, one, for example, that produces the square of the input, it will "mix" the frequencies. Expand (Asin(ω₁t) + Bsin(ω₂t ))², what do you get? ( there will be components with frequencies of 2ω₁, 2ω₂, ω₁+ω₂, ω₁-ω₂).
Semiconductor diodes, transistors have nonlinear characteristics, and can be used for "mixing frequencies"

 

[Prashasti]

If the sum of two signals of different frequencies ω₁ and ω₂ is the input of a linear network, the output would be again the sum of two signals with the same frequencies.
Using a nonlinear device, one, for example, that produces the square of the input, it will "mix" the frequencies. Expand (Asin(ω₁t) + Bsin(ω₂t ))², what do you get? ( there will be components with frequencies of 2ω₁, 2ω₂, ω₁+ω₂, ω₁-ω₂).
Semiconductor diodes, transistors have nonlinear characteristics, and can be used for "mixing frequencies"

Ok.
So, the problem lies in the arrangement of all that in my textbook..

 

[Prashasti]

They mentioned the result before deducing the expression for the same.
I got it!
Thanks!