MDS said:
The part in bold is why this myth persists. There is ONE signal.
A musical waveform is a (potentially) infinite number of sine waves superimposed into one. Use an audio editor like Sound Forge and generate a 30 Hz sine wave at whatever amplitude you desire. Then generate a 1 khz sine wave at any amplitude you desire. Paste Mix them together. What do you get? One sine wave that contains both of the frequencies.
You have a new definition of sine functions I think
The generalization of a sine wave is A sin (w t +P) w/ amplitude A, frequency w and phase P w/ t being time. Phase just makes things
complicated so ignoring the phase of the two original sine waves, there is no way you can find a C3, w3, P3 such that :
C3 sin (w3 t + P3) = C1 sin (w1 t) + C2 sin (w2 t)
is true.
That is you can not find a new generalized sine function that is the sum of two sine waves where the new sine function is "composited" such that
C1,C2,C3,w1,w2,w3, P3 are *constants*.
What I think you are thinking of is the fact that you can write
cos((w1-w2)/2 t) sin ( (w1+w2)/2 t) = sin (w1 t) + sin (w2 t)
which people sometimes write the left side as
A(t) sin( w' t)
so A(t) is the new analogous amplitude of a sine wave w/ frequency w'.
This isn't really a sine wave because A(t) is a function of time.
Really you get the product of two sine waves w/ one phase shifted by pi/2 (since cos(x-pi/2) = sin(x) ). This can be thought of as a big wave w/ slow freq (w1-w2)/2 but superimposed inside exists a fast wave w/ freq (w1+w2)/2. That big wave w/ a slow frequency is the beat frequency since
if w1 is close to w2 then the resulting frequency is so slow you can hear it oscillate.
Another way to think of this is that your first statement is true. Sound in general can be written as a sum of an infinite number of sine waves
i.e
Sound(t) = Sum_i A_i sin (w_i t) + B_i sin (w_i t - pi/2)
where I am summing over an infinite of sine functions of frequencies
w_i (i.e. w_1, w_2, w_3 ...) and corresponding amplitudes A_i (i.e. A_1, A_2, A3 ... ) and amplitudes B_1, B_2, B3 .. The second term are just
sine functions phase shifted by pi/2 (cosines). In fact there is a very simple recipe, given a frequency w_i to find the corresponding amplitude A_i that that frequency contributes to your sound(t). That recipe is actually how real time analyzers on graphic equalizers work.
The reason for this has nothing to do with sound but because the right hand side is known fourier series that can be used to write any (periodic or finite) function. That is I can write any arbitrary function f(t) as
f(t) = Sum_i { A_i sin (w_i t) + B_i sin (w_i t - pi/2) }
where f(t) could be a square wave, the outline of my pet dog, or whatever
else you fancy.
so if I take your argument that Sound(t) is a sine function because it
can be expressed as an "potentially infinite number of sine functions" I have to also agree that *any* function such as a square wave or my dog is also a sine function for the same reason. But it would be silly to define *any* function as a sine wave just because it can be written as a superposition of an infinite number of sine waves.
Now what this has to do w/ speakers I'll leave to the experts
