# Sound wave propagation

How does a mixture of sound frequencies travel through the same medium? I know how it works for single frequency. I don't understand how it works for mixture of different frequencies(the movement of the particles should be different for different frequencies.I am confused here).

• Just like any other wave. In the small amplitude limit the medium is linear, and wave of different frequency can be superimposed. Jan 6, 2017 at 18:27

Thanks to the superposition principle of waves. What you call a "mixture of sound frequencies" is really a superposition of waves of different frequencies; for example your wave could look something like this: $$f(\textbf{x},t) = A \hbox{cos}(\textbf{k} \cdot \textbf{x} - 2 \pi f_1 t) + B \hbox{cos}(\textbf{k} \cdot \textbf{x} - 2 \pi f_2 t)$$ This means that at every point x in space, at every time t, the sound wave is the linear combination of two waves of different frequencies $f_1$ and $f_2$.

Thus the displacement of the particles in the medium of propagation is the sum of these two (in general, an arbitrary number) of displacements.

As overlapping waves can increase amplitude or cancel, so can waves of different frequencies.

Waves do exist, despite the media demand-driven ideology that all is particles, though they may be compression waves for instance.

Think of light through a lens, lights of different colors aren't travelling at different speeds, simply the different colors which are different frequencies are all still travelling at the speed of light in that medium.

Sound can be compared to the light, being a wave (The equations fits as a wave, which is also the only argument for both particles and quantum mechanics), the higher pitches blue, the lower pitches red, the speed of sound is also the same in whatever medium, it's the level of energy that's associated with that sound that changes the pitchs or frequency, and energy is conserved and remains the same.

While it's possible to filter certain colors, in general in a medium the type of wave travelling it stays the same within it.

But if you wish to stand by the particle motif:

Hook's Law which represent weights on a spring have that wave pattern in their motion. That motion of those "Particles" vibrating together each have different frequencies and those frequencies add on each other.

F(hooke) = F(x+2h) - F = k[u(x+2h,t)-u(x_h,t)]-k[u(x+h,t)-u(x,t)]

Sinusoidal waves are a man-made model, a useful fiction. Resolving a disturbance such as a wave or pulse or noise into sinusoidal components of different frequencies (Spectral Analysis) is a man-made technique for analyzing such motion mathematically.

The disturbance does not actually consist of a superposition of those components. So there is really nothing to explain.

Concepts such as group velocity and dispersion result from our attempt to fit sinusoidal waves as fundamental entities to the phenomena which we observe in nature.

Particularly in fluids, the propagation of the disturbance may have only a fleeting connection with the motion of individual particles within the fluid. So even a sinusoidal propagation does not necessarily match what is happening in the medium.

• How is that useful? You could say everything in physics is "a man-made model, a useful fiction", but I don't think it is the best answer to a question like this. Answers like mine can surely be improved by adding information about the approximations made in the sinusoidal wave model, but after all physicists think in terms of approximate models so really what is the point of your answer? I think that it is not fair stating that "The disturbance does not actually consist of a superposition of those components"; it does, in the limits of the wave model, so why disregard it this way? Jan 7, 2017 at 7:17
• @tomph My answer points out that the difficulties in explaining how superposition works are caused by the model, not by nature. I don't think the difficulty is approximations. Nevertheless I take your point that this is not providing an answer within the model. Jan 7, 2017 at 14:30