If frequency of a sound wave increases, why does the speed of sound remain the same? [duplicate]

Could someone explain why speed of sound doesn't change when frequency is increased?

My thinking is, if frequency increases then the period will decrease, thus air molecules will be oscillating faster. Shouldn't this speed up the transfer of the sound wave through air?

I don't understand why velocity doesn't increase as the frequency increases...

• You obviously know that 𝑓 λ = 𝑣 where 𝑣 is the speed of sound. So are you asking why is the speed of sound constant when the frequency changes? Jan 3 at 8:12
• yes. I don't understand the logic of it. Jan 3 at 8:13
• In that case you should change your question to explicitly ask "Why is the speed of sound independent of the frequency". As your question currently written it is not obvious that is what you're asking. Jan 3 at 8:15

Higher frequencies correspond to shorter wavelengths. Thus, in a case of a dispersionless media the frequency and the wave length are inversely proportional to each other as $$f=\frac{c}{\lambda},$$ where the proportionality constant is called speed of sound/light/etc. (depending on what kind of waves we are talking about.)

More formally, the frequency of waves can depend on a wave length in a more complex manner, as a function $$\omega(\mathbf{k})$$ (where I use usual definitions of $$\omega=2\pi f$$, $$k=|\mathbf{k}|=2\pi/\lambda$$). In this case one usually defines phase velocity, $$\mathbf{v}_{ph}(\mathbf{k})=\frac{\omega(\mathbf{k})}{k}\frac{\mathbf{k}}{k}$$ and group velocity $$\mathbf{v}_g(\mathbf{k})=\nabla_\mathbf{k}\omega(\mathbf{k})= \frac{\partial\omega(\mathbf{k})}{\partial\mathbf{k}}$$ The speed of waves can be the magnitude of either of these two velocities - depending on the context. We then call a medium dispersionless, if $$\omega(\mathbf{k})=c|\mathbf{k}|$$ - in which case the two speeds are equal.

• Thanks. I'm stuck on the logic that increased frequency of air molecule --> decreased period of air molecule --> air molecules are oscillating faster --> sound wave is transferred through the air faster --> sound wave covers more distance --> increased wavelength . Jan 3 at 9:11
• @mar01 wave is not particle, it does not "cover distance" - instead the wave exists everywhere and has frequency and wavelength. The product of these two has dimensionality of speed/velocity and is called this way, but it is not velocity of some particles. Now, when the wave is generated, its front propagates with group velocity, which in dispersionless media (and air is nearly dispersionless) is the same as the product of frequency and wavelength (i.e., phase velocity). All this has to do with the properties of waves - it is not logical consequence of only how fast molecules oscillate. Jan 3 at 9:17

Sound is a periodic disturbance of a medium, we can generalize to any disturbance of a medium.

For a thought experiment, imagine a very long wire in tension. About the same tension as a guitar string, but much longer.

You give that wire a whack with a hammer, so the initial disturbance is local. That disturbance starts to propagate. What factors determine the speed of that propagation?

Tension of the wire
Wire that is away from the equilibrium position pulls on adjacent wire, pulling it away from equilibrium. The higher the tension, the more force is available to pull on neighbouring wire. The tension in the wire pulls back to the equilibrium position, and then (because of velocity) the motion overshoots, which is also communicated to neighbouring wire.

Opposition to change of velocity
In this thought experiment the wire is composed of matter, it has inertial mass. For the same amount of tension force: a thicker wire will have a more sluggish response to exerted force, because it has more inertia. Once the wire has velocity it is harder to reverse that velocity, because it has more inertia.

Translating that to propagation of sound in air:

Every gas has an elasticity; air has elasticity. The elasticity of air provides restoring force.

In a gas with a higher or lower density sound propagates at a higher/lower velocity, because with higher/lower density you have more/less inertia per unit of volume.

I started with the thought experiment of a long wire under high tension because with a wire you can have a disturbance with the amplitude of the disturbance at right angles to the direction of propagation.

I figure that makes it easier to build an understanding of why (in general) speed of propagation of a disturbance is independent of frequency.

In the case of the wire: the speed of the sideways motion (frequency of oscillation) is of minor influence; it is because of the tension in the wire that a disturbance is communicated along the length of the wire.

"Why" questions in physics are tricky, since ultimately there is no answer, things just are the way they are. There are a few things to note, however:

• The speed of sound is not independent of the frequency/wavelength in general, it is only approximately true in certain ranges of frequencies and amplitudes. For example, if the wavelength is shorter than the typical distance between the particles in the gas, or if the period is shorter than the typical scattering times between the particles, then obviously the whole description fails. Similarly, the wavelengths have to be shorter than large scale features of the gas, such as the size of the container it is in, or its large-scale variability.

• In general, the theory of sound waves with constant speed works under the assumption that the transformations of the gas elements are adiabatic. This means that the perturbations have to be small enough so that the gas is always limitingly close to thermal equilibrium. But a gas in thermal equilibrium is perfectly homogeneous, so this is obviously never exactly true for sound waves. In theoretical physics, we say that this description of the sound wave is only asymptotic.

• Similarly, the speed of sound is not independent of frequency for large perturbations, see solitons and shock waves.

• A good way to visualize waves with no dispersion (speed of sound independent of frequency) is to transform into the frame comoving with the wave train. Additionally, it is good to think about the wave simply as a shape in position space, it can have any shape with various features, no need to visualize it only as a sinusoid. In the comoving frame the wave simply keeps its shape. Small features of the wave train stay the same, large features of the wave stay the same. One can perform a Fourier transform of this shape. Small features correspond to contributions from short wavelengths and large features to long wavelengths. In theoretical physics one would say the dynamics of the wave train are scale-free. As already discussed, this will be true only as long as the features do not meet even by far the characteristic microscopic scales of the gas. In particular, the linear sound wave only applies in the limit when the motion and features of individual molecules are completely washed out in the collective statistical behaviour of the gas.

• Speed of sound is the speed of propagation of a perturbation in the medium, as a characteristic of the medium itself.

• You can think at the frequency as the inverse of the time between the creation of two successive perturbations, as a characteristic of the source of sound.

The higher the frequency, the closer the perturbations are (i.e. the smaller the wavelength, the "distance in space"), the higher the pitch you hear.

But once produced, in a non-dispersive medium, the velocity of each perturbation and thus the distance in space between the two perturbations are constant.