# Speed of light and speed of sound in the same medium

Can anybody please tell me if there is an established relationship between the speed of sound and speed of light when they are traveling in the same medium? Are they related to each other in some way? One finds that both, when propagating in a medium, are affected by the medium and its properties.

• Light is affected by the interference of the electromagnetic fields of the atoms and other electric phenomena and interference. I think density of material and the nature of bonding in a way can also contribute.

• Sound is affected by the state, tension, density, structure, temperature, pressure, etc., i.e, the material's properties.

But I can't see any clear relation between them for a particular material. Hence please help me in suggesting other factors that may affect. And is there a relation between the two speeds in a medium?

• Please provide some explanation or equation if a user wishes to answer – Tanmay Siddharth Jan 3 '20 at 7:11
• good question. I almost posted an answer until I realized that I needed to think about this a lot more... – niels nielsen Jan 3 '20 at 8:19
• A relation can be there but buried. I guess density plays a role at least on defining how much of the stuff is in the path. The dielectric constant should be affected by d – Alchimista Jan 3 '20 at 8:20
• Thank you, for answering the question via Equations and characteristics. I agree quite well with the answers provided til now. But can somebody emphasize on/ explain these answers via providing a more theoretical explanation? – Tanmay Siddharth Jan 3 '20 at 8:27
• Even within a single medium the propagation of light has differing speeds for different wavelengths (dispersion). Maybe one should start with an understanding of what causes that before trying to link the speeds of a mechanical longitudinal wave and an EM wave. On the other hand, there is piezoluminescence ... – Bill N Jan 3 '20 at 15:08

No direct relationship

Speed of light in medium depends only on material's electromagnetic properties: $$v_{\textrm{light}}={\frac {1}{\sqrt {\mu \varepsilon }}}$$ Where as speed of sound in general depends on how pressure in material changes in relation to density change: $$v_{\textrm{sound}}={\sqrt {\left({\frac {\partial p}{\partial \rho }}\right)_{s}}}$$

There is Clausius–Mossotti equation, which relates material permittivity to molecular polarizability $$\alpha$$ and number density of the molecules $$N$$ :

$${\frac {\varepsilon _{\mathrm {r} }-1}{\varepsilon _{\mathrm {r} }+2}}={\frac {N\alpha }{3\varepsilon _{0}}}$$

Number density relates to mass density in such way : $$N={\frac {N_{\rm {A}}}{M}}\rho _{\mathrm {m} }$$ where $$N_A$$ is Avogadro constant, M - molar mass.

• Isn't to neatly presented? I mean, the formula are correct but no direct link to the mechanism is apparent. The point is the behavior of epsilon and dp/dt in the material. – Alchimista Jan 3 '20 at 8:24
• see edit, I've added weak links – Agnius Vasiliauskas Jan 3 '20 at 9:34
• Thank you for the equation you have provided I was not aware of it – Tanmay Siddharth Jan 3 '20 at 17:17

NO. There is no relation at all. Sound waves are mechanical waves whose speed is function of mechanical elastic properties of the medium, e.g. $$v=\sqrt{\frac{E}{D}}$$ in solids and $$v=\sqrt{\frac{B}{D}}$$ in gases and liquids. On the other hand light waves are e.m.w. whose speed is function of electric and magnetic properties of the medium or even vacuum given by $$v=\sqrt{\frac{1}{\mu\epsilon}}$$, where $$E=$$Young's modulus, $$B=$$Bulk modulus, $$D=$$density, $$\mu=$$magnetic permeability, $$\epsilon=$$electric permittivity.

• I agree. But Aren't the elastic properties (spiring force or the elastic force) a function of electromagnetic forces. Can't they effect each other? – Tanmay Siddharth Jan 3 '20 at 8:11
• Tanmay Siddharth -,Sound waves are longitudinal mechanical waves while light is em transverse waves. No mutual effect is likely to happen. – user248651 Jan 3 '20 at 8:26
• Okay I agree, but please tell how density of medium would not be relating them to each other. – Tanmay Siddharth Jan 3 '20 at 8:31
• @TanmaySiddharth You are right in a way .But if you consider that threshold of pain (Max Allowed sound intensity for Human Ear)is 120 decibel corresponding to few millionth change of density of air, then the effect on mu and epsilon(hence the speed of emw) is extremely small.and normally not considered.The idea is smart but the magnitudes are toooo small to consider or calculate, – user248651 Jan 3 '20 at 8:42
• To my knowledge no one did,. Allow me to film the change in pressure and intensity of room air at NTP by a loud human voice sound wave in the NPT air by changing theater temperature when smoking a cigarette. Unless you're talking about a sound wave originating from an upward explosion of a heavy bomb.@TanmaySiddharth – user248651 Jan 3 '20 at 9:11

If there IS a relationship between them, here is what it might look like.

The speed of light in air is set by the permeability and the permittivity of air, as expressed in the equation cited above by Agnius Vasiliauskas.

Those characteristics are set by the details of how electric charge is distributed around the gas molecules in the air, which determine how the molecules respond to changing electromagnetic fields in their vicinity.

Changes in the density of the air and/or its temperature will change the permeability and the permittivity of the air in bulk, and hence the speed of light going through it. Now note that changes in the density and the temperature of the air will also change the speed of sound going through it, as cited by Lionheart above.

The coupling that Tanmay seeks then looks like a relationship between the permittivity and permeability of air as functions of pressure and temperature, and the elasticity and mass density of air as functions of pressure and temperature.

This question is very broad. There are many media and there are many wavelengths. It is quite conceivable that two completely different media exhibit the same speed of sound. Obviously there are factors that affect both sound and light but these do not allow to establish a general one to one relation between the two speeds, unless perhaps for very specific cases.

There can be a relation between electromagnetic waves and acoustic (sound) waves. This coupling originates from the Lorentz force. This force can be interpreted as a mechanical body force acting on the medium which depends on the electric and magnetic field. The coupling is mathematically described by the Maxwell stress tensor (https://en.wikipedia.org/wiki/Maxwell_stress_tensor). This interaction can give rise to electroelastic or magnetoelastic waves in solids.