Increased Density Results in Decreased speed of sound?

[Da Apprentice]

According to the equation c=√(C/ρ) where c is the speed of sound, C is the coefficient of stiffness and ρ is the density of the medium throughout which a sound is played the speed of sound should acctually decrease with an increase in density. Why is this so? I would've thought that increasing the amount of particles within a given area that sound can be transmitted along would result in the increased speed of the sound. I thought it was exactly due to the increased density of water in relation to ait that it transmitted sound faster. Is this formula wrong or am I reading it wrong? Also could someone please explain the coefficient of stiffness part of the equation and where this relates to the physical properties of a medium?

Thanks,

 

[TR_KY]

I am not the best to explain this, I work in applied work. The wave equation for acoustic wave propagation is derived from our understandings of stress and F=ma. What we are really talking about is very small disturbances in matter, as such the mass is an important factor but in the derivation of the wave propagation equation the volume cancels out leaving only the density.

The coefficient of stiffness is what would be called in seismic work the bulk modulus which is the resistance of a material to compression, at least in the acoustic case but not in the elastic case where it would be multiple moduli. Pretty much any book on geophysics should give you a good explanation of elastic wave propagation, sorry mine are all in storage.

 

[olivermsun]

Think of a simple harmonic oscillator (e.g., weight m dangling from a spring with constant k). Intuitively, what do you expect to happen to the oscillations if you increase the mass m (or more to the point, the ratio m/k)?

 

[cjl]

The formula is exactly right. The stiffness portion of the equation is the reason why water has a much higher speed of sound than air, not the density. The relevant stiffness for sound is compression, since sound is a compression wave. Think of how hard water is to compress compared to air. That is the reason why water has a very high sound speed. As oliver said, a good way to think about it is a simple mass spring system - if the mass is increased, and the spring is kept the same, the oscillations slow down.

 

[PJC01]

I understand that it may seem counterintuitive that increased density would result in a decreased speed of sound. However, this is actually in line with the physics behind sound propagation. Let me explain.

The equation you mentioned, c=√(C/ρ), is known as the acoustic wave equation. It describes the relationship between the speed of sound (c), the coefficient of stiffness (C), and the density of the medium (ρ). The coefficient of stiffness refers to the ability of a medium to resist deformation under stress. In simpler terms, it is a measure of how rigid a material is.

Now, let's consider the physical properties of a medium. When a sound wave travels through a medium, it causes the particles of the medium to vibrate. These vibrations then propagate as sound waves. In a less dense medium, there is more space between particles, allowing them to vibrate more easily and transmit sound faster. On the other hand, in a denser medium, there is less space for particles to move, making it more difficult for them to vibrate and transmit sound. This is why the speed of sound decreases with an increase in density.

To address your question about water and air, it is true that sound travels faster in water than in air. However, this is not solely due to the density of water being higher than air. Water also has a higher coefficient of stiffness, meaning it is more rigid and can transmit sound waves more efficiently.

So, in summary, the acoustic wave equation is not wrong, and you are not reading it wrong. It is simply describing the relationship between density, coefficient of stiffness, and the speed of sound. I hope this explanation helps to clarify any confusion.