# Increased Density Results in Decreased speed of sound?

1. Aug 22, 2011

### 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 particals 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,

2. Aug 22, 2011

### 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.

3. Aug 22, 2011

### 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)?

4. Aug 22, 2011

### 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.