How Is the Speed of Sound Derived Using Bulk Modulus and Density?

[azure kitsune]

Homework Statement

Show that $$v = \sqrt{B/\rho}$$

Homework Equations

The Attempt at a Solution

Let the density of the medium be ρ.

Suppose you have a single pulse of a longitudinal wave inside a pipe traveling with speed v.

At any time, consider the location of the pulse. Originally, the mass at the location was m. Suppose the pulse increases the mass of its current location by Δm. With the pulse the total mass there becomes m + Δm.

The mass at the location of the pulse must be compressed; otherwise, its volume would expand by ΔV to keep the density constant. We have

$$\frac{m+\Delta m}{V + \Delta V} = \frac{m}{V} \implies \rho = \frac{\Delta m}{\Delta V}$$

So the mass at that location must be compressed by ΔV. From the definition of the bulk modulus, we have

$$B = \frac{\Delta P}{-\Delta V / V} = \frac{V}{\Delta m} \cdot \Delta P \cdot \frac{\Delta m}{-\Delta V}$$

Then we have

$$B = \frac{V}{\Delta m} \cdot \Delta P \cdot \rho \implies \frac{B}{\rho} = \frac{V}{\Delta m} \cdot \Delta P$$

ΔP must be the pressure difference between a stationary pulse and a moving one, so we have by Bernoulli that

$$P_{stationary} = P_{pulse} + \frac{1}{2} \left(\frac{m+\Delta m}{V}\right)v^2$$

so

$$\Delta P = \frac{1}{2} \left(\frac{m+\Delta m}{V}\right)v^2$$

We end up with:

$$\frac{B}{\rho} = \frac{V}{\Delta m} \cdot\frac{1}{2} \left(\frac{m+\Delta m}{V}\right)v^2 = \frac{m+\Delta m}{2\Delta m}v^2$$

Something must have gone wrong somewhere.

 

[jbsteele]

However, if we make the assumption that Δm is much smaller than m, then this equation simplifies to \frac{B}{\rho} = \frac{m}{2\Delta m}v^2 so v = \sqrt{\frac{B}{\rho}} as desired.

 

[kerimek]

Great work on deriving the speed of sound equation! However, there is a slight error in your attempt. In the Bernoulli equation, the pressure difference should be between the stationary pulse and the moving pulse, not the stationary pulse and the moving medium. This means that the equation should be P_stationary = P_pulse + (1/2)(m/V)v^2 instead of P_stationary = P_pulse + (1/2)(m+Δm/V)v^2. This will lead to the correct equation of v = √(B/ρ). Keep up the good work!