(a) With neglect of viscosity and heat conductivity, small...

(a) With neglect of viscosity and heat conductivity, small disturbances in a fluid propagate as undamped sound waves. Given the relation $p=$ $p(\rho, S)$, where $p$ is pressure, $\rho$ is the mass density, $S$ is the entropy, derive an expression for the sound wave speed $v$. (b) As an example of such a fluid, consider a system of identical, noninteracting spin $1 / 2$ particles of mass $m$ at the absolute zero of temperature. The number density is $n$. Compute the sound speed $v$ in such a system.

(a) With neglect of viscosity and heat conductivity, small disturbances in a fluid propagate as undamped sound waves. Given the relation $p=$ $p(\rho, S)$, where $p$ is pressure, $\rho$ is the mass density, $S$ is the entropy, derive an expression for the sound wave speed $v$.
(b) As an example of such a fluid, consider a system of identical, noninteracting spin $1 / 2$ particles of mass $m$ at the absolute zero of temperature. The number density is $n$. Compute the sound speed $v$ in such a system.

Key Concepts

Equation of State for a Degenerate Fermi Gas

For a system of noninteracting spin 1/2 particles at absolute zero, quantum statistics lead to a specific equation of state where the pressure arises from the degeneracy pressure of fermions filling states up to the Fermi energy. Typically, the pressure depends on the number density raised to the 5/3 power in a three-dimensional system, which is key to determining dynamical quantities such as the sound speed.

Sound Wave Propagation in Fluids

Sound waves are small, adiabatic disturbances that propagate through a fluid by locally compressing and expanding the medium. In the absence of viscosity and heat conductivity, the governing equations (like Euler’s equations) can be linearized, leading to a relation for sound speed in terms of the derivative of pressure with respect to density under constant entropy conditions.

Adiabatic Pressure-Density Relationship

Under adiabatic processes, where entropy remains constant, the speed of sound in a fluid is determined by how pressure changes with density. Mathematically, this is expressed as v² = (?p/??)_S. This concept links the mechanical properties of the medium (through its equation of state) to dynamic processes like wave propagation.

Fermi Energy and Sound Speed Relation

In a degenerate Fermi gas, the Fermi energy sets a natural velocity scale (the Fermi velocity, v_F). Through the relation v² = (1/m)(dp/dn), where m is the particle mass and n is the number density, one can show that the sound speed is proportional to the Fermi velocity, often found to be v = v_F/?3. This connection illustrates how quantum mechanical effects influence classical wave propagation in such systems.

Transcript

00:01 So question a asked for the speed of the sound for an ideal gas with molar mass being m.

00:09 The speed of the sound is because of the square root of this quantity, bulk modulus over the density, where for ideal gas, the adjabatic bulb modulus is equals to this negative of the volume multiplied by dp over the value.

00:34 Or the change in pressure with respect to volume.

00:43 And in a jabatic process, this expression, the product of p and v raised to this gamma, which is the ratio of heat capacities is constant.

01:18 Ratio of heat capacity cross -npressure process over the constant volume process.

01:23 And taking the full derivative of both sides of the equation, then we have this.

01:31 So first with respect to pressure, then with respect to volume, and we know that the derivative of constant is zero.

01:48 Then we have this.

02:04 Simplifying or moving to the rights of the equation, this, then isolating dp over dv, which is also equals to this, which we can cancel this out, then we can find that dp over div is equal to this.

02:49 Since v -raised negative 1 is 1 over v.

02:54 Then inserting it in our equation for b, then we have, don't forget the negative -stern.

03:12 We can see that the jubatic bulk modules for ideal as well be just equals to the product of the ratio -fit capacities and the pressure.

03:23 Then inserting it our equation for the speed of the sound, we have this.

03:31 And we know that the density is equal to the mass over the volume.

03:41 Then our equation becomes like this...

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