Understanding Pressure Wave Velocity in Rigid Pipes

The question asks how the velocity of a pressure wave varies in a rigid pipe containing a fluid with density $\rho$ and viscosity $\mu$. A pressure wave, in this context, is essentially an acoustic wave or sound wave traveling through the fluid.

Factors Affecting Wave Speed in Fluids

The speed of a pressure wave (or sound) in a fluid primarily depends on two properties of the fluid:

1. Its compressibility (how much its volume changes under pressure), represented by the bulk modulus ($K$).
2. Its inertia (resistance to acceleration), represented by its density ($\rho$).

The standard formula for the speed of sound ($c$) in a fluid is given by:

$\qquad c = \sqrt{\frac{K}{\rho}}$

Here:

• $c$ is the speed of the pressure wave.
• $K$ is the bulk modulus of the fluid.
• $\rho$ is the density of the fluid.

Role of Rigid Pipe

The term 'rigid pipe' is important. It means that the walls of the pipe do not deform significantly when the pressure changes. This simplifies the problem, ensuring that the wave speed is determined mainly by the fluid properties, particularly its bulk modulus and density, without significant influence from the pipe's elasticity.

Dependence on Density

Looking at the formula $c = \sqrt{\frac{K}{\rho}}$, we can see the relationship between the wave speed $c$ and the fluid density $\rho$. Assuming the bulk modulus $K$ of the fluid remains constant (which is often a reasonable assumption for a given fluid under specific temperature and pressure conditions, although $K$ itself can sometimes depend on pressure), the velocity $c$ is inversely proportional to the square root of the density $\rho$.

Mathematically, this proportionality can be written as:

$\qquad c \propto \frac{1}{\sqrt{\rho}}$

Considering Viscosity

The question also mentions viscosity ($\mu$). Viscosity is the fluid's resistance to flow and internal shear stresses. While viscosity plays a significant role in how waves attenuate (lose energy) as they travel through the fluid, its primary effect on the fundamental wave speed formula $c = \sqrt{\frac{K}{\rho}}$ is typically secondary or negligible in many common scenarios, especially when compared to the influence of bulk modulus and density. The standard formula for acoustic speed relies on the elastic and inertial properties, not viscous ones. Therefore, for the "variation" asked in the question, the dominant term is the inverse square root dependence on density.

Comparing with Options

We found that the velocity of the pressure wave varies as $1/\sqrt{\rho}$. Let's look at the given options:

• Option 1: $\rho$
• Option 2: $\sqrt{\rho}$
• Option 3: $\rho/\mu$
• Option 4: $1/\sqrt{\rho}$

Our derivation shows that the velocity is proportional to $1/\sqrt{\rho}$. This matches Option 4.

Conclusion on Pressure Wave Velocity

Based on the fundamental formula for the speed of sound in a fluid and the negligible impact of viscosity on the fundamental speed formula compared to density and bulk modulus, the velocity of a pressure wave in a rigid pipe carrying a fluid varies as $1/\sqrt{\rho}$.

Revision Table: Fluid Wave Properties

Additional Information: Bulk Modulus and Fluid Compressibility

The bulk modulus ($K$) is a measure of how resistant a substance is to compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease in volume. For a fluid, a higher bulk modulus means it is less compressible, and pressure changes propagate faster through it.

The formula for bulk modulus is:

$\qquad K = -\frac{\Delta P}{\Delta V / V}$

Where:

• $\Delta P$ is the change in pressure.
• $\Delta V$ is the change in volume.
• $V$ is the original volume.

The negative sign is included because an increase in pressure ($\Delta P > 0$) typically causes a decrease in volume ($\Delta V < 0$), making $K$ positive. For many liquids, the bulk modulus is very high, indicating they are relatively incompressible compared to gases.

In the context of wave speed, a higher bulk modulus ($K$) leads to a higher wave speed, as the pressure disturbance is efficiently transmitted through the less compressible medium. Conversely, a higher density ($\rho$) means more inertia, requiring more force to accelerate the fluid particles, thus slowing down the wave propagation.