Speed of Sound Waves

Waves in a Harmonic Oscillator Chain

The speed of sound in solids can be modeled using a chain of masses connected by springs, representing atoms in a crystal lattice. This model helps us understand how mechanical waves propagate through discrete systems.

• Discrete Model: Consider an infinite chain of masses m connected by springs of stiffness K and rest length a.

• Displacement: The displacement of the nth mass from equilibrium is yn.

• Force on nth Mass:

• Equation of Motion:

• Wave Speed (Dimensional Analysis): The only combination of K, m, and a with units of speed is

Polynomial Approximation of Functions

To analyze wave propagation, we often approximate functions using Taylor series expansions near a point x0.

• Approximation:

• Purpose: This allows us to treat discrete variables as continuous, simplifying the analysis of wave equations.

Continuous Limit of Harmonic Oscillator Chain

When the spacing a is much smaller than the wavelength λ, the system can be treated as a continuous medium.

• Position:

• Displacement Function:

• Taylor Expansion:

• Force in Continuous Limit:

• Wave Equation:

• Wave Speed:

Speed of Sound in Liquids and Gases

Sound Propagation in Liquids

Sound waves in liquids can be modeled by dividing the liquid into small segments, each acting as a mass connected by an elastic spring.

• Model: Uniform cylinder of cross-sectional area S, mass density ρ, and equilibrium pressure P.

• Segment Mass:

• Force from Pressure Change:

Bulk Modulus and Hooke's Law for Liquids

The pressure change in a liquid due to compression is described by Hooke's law, using the bulk modulus B.

• Pressure Change: , where ,

• Force:

• Spring Constant:

• Speed of Sound:

Speed of Sound in Gases

In gases, sound waves cause rapid pressure variations, and the bulk modulus is related to the adiabatic process.

• Bulk Modulus:

• Pressure:

• Speed of Sound:

• Adiabatic Index (γ):

  • Monatomic gases:

  • Diatomic/linear molecular gases:

  • Non-linear molecular gases:

Exercises

Exercise 1: Speed of a Longitudinal Wave in a Slinky

A slinky is a long spring of mass m, equilibrium length L0, spring constant K, and N windings of radius R. Find the speed of a longitudinal wave in a slinky stretched to length L and the time for a pulse to travel from one end to the other.

• Wave Speed: (Additional info: Derived using the stretched length and mass distribution.)

• Travel Time:

Exercise 2: Speed of Sound in Water

Compressing water in a rigid cylinder increases pressure by MPa, decreasing volume by 2.25%. Use this to predict the speed of sound in water.

• Bulk Modulus:

• Speed of Sound:

Exercise 3: Speed of Sound in Air at Different Temperatures

Given air density at 300 K is 1.177 kg/m3 and atmospheric pressure is 1 atm = 101.3 kPa, calculate the speed of sound in air at T = 250, 300, and 350 K.

• Speed of Sound:

• Application: Use γ for diatomic gases (air is mostly N2 and O2), .

Additional info: These exercises reinforce the application of wave speed formulas in real-world scenarios.