Lecture 15: Sound Waves: Sound Speed

Sound as pressure wave

• Another view on sound is to treat it as a pressure wave
• This view is "macroscopic" and simplier, since pressure is already a quantity averaged over motions of atoms in some volume. So wavefronts are surfaces of constant pressure, etc.
  
• Units of pressure are Pascales Pa = N/m² = kg/(m s²)
• Standard atmospheric pressure is 101.325 kPa ~ 10⁵Pa

What pressure oscillation results from oscillating displacements of atoms ?

• We shall consider plane waves for simplicity. The displacement wave is given by the formula y(x,t) = A cos( k_x x - ω t + φ₀ ) where y(x,t) is a displacement in x of the plane
• Last time we have showed on the board that the displacement wave causes local volume compression/rarification with
• In bulk materials, the volume compression corresponds to pressure increase over the ambient pressure, Δ p = P - P_a as described by bulk modulus B
• B plays the role of spring constant in Hooke's Law, but now for volumetric deformations (strains). Units of bulk modulus are Pascales, as that for pressure. Please browse through Chapter 11.4 for discussion of deformations in bulk materials.
  Notice the difference with Hooke's law. Here it is the relative change of the volume that is involved, while in Hooke's law we deal with the absolute change in the spring length.
  
  
• If we revert the relation between the volume and pressure we get So bulk modulus describes how local pressure responds to the change of the local density.

• For solids, B is essentially constant, depending on composition. It is typically in the range of 10¹¹ Pa .
• For gases, bulk modulus depends on density and temperature of the gas. So it matters how gas is compressed in the wave - is it isothermal compression ? adiabatic compression ? Typical value in air at room temperature is ~ 10⁵ Pa

Properties of pressure waves

• Now we ready to write the formula for the pressure in plane sound wave.
• Pressure wave has the same frequency, wavelength and velocity as displacement wave. It is description of the same sound wave !
• Pressure wave is shifted relative to displacement in phase by ½&pi. Peak of displacement is at zeroes of pressure change, maximum/minimum pressure is at zero desplacement.
• Pressure wave amplitude is Δ p_max = A B k . Pressure fluctuation in the air that a human can hear is ~ 10^-4 Pa which is 10^-9 fraction of the atmospheric one ! Exact threshold of hearing depends on frequency and is lower at lower frequencies.

Sound Speed

• Sound speed is expected to be determined by the properties of the material in which sound propagates. Physically, by experience from springs, restoring force and inertia should be involved. Dimensional analysis ? We have B ~ kg/(m s²) (force per unit area) density ρ ~ kg/m³ (mass per unit volume), so v ~ (B/ρ)^½ ?

• Let us look in detail on the board. Consider piston that sends a pulse by moving into the tube with fluid with velocity v_y The front of the pulse moves ahead with velocity v (sound speed). Steps:
  • Fluid of mass M = v t S ρ accuires velocity v_y in time t . That means it was subject of the force F = M v_y/t which came from pressure increase Δ p acting on surface S . I.e Δ p = M v_y/t , so Δ p = v v_y ρ
  • On the other hand the volume of fluid V = v t S experience contraction by Δ V = - v_y t S , hence Δ V / V = - v_y/v
  • Pressure increase and volume decrease are related by the bulk modulus v v_y ρ = B v_y/v , which solving for sound speed v gives indeed v = (B/ρ)^½
• Typical values of sound speed for metals are v ~ 2000-6000 m/s

Sound speed in a solid rod

Solid rod behaves itself somewhat different that the bulk material - it can bulge when sound propagates in it. Its compression properties are described by Young modulus Y instead of bulk modulus B . Y also has unit of pressure, Pa sound speed along the length of the solid rod is v = (Y/ρ)½ Just get the right modulus for the problem ! For the same solid material Young and bulk moduli are typically of the same order, but which one is larger depends on material (see Table 11.1 in the textbook)

Sound speed in gases

• Sound speed in gases is quite different from sound speed in solids.
• It depends quite notably on the temperature.
• And how gas is compressed in the wave

• If compression is adiabatic , pressure reflects the density according to P ~ ρ^γ where γ is the ratio of heat capacities of the gass.
• Then B = γ P_am
• For ideal gas, pressure, density and temperature are related by ideal gas law (father of all well known gas laws) P = ( R/M) ρ T
• Which give for sound speed in a gas v = (B/ρ)^½ = (γ P/ρ)^½ = (γ T R/M )^½