17-1 Speed of Sound

17-1 Speed of Sound#

Prompts

  • Sound waves are longitudinal. How do air molecules move relative to the direction of propagation? Contrast with transverse waves on a string.

  • What is the bulk modulus \(B\)? How does it relate to compressibility? Why does a stiffer (less compressible) medium support faster sound?

  • Write the formula for the speed of sound in a medium. What inertial and elastic properties does it depend on?

  • Why is the speed of sound in water (1482 m/s) greater than in air (343 m/s) even though water is much denser?

Lecture Notes#

Overview#

  • Sound waves are longitudinal mechanical waves—air (or other medium) oscillates parallel to the direction of propagation.

  • The speed of sound \(v\) depends on the medium’s bulk modulus \(B\) (elasticity) and density \(\rho\) (inertia): \(v = \sqrt{B/\rho}\).

  • In air at 20°C, \(v \approx 343\;\text{m/s}\). Sound travels faster in water and solids than in air—despite higher density, the bulk modulus increases even more.

Sound waves: longitudinal and mechanical#

Sound is a longitudinal mechanical wave. As the wave passes:

  • Air elements oscillate back and forth along the direction of travel (compression and rarefaction).

  • Energy is stored as kinetic (motion) and potential (compression/expansion).

Wavefronts are surfaces of constant phase; rays are perpendicular to wavefronts and indicate direction of travel. Near a point source, wavefronts are spherical; far away they approximate planes.

Bulk modulus#

The bulk modulus \(B\) measures resistance to volume change under pressure:

(119)#\[ B = -\frac{\Delta p}{\Delta V/V} \]

  • \(\Delta p\) = change in pressure; \(\Delta V/V\) = fractional change in volume.

  • The minus sign ensures \(B > 0\): increasing pressure (\(\Delta p > 0\)) decreases volume (\(\Delta V < 0\)).

  • Stiffer (less compressible) medium → larger \(B\).

Speed of sound#

For any mechanical wave, speed depends on an elastic property (stores potential energy) and an inertial property (stores kinetic energy). For sound in a fluid or solid:

(120)#\[ v = \sqrt{\frac{B}{\rho}} \]

  • \(B\) = bulk modulus (elasticity—resistance to compression).

  • \(\rho\) = density (inertia—mass per unit volume).

Analogy to string waves

For a string: \(v = \sqrt{\tau/\mu}\)—tension \(\tau\) (elasticity), linear density \(\mu\) (inertia). Same structure: \(v = \sqrt{\text{elastic}/\text{inertial}}\).

Speed in different media#

Medium

Speed (m/s)

Air (0°C)

331

Air (20°C)

343

Water (20°C)

1482

Steel

5941

Aluminum

6420

Why is sound faster in water than in air? Water is ~1000× denser than air, so inertia alone would suggest slower speed. But water’s bulk modulus is far more than 1000× that of air—water is much less compressible. The ratio \(B/\rho\) is larger for water, so \(v\) is higher.

Summary#

  • Sound: longitudinal mechanical wave; air oscillates parallel to propagation.

  • Bulk modulus \(B = -\Delta p/(\Delta V/V)\); measures compressibility.

  • Speed of sound \(v = \sqrt{B/\rho}\); in air at 20°C, \(v \approx 343\;\text{m/s}\).