In general, the speed of a mechanical wave is given by:

v

=

(

elastic property inertial property

)

½

e.g., For a wave on a string:     v

=

(

T μ

)

½

For sound waves the inertial factor is the density, ρ.

The relevant elastic property is the bulk modulus, B. This tells us how much a medium's volume changes when the pressure on it changes.

B

= -

Δp ΔV/V

Therefore the speed of sound is     v

=

(

B ρ

)

½

The bulk modulus is a measure of how incompressible a material is. The higher the value, the less its volume changes when the pressure changes. Gases generally have small values of B, liquids have higher values, and solids even higher.

Some sample values of the speed of sound are:

Humans are sensitive to a particular range of frequencies, typically from 20 Hz to 20000 Hz. Whether you can hear a sound also depends on its intensity - we're most senstive to sounds of a couple of thousand Hz, and considerably less sensitive at the extremes of our frequency range.

We generally lose the top end of our range as we age.

Other animals are sensitive to sounds at lower or higher frequencies. Anything less than the 20 Hz that marks the lower range of human hearing is classified as infrasound - elephants, for instance, communicate using low frequency sounds. Anything higher than 20 kHz, our upper limit, is known as ultrasound. Dogs, bats, dolphins, and other animals can hear sounds in this range.

For a transverse wave like a wave on a string, when the wave is traveling in the x-direction the pieces of string oscillate back and forth in the y-direction. For a longitudinal wave like a sound wave the oscillations are parallel to the direction the wave travels.

For a wave traveling in the +x direction, for instance, the oscillation of the particles in the medium in the x-direction can be described by:

s(x,t) = s_max sin(ωt -kx)

The oscillations of the particles produces small changes in pressure in the medium. A higher density of particles corresponds to a higher pressure, and a lower density corresponds to a lower pressure. Examining the simulation closely, it can be seen that when a particle has a displacement of zero, the pressure is at a maximum or minimum. High pressure at a point occurs when neighboring particles come toward the point; low pressure when the neighboring particles move away.

If the particles at a given point fluctuate following a sine, the pressure there fluctuates like a cosine, 90 degrees out of phase with the displacement.

ΔP = ΔP_max cos(ωt - kx)

The relationship between the maximum pressure change ΔP_max and the maximum displacement amplitude of the particles s_max is:

ΔP_max = (vρω)s_max

This is derived in the book.