Propagation in Elastic Materials
the previous pages, it was pointed out that sound waves propagate
due to the vibrations or oscillatory
motions of particles within a material. An ultrasonic wave
may be visualized as an infinite number of oscillating masses
or particles connected by means of elastic springs. Each individual
particle is influenced by the motion of its nearest neighbor and
restoring forces act upon each particle.
A mass on a spring has a single resonant frequency determined
by its spring constant k and its mass m. The spring
constant is the restoring force of a spring per unit of length.
Within the elastic limit of any material, there is a linear relationship
between the displacement of a particle and the force attempting
to restore the particle to its equilibrium position. This linear
dependency is described by Hooke's Law.
terms of the spring model, Hooke's Law says that the restoring
force due to a spring is proportional to the length that the spring
is stretched, and acts in the opposite direction. Mathematically,
Hooke's Law is written as F =-kx, where F
is the force, k is the spring constant, and x is
the amount of particle displacement. Hooke's law is represented
graphically it the right. Please note that the spring is applying
a force to the particle that is equal and opposite to the force
pulling down on the particle.
The Speed of Sound
Hooke's Law, when used along with Newton's Second
Law, can explain a few things about the speed of sound. The speed
of sound within a material is a function of the properties of
the material and is independent of the amplitude of the sound
wave. Newton's Second Law says that the force applied to a particle
will be balanced by the particle's mass and the acceleration of
the the particle. Mathematically, Newton's Second Law is written
as F = ma. Hooke's Law then says that this force will be
balanced by a force in the opposite direction that is dependent
on the amount of displacement and the spring constant (F =
-kx). Therefore, since the applied force and the restoring
force are equal, ma = -kx can be written. The negative
sign indicates that the force is in the opposite direction.
Since the mass m and the spring constant
k are constants for any given material, it can be seen
that the acceleration a and the displacement x
are the only variables. It can also be seen that they are directly
proportional. For instance, if the displacement of the particle increases,
so does its acceleration. It turns out that the time that it takes
a particle to move and return to its equilibrium position is independent
of the force applied. So, within a given material, sound always
travels at the same speed no matter how much force is applied
when other variables, such as temperature, are held constant.
What properties of material
affect its speed of sound?
Of course, sound does travel at different speeds
in different materials. This is because the mass of the atomic
particles and the spring constants are different for different
materials. The mass of the particles is related to the density
of the material, and the spring constant is related to the elastic
constants of a material. The general relationship between the
speed of sound in a solid and its density and elastic constants
is given by the following equation:
Where V is the speed of sound, C is
the elastic constant, and p is the material density.
This equation may take a number of different forms depending on
the type of wave (longitudinal or shear) and which of the elastic
constants that are used. The typical elastic constants of a materials
- Young's Modulus, E: a proportionality constant between
uniaxial stress and strain.
- Poisson's Ratio, n:
the ratio of radial strain to axial strain
- Bulk modulus, K: a measure of the incompressibility
of a body subjected to hydrostatic pressure.
- Shear Modulus, G: also called rigidity, a measure of
a substance's resistance to shear.
- Lame's Constants, l and
material constants that are derived from Young's Modulus
and Poisson's Ratio.
When calculating the velocity of a longitudinal
wave, Young's Modulus and Poisson's Ratio are commonly used. When
calculating the velocity of a shear wave, the shear modulus is
used. It is often most convenient to make the calculations using
Lame's Constants, which are derived from Young's Modulus and Poisson's
It must also be mentioned that the subscript ij
attached to C in the above equation is used to indicate
the directionality of the elastic constants with respect to the
wave type and direction of wave travel. In isotropic materials,
the elastic constants are the same for all directions within the
material. However, most materials are anisotropic and the elastic
constants differ with each direction. For example, in a piece
of rolled aluminum plate, the grains are elongated in one direction
and compressed in the others and the elastic constants for the
longitudinal direction are different than those for the transverse
or short transverse directions.
Examples of approximate compressional sound velocities in materials
- Aluminum - 0.632 cm/microsecond
- 1020 steel - 0.589 cm/microsecond
- Cast iron - 0.480 cm/microsecond.
Examples of approximate shear sound velocities in materials are:
- Aluminum - 0.313 cm/microsecond
- 1020 steel - 0.324 cm/microsecond
- Cast iron - 0.240 cm/microsecond.
When comparing compressional and shear velocities, it can be noted
that shear velocity is approximately one half that of compressional velocity.
The sound velocities for a variety of materials can be found in
the ultrasonic properties tables in the general resources section
of this site.