# 3 types of modulus, 3 types of soundwaves in solids

I noticed there are 3 types of elastic modulus and 3 types of waves in solids.We have Young modulus that describes tensile stiffness,shear modulus about shearing stiffness and bulk modulus that is about compression stiffness.

We have three types of sound wave in solids,longitudinal,shear and flextural.Are the velocities of 3 types of waves dependent on the value of 3 types of modulus?

For example,longitudinal wave,is compression wave,so its velocity in solid depends on bulk modulus and not other types of modulus becose bulk is the specific type of modulus that describes compression stiffness of solid. Likewise,flextural wave depends on Young modulus of solid becose Young is the one who describes solids stiffness to flexing.

Longitudinal wave = bulk modulus ... Shear wave = shear modulus ... Flextural wave = Young modulus ...

Is what I have written correct? Another example,sound velocity depends on density and modulus,so if solid have 1 gram cm3 density,100 GPa Young modulus,10 GPa shear modulus and 1000 GPa bulk modulus,then its sound velocity for three types of waves will be like this :

Flextural - 10000 m/s ... Shear - 3162 m/s ... Longitudinal - 31622 m/s ...

Are my calculations correct? Is my understanding right?

• Sadly, it is worse than that. In general, the stiffness (or compliance) tensor connecting stress to strain can have 81 coefficients, although crystal symmetry tends to reduce this. Cubic and isotropic systems have only 3. Triclinic crystals have many more... Mar 16 '18 at 15:03
• What coefficient do you mean? What are Triclinic crystals? What is Cubic system? I know what isotropic means but rest of your comment I dont understand.If you could teach me what these things mean I would be super grateful. Mar 16 '18 at 15:09
• en.wikipedia.org/wiki/Hooke%27s_law is a good place to start. Mar 16 '18 at 15:32
• And a longitudinal wave in a rod would depend on Young's modulus.
– user137289
Mar 16 '18 at 15:52
• The 3 moduli you mentioned are all inter-related. There are really only two independent parameters: the Young's modulus and the Poisson ratio. The bulk modulus and the shear modulus are determined from these. Mar 18 '18 at 3:22

## 1 Answer

Your question is a little bit confused, so let me try to explain things slightly more systematically.

1. It is important to understand that there are actually many different types of waves, not three and not four. However, many waves are enabled by specific geometric features of the problem at hand, so one usually separates so-called body waves from all the other kinds of waves. Body waves can propagate in an infinite elastic space. There are exactly 2 types of body waves: longitudinal and shear waves. Generally speaking, in anisotropic materials, the velocity of these waves can depend on the direction of propagation and these waves themselves can couple to each other. However, assuming isotropic materials, you will have them uncoupled and their velocities will be direction-independent and given by $$c_1^2=\frac{\lambda+2\mu}{\rho},\quad c_2^2=\frac{\mu}{\rho},$$ where $c_1$ is the longitudinal wave speed and $c_2$ is the shear wave speed. You can see that the shear wave speed is proportional to the square root of the shear modulus. However, the velocity of longitudinal waves is not related to one specific modulus.
2. Any isotropic material can be characterized by exactly two modulae. $\lambda$ and $\mu$ that I used just above are often called Lame parameters. In some situations people prefer to use other modulae too. Modulae most commonly used for isotropic materials are: $$E=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu},\quad \nu=\frac{\lambda}{2(\lambda+\mu)},\quad \kappa=\lambda+\frac{2}{3}\mu.$$ $\kappa$ is the bulk modulus, it characterizes stiffness for volume deformations; $E$ the Young modulus, it predicts the stiffness of a thin bar, and $\nu$ the Poisson ratio that characterizes how much thin bar contracts transversally when it is stretched.

3. Note the emphasis on specific geometries in the last two examples. Some combinations of modulae were introduced precisely because they are convenient to use for specific geometries. Since thin bars are convenient objects for testing, a number of parameters are defined with thin bar testing in mind. In this sense, you may find it instructive that velocity of sound in a thin elastic bar is given by $$c_{\mathrm{bar}}^2=\frac{E}{\rho},$$ i.e. that the Young modulus is directly proportional to the square of velocity of longitudinal waves in bars. However, outside of this specific geometry, the interpretation of Young modulus becomes much less intuitive.