Stress Waves in Solids
============================

```{admonition} The Wave equation
$$
\frac{\partial^2 u}{\partial t^2} = {\color{blue}{C}}^2\sum_i \frac{\partial^2 u}{\partial x_i^2} 
$$
```

Consider a "1D" (i.e. $L>>\phi$) rod whose end is extended following $V(t)=\dot{u}$

Since the body can not be said to be in rest the sum of forces on the body is given by 

$$
\sum F = ma =m\dot{V} = m\ddot{u}
$$

Looking at an infinitesimal element within our rod we can construct the free body diagram:

```{image} ../WV/SW.png
:alt: free body diagram
:width: 600px
:align: center
```
from which we can see that 

$$
\left [ \sigma(x+dx) -\sigma(x) \right ]dyxz = \rho dx dy dz \ddot{u}
$$

taking $dx \to 0$ we obtain:

$$
\frac{\partial \sigma}{\partial x} = \rho \ddot{u}
$$

Assuming linear elasticity , and recalling that $\epsilon=\frac{\partial u}{\partial x}$ the above equation can be written as 

$$
{\color{blue}E}\frac{\partial^2 u}{\partial x^2} = {\color{blue}\rho} \frac{\partial^2 u}{\partial t^2}
$$

or

$$
\frac{\partial^2 u}{\partial t^2} = {\color{blue}{C_L}}^2 \frac{\partial^2 u}{\partial x^2} \ \, \ \ \text{with} \ \ {\color{blue}{C_L=\sqrt{\frac{E}{\rho}}}}
$$

```{note}
1. Under shear stresses a similar expression can be derived leading to ${\color{red}{C_S=\sqrt{\frac{G}{\rho}}}}$
2. For those of you interested in stress waves ME036006 is highly recommended.
3. [Here](https://www.acs.psu.edu/drussell/Demos/waves/wavemotion.html) and [here](https://acoustics.byu.edu/animations-propagation) you can find more information as well as some animations of different waves. 
```

```{admonition} Question
When we conduct a simple tensile test, waves are constantly running through the system. Still we treat this as a **Quasi-static** problem. Why? 
```

We can define a (scale dependent) characteristic time by considering

$$
\Delta t = \frac{L}{\color{blue}{C_L}}
$$

For the same bar, we can write the natural frequencies of longitudinal vibration as:

$$
f_n = \frac{n{\color{blue}{C_L}}}{2L}
$$

```{dropdown} What should we require of the natural frequencies in our system so that they will not interfe with our measurement?

given that our measured phenomena is characterized by a frequency $f$ we can require that 

$$
f<<\frac{{\color{blue}{C_L}}}{2L}
$$

(why $n=1$?) 

or 

$$
\frac{1}{\Delta t}<<\frac{{\color{blue}{C_L}}}{2L}
$$
```