Speed of sound

Given equation:

$$\dot{m}=\rho A\overrightarrow{v}$$

 

$$\rho A\overrightarrow{v}=(\rho +\mathrm{\Delta }\rho )A(c-\mathrm{\Delta }\overrightarrow{v})$$

$$\rho c=(\rho +\mathrm{\Delta }\rho )(c-\mathrm{\Delta }\overrightarrow{v})$$

$$\rho c=\rho c-\rho \mathrm{\Delta }\overrightarrow{v}+\mathrm{\Delta }\rho c-\mathrm{\Delta }\rho \mathrm{\Delta }\overrightarrow{v}$$

 

We can neglect the last term of equation

$$0=-\rho \mathrm{\Delta }\overrightarrow{v}+\mathrm{\Delta }\rho c$$

$$c\mathrm{\Delta }\rho =\rho \mathrm{\Delta }\overrightarrow{v}$$

 

$$c\frac{\mathrm{\Delta }\rho }{\rho }=\mathrm{\Delta }\overrightarrow{v}\cdots (1)$$

 

Given equation:

$$PA+\rho \overrightarrow{v}\overrightarrow{v}A=(p+\mathrm{\Delta }p)A+(\rho +\mathrm{\Delta }\rho )(\overrightarrow{v}-\mathrm{\Delta }\overrightarrow{v})(\overrightarrow{v}-\mathrm{\Delta }\overrightarrow{v})A$$

 

As we can rearrange the two term like below:

$$(\rho +\mathrm{\Delta }\rho )(\overrightarrow{v}-\mathrm{\Delta }\overrightarrow{v})=\rho \overrightarrow{v}$$

 

$$PA+\rho \overrightarrow{v}\overrightarrow{v}A=(p+\mathrm{\Delta }p)A+\rho \overrightarrow{v}(\overrightarrow{v}-\mathrm{\Delta }\overrightarrow{v})A$$

 

We can omit $A$ and others...

$$P+\rho \overrightarrow{v}\overrightarrow{v}=(p+\mathrm{\Delta }p)+\rho \overrightarrow{v}(\overrightarrow{v}-\mathrm{\Delta }\overrightarrow{v})$$

$$\rho \overrightarrow{v}\overrightarrow{v}=\mathrm{\Delta }p+\rho \overrightarrow{v}(\overrightarrow{v}-\mathrm{\Delta }\overrightarrow{v})$$

 

$$\rho \overrightarrow{v}\overrightarrow{v}=\mathrm{\Delta }p+\rho \overrightarrow{v}\overrightarrow{v}-\rho \overrightarrow{v}\mathrm{\Delta }\overrightarrow{v}$$

$$0=\mathrm{\Delta }p-\rho \overrightarrow{v}\mathrm{\Delta }\overrightarrow{v}$$

 

$$\rho \overrightarrow{v}\mathrm{\Delta }\overrightarrow{v}=\mathrm{\Delta }p$$

$$\mathrm{\Delta }\overrightarrow{v}=\frac{1}{\rho \overrightarrow{v}}\mathrm{\Delta }p$$

 

We can replace $\overrightarrow{v}$ to $c$ which means 'speed of sound'

$$\mathrm{\Delta }\overrightarrow{v}=\frac{1}{\rho c}\mathrm{\Delta }p\cdots (2)$$

 

Now use equation $(1)$ and $(2)$$$\mathrm{\Delta }\overrightarrow{v}=c\frac{\mathrm{\Delta }p}{\rho }\cdots (1)$$$$\mathrm{\Delta }\overrightarrow{v}=\frac{1}{\rho c}\mathrm{\Delta }p\cdots (2)$$

 

$$\frac{1}{\rho c}\mathrm{\Delta }p=c\frac{\mathrm{\Delta }\rho }{\rho }$$$$\frac{1}{c}\mathrm{\Delta }p=c\mathrm{\Delta }\rho$$$$\frac{\mathrm{\Delta }p}{\mathrm{\Delta }\rho }=c^2$$

 

$$a^2={\left(\frac{\mathrm{\Delta }p}{\mathrm{\Delta }\rho }\right)}_s={\left(\frac{\partial p}{\partial \rho }\right)}_s$$