On page $519$ of the book Engineering Vibration (which can be downloaded from here), the following wave equation of a variable cross-section rod is derived:
$${ {\partial}\over{\partial}x} \Big( EA(x){ {\partial}w(x,t)\over{\partial}x} \Big)={\rho}A(x) { {\partial}w^2(x,t)\over{\partial}t^2}\tag{1}$$
where:
- $x$ is the spatial coordinate
- $t$ is time
- $E$ is the Young's modulus of the rod
- $A(x)$ is the variable cross-section function of the rod
- $\rho$ is the density of the rod
- $w(x,t)$ is the wave function
Because some of the steps are missing from the derivation process, I tried deriving the same equation myself but failed. Please point out my mistake.
For the variable cross-section rod in the picture above, I wrote the following dynamics equation: $$\Big(F(x,t)+dF(x,t)\Big) - F(x,t) = \Big(dm(x)\Big){ {\partial}w^2(x,t)\over{\partial}t^2}\tag{2}$$ where $dm(x)$ is the mass of the Infinitesimal element of the rod. Next, I crossed out the forces $F(x,t)$ and defined the mass $dm(x)$ as a product of volume $dV(x)$ and density: $$dF(x,t) = {\rho}\Big(dV(x)\Big){ {\partial}w^2(x,t)\over{\partial}t^2}\tag{3}$$ Since the volume $dV(x)$ is: $$dV(x) = A(x)dx\tag{4}$$ equation $(3)$ can be rewritten as: $$dF(x,t) = {\rho}A(x)dx{ {\partial}w^2(x,t)\over{\partial}t^2}\tag{5}$$ After this, I defined $dF(x,t)$ as: $$dF(x,t) = d\Big(P(x,t)A(x)\Big)=\Big(dP(x,t)\Big)A(x)+ P(x,t)\Big(dA(x)\Big)\tag{6}$$ where $P(x,t)$ is the pressure. By expanding the total derivatives, I obtained: $$dF(x,t) = \Big( { \partial P(x,t) \over \partial x}dx + { \partial P(x,t) \over \partial t}dt \Big)A(x) + P(x,t){ \partial A(x) \over \partial x}dx \tag{7}$$ Hook's law states: $$P(x,t) = E{ {\partial}w(x,t)\over{\partial}x}\tag{8} $$ and because of that, I rewrote equation $(7)$ as: $$dF(x,t) = E\Bigg(\Big( { \partial ^2 w(x,t) \over \partial x^2}dx + { \partial ^2 w(x,t) \over \partial x \partial t}dt \Big)A(x) + w(x,t){ \partial A(x) \over \partial x}dx\Bigg) \tag{9}$$ In the end, I derived the expression: $$E\Bigg(\Big( { \partial ^2 w(x,t) \over \partial x^2}dx + { \partial ^2 w(x,t) \over \partial x \partial t}dt \Big)A(x) + w(x,t){ \partial A(x) \over \partial x}dx\Bigg) = {\rho}A(x)dx{ {\partial}w^2(x,t)\over{\partial}t^2}\tag{10}$$ In equation $(10)$ there is an extra term. Namely, the ${ \partial ^2 w(x,t) \over \partial x \partial t}dt$ term. If that did not exist, I would get equation $(1)$. So, my question is, why does that term not exist in equation $(1)$?
