PX154 - G2 - longitudinal waves along a bar (1D)

when the end of a bar is hit, a compression is created within the bar that then will propagate along it
let's say the element $d x$ is compressed by an amount $d σ$
- [YF fig 11.14] stress/strain
  - $s t r a i n = \frac{Δ l}{l_{0}}$
- force creates a stress:
$s t r e s s = \frac{F}{A}$
- and a strain:
$s t r a i n = \frac{d σ}{d x}$
- the stress and strain are linked by young's modulus:
$Y = \frac{s t r e s s}{s t r a i n}$ $o r, s t r e s s = \frac{F}{A} = Y \frac{d σ}{d x}$
- if we follow yesterday's derivation:
$\frac{Y}{ρ} \frac{\partial^{2} σ}{\partial x^{2}} = \frac{\partial^{2} σ}{\partial t^{2}}$ - another wave equation
- wave speed: $v = \sqrt{\frac{Y}{ρ}}$
- for steel: $Y = 2 \times 10^{11} N m^{- 2}$ ; $ρ = 8 \times 10^{3} k g m^{- 3}$
- $v = 5 \times 10^{3} m s^{- 1}$