Physics > Sound Waves > 2.0 Displacement and pressure Waves
Sound Waves
1.0 Introduction
2.0 Displacement and pressure Waves
2.1 Relation between displacement wave and pressure wave
2.2 Relation between pressure wave and density wave
3.0 Speed of a longitudinal Wave
4.0 Doppler's Effect
5.0 Application of doppler's effect in different situations
6.0 Doppler's effect in two dimension
6.1 When Medium is at rest while source and observer is moving
6.2 When the medium also moves with source and observer
6.4 Questions
7.0 Characteristic of Sound waves
2.1 Relation between displacement wave and pressure wave
2.2 Relation between pressure wave and density wave
6.2 When the medium also moves with source and observer
6.4 Questions
Consider a harmonic displacement wave moving through air contained in a long tube of cross-sectional area $S$.
The volume of gas that has a thickness $\Delta x$ in the horizontal direction is, $${V_i} = S.\Delta x$$ The change in volume $\Delta V$ is, $$\Delta V = S.\Delta y$$ where $\Delta y$ is the difference between the value of $y$ at $x + \Delta x$ and the value of $y$ at $x$.
From the definition of bulk modulus, the pressure variation in the gas is,$$\begin{equation} \begin{aligned} \Delta P = - B\frac{{\Delta V}}{{{V_i}}} \\ \Delta P = - B\left( {\frac{{S.\Delta y}}{{S.\Delta x}}} \right) \\ \Delta P = - B\left( {\frac{{\Delta y}}{{\Delta x}}} \right) \\\end{aligned} \end{equation} $$
As $\Delta x$ approaches zero, $$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \frac{{\delta y}}{{\delta x}}$$ Therefore, $$\frac{{\delta y}}{{\delta x}} = - \frac{{\Delta P}}{B}\quad ...(i)$$
Now, the displacement equation of wave is $$y = A\cos (kx - \omega t)....(ii)$$Differentiating the above equation with respect to $x$ we get, $$\frac{\delta y}{\delta x} = - kA\sin (kx - \omega t)....(iii)$$ From equation $(i)$ and $(iii)$, $$\begin{equation} \begin{aligned} \Delta P = BAk\sin (kx - \omega t) \\ \Delta P = {(\Delta P)_m}\sin (kx - \omega t) \\\end{aligned} \end{equation} $$Here,$${(\Delta P)_m} = BAk$$ where,
${(\Delta P)_m}$ is the maximum amplitude of pressure variation or the maximum amplitude of the pressure wave.
$B=$ is the bulk modulus
$k=$ is the wave number
Note: Pressure wave is $90^\circ $ out of phase with the displacement wave, which means that the pressure is maximum when displacement is minimum and vice-versa.
