PHYS212 : Lectures

Subsections

• 4.1.1 Displacement Waves
• 4.1.2 Pressure Waves
• 4.1.3 The Speed of Sound in an Ideal Gas

4.1 The Wave Equation

4.1.1 Displacement Waves

Figure 4.1: A cubic volume $dV_o = dx dy dz$ of material stretched along $\hat{\imath}$ a distance $d\delta_x$.

• Consider a small cubic volume $dV_o = dx dy dz$ of material.^4.1If we stretch the volume along $\hat{\imath}$ by displacing one side of the cube by a distance $d\delta_x$ as shown in Fig. 4.1, the change in volume can be written
  

  $$dV = (dx + d\delta_x) dy dz = dV_o + d\delta_x dy dz$$

  $$= dV_o (1 + \frac{d \delta_x}{dx}) = dV_o (1 + \delta'_x)$$ (4.1)

  
  

• We will use $\delta_x$ to describe the displacement of the medium from equilibrium along $\hat{\imath}$. It is a function of $x$, $y$, $z$, and $t$, not an independent variable.

• Having stretched the cube, we have not changed the number of particles in it, so its mass $dm$ is constant. Hence, the local density of the material can be expressed (in the limit as $dV$ becomes infinitesimal) as
  

  $$\rho = \frac{dm}{dV} = \frac{\rho_o} {\left( 1 + \delta_x' \right)}$$ (4.2)

  
  

• Newton's second law describing the displacement of a small volume element is
  

  $$dm \ddot{\delta}_x = -dP_x A = - \frac{\partial P}{\partial x} dx A$$ (4.3)

  
  
  where $A$ is the cross sectional area $dy dz$ of the element in the $\hat{\jmath}$-$\hat{k}$ plane. Recognizing $A dx$ as $dV$, we can conclude
  

  $$\rho_o \ddot{\delta}_x = - P'$$ (4.4)

  
  

• If waves are propagating in the medium, its density varies with $\delta'_x$ (Eq. 4.2) and hence with position. The pressure varies with the density, so
  
  $$P' = \frac{\partial P}{\partial \rho} \frac{\partial \rho}{\partial x}$$
  
  
  and we can make Eq. 4.4 more explicit
  
  $$\ddot{\delta}_x = - \frac{\partial P}{\partial \rho} \frac{\rho'}{\rho_o}$$
  
  
  which with Eq. 4.2 becomes
  

  $$\ddot{\delta}_x = \frac{\partial P}{\partial \rho} \frac{ \delta''_x }{( 1 + \delta'_x)^2}$$ (4.5)

  
  
  a non-linear second order partial differential equation.

• In the acoustic limit we assume $\delta'_x « 1$ - that displacements are small relative to the length scale on which they vary ($\lambda$). In this limit, the factor $( 1 + \delta'_x)^{-2}$ in Eq.'s 4.2 and 4.5 becomes 1, yielding the wave equation
  

  $$\ddot{\delta}_x = \left. \frac{\partial P}{\partial \rho} \right\vert _{\rho_o} \delta''_x$$ (4.6)

  
  
  with phase velocity
  

  $$v_p = \frac{\omega}{k} = \sqrt{\left. \frac{\partial P}{\partial \rho} \right\vert _{\rho_o}}$$ (4.7)

  
  

• Note that Eq. 4.7 gives a constant phase velocity, and hence that sound waves in the acoustic limit are non-dispersive ($v_g = v_p$).

• Also note that there is nothing special about our choice to stretch the medium along $\hat{\imath}$. Identical arguments lead to wave equations of the same form for $\delta_y$ and $\delta_z$,
  

  $$\ddot{\delta}_y = \left. \frac{\partial P}{\partial \rho} \right\vert _{\rho_o} \frac{\partial^2 \delta_y}{\partial y^2}$$

  $$\ddot{\delta}_z = \left. \frac{\partial P}{\partial \rho} \right\vert _{\rho_o} \frac{\partial^2 \delta_z}{\partial z^2}$$ (4.8)

  
  
  which give the same speed in all directions, as one should expect for an isotropic medium.

• Plane wave solutions to Eq. 4.6 take the usual form
  

  $$\delta_x(x, t) = Re\left\{ \delta_{xo} e^{i(kx - \omega t + \phi)} \right\}$$ (4.9)

  
  
  with $k$ and $\omega$ related by Eq. 4.7.

• In general displacements of a three dimensional medium are vectors
  

  $$\vec{\delta}(\vec{r}) = \delta_x(\vec{r}) \hat{\imath} + \delta_y(\vec{r}) \hat{\jmath} + \delta_z(\vec{r}) \hat{k}$$ (4.10)

  
  
  Only a plane wave traveling in $\hat{\imath}$ has a displacement function independent of $y$ and $z$.

4.1.2 Pressure Waves

• We obtained a vector displacement wavefunction in Section 4.1.1, which can render descriptions of three dimensional waves rather awkward.

• It would simplify matters if we had a scalar instead of a vector wavefunction. The (scalar) pressure and displacement of the medium are linked (Eq. 4.4). It turns out that they are both governed by the wave equation, but we need to prove it.

• Changes in the pressure and volume of a material are related via
  

  $$dP = -B \frac{dV}{V}$$ (4.11)

  
  
  a form of Hooke's law. The bulk modulus $B$ is a characteristic of the medium and sets the strength of the restoring pressure counteracting a volume compression $\frac{dV}{V}$.

• Defining $p = P-P_o$, the variation in the pressure relative to the equilibrium value $P_o$, if we compress our infinitesimal volume element $dV_o$ by stretching it along $\hat{\imath}$ a distance $d\delta_x$, Eq. 4.11 gives
  

  $$p = -B \frac{d\delta_x dy dz}{dx dy dz} = -B \frac{\partial \delta_x}{\partial x} = -B \delta_x'$$ (4.12)

  
  

• Taking the second time derivative of Eq. 4.12
  
  $$\ddot{p} = -B \ddot{\delta_x}'$$
  
  
  and making use of Eq. 4.4, we obtain the pressure wave equation
  

  $$\ddot{p} = \frac{B}{\rho_o} p''$$ (4.13)

  
  

• This second wave equation gives a phase velocity
  

  $$v_p = \frac{\omega}{k} = \sqrt{\frac{B}{\rho_o}}$$ (4.14)

  
  
  which is constant and hence also the group velocity. Eq.'s 4.7 and 4.14 must both describe the same velocity, so
  

  $$\left. \frac{\partial P}{\partial \rho} \right\vert _{\rho_o} = \frac{B}{\rho_o}$$ (4.15)

  
  
  You will consider this relationship for homework.

• The displacement plane wave solutions of Eq. 4.9 lead us, via Eq. 4.12, to pressure plane waves
  

  $$p(x,t) = - ikB \delta_x(x, t)$$

  $$= Re\{ -ikB \delta_{xo} e^{i(kx - \omega t + \phi)}\}$$

  $$= kB \delta_{xo} \sin(kx - \omega t + \phi)$$ (4.16)

  
  
  The factor of $-i = e^{-i \frac{\pi}{2}}$ indicates that the pressure and displacement waves are $-\frac{\pi}{2}$ out of phase with each other.

• It should also be noted that
  

  $$p_o = kB \delta_{xo}$$ (4.17)

  
  

4.1.3 The Speed of Sound in an Ideal Gas

• The problem of calculating the speed of sound in a gas via Eq. 4.7 is finding the density dependence of the pressure. The corresponding problem using Eq. 4.14 is finding the bulk modulus of the gas, which varies with pressure and temperature.

• Changes in the pressure, volume, and density of a medium due to an acoustic wave are adiabatic. They happen too rapidly to allow significant heat flow in the medium. For ideal gases, adiabatic processes are described by
  

  $$PV^\gamma = \mathrm{constant}$$ (4.18)

  
  
  where
  

  $$\gamma = \frac{C_p}{C_V}$$ (4.19)

  
  
  See the Appendix (Section 4.5) on adiabatic expansion for the background to this result.

• Eq. 4.18 can be expressed alternatively as
  
  $$P V^\gamma = P_o V_o^\gamma$$
  
  
  which can be rearranged to give the density dependence of the pressure of an ideal gas
  

  $$P = P_o \left( \frac{\rho}{\rho_o} \right)^\gamma$$ (4.20)

  
  

• With Eq. 4.20 we can evaluate Eq. 4.7 to find
  

  $$v_p = \sqrt{\gamma \frac{P_o}{\rho_o}}$$ (4.21)

  
  
  which requires us to know the absolute equilibrium pressure $P_o$ of the gas, not the relative $p$ used to describe pressure waves which has an equilibrium value of zero.

• Air has^4.2$\gamma = 1.40$ and $\rho_o = 1.29$ kg/m$^3$. At STP, $P_o$ = 1.00 atm = $1.013\mathrm{e}3$ N/m$^2$, and the speed of sound predicted by Eq. 4.21 is 332 m/s, in excellent agreement with the observed speed (331.0 m/s).

• By dividing by the mass of the volume of gas, the ideal gas law can be written in terms of the density $\rho = \frac{m}{V}$ and the molar mass $M = \frac{m}{n}$ as
  
  $$\frac{P}{\rho} = \frac{RT}{M}$$
  
  
  yielding an alternative expression for the phase velocity
  

  $$v_p = \sqrt{\gamma \frac{RT}{M}}$$ (4.22)

  
  

• Replacing $T$ with $T+273$ yeilds an expression of the phase velocity in terms of the Celcius temperature
  

  $$v_p = \sqrt{\gamma \frac{R(273 \mathrm{K})}{M}} \sqrt{1 +... ...{T}{273 \mathrm{K}}} \approx (332 + 0.60 T) \mathrm{m/s}$$ (4.23)

  
  

Copyright © 2003-2009, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006