EXERCISES
4.2 Physics of Sound
4.2.2 The Wave Equation
4.2 Physics of Sound 115
EXAMPLE4.1 Suppose the frequency of a sound wavef = 50 cycles/s (or Hertz, denoted by Hz) and assume that the velocity of sound at sea level isc = 344 m/s. Then the wavelength of the sound wave λ = 6.88 m. For f = 1000 Hz, λ = 0.344 m, while for f = 10000 cycles/s, λ = 0.0344 m. This wide range of wavelengths occurs in the propagation of speech
sounds. 䉱
In describing sound propagation, it is important to distinguish pressure change due to variations that occur rapidly or slowly [2]. Anisothermalprocess is a slow variation that stays at constant temperature. Isothermal compression of a gas results in an increase in pressure because a given number of molecules are forced into a smaller volume and will necessarily collide with each other (and the boundaries of an enclosure) more often, thus increasing the time rate of change of momentum. However, because the variation is slow, there is time for the heat generated by collisions to flow to other parts of the air medium; thus the temperature remains constant. On the other hand, anadiabaticprocess is a fast variation in which there is no time for heat to flow away and thus there occurs a temperature change in the medium. The heat generated causes an even greater number of collisions. The molecules get hotter faster and collide with each other (or the boundaries of an enclosure) more frequently. They have greater momentum (they are moving faster) and thus transfer more momentum to each other during collisions. It follows that an adiabatic gas is “stiffer” (than an isothermal gas) since it takes more force to expand or compress it.3For most of the audible range, a sound wave is an adiabatic process,4a property that we use in deriving the wave equation.
Infinite Vibrating Wall
Δy Δx
Δz p
x
Δx p + ∂P
∂x
Figure 4.3 Cube of air in front of an infinite vibrating wall. Cube configuration is shown with first-order pressure change across the cube. Arrows indicate forces on two vertical surfaces.
sensitive to pressure changes. The particle velocity is the rate of change in the location of an air particle and fluctuates about zero average velocity. The particle velocity is measured in m/s and is denoted byv(x, t ). The density of air particles, denoted by ρ(x, t ), is the mass per unit volume and is measured in kg/m3 around an average density ρo, the total density being ρo+ρ(x, t ).
There are three laws of physics used to obtain the desired relations between pressure and velocity of the air particles within the cube [2]. Newton’s Second Law of Motion states that the total force on the cube is the mass times the acceleration of the cube and is written asF =ma. This law predicts that a constant applied force produces a constant acceleration of the cube.
The Gas Law, from thermodynamics, relates pressure, volume, and temperature and, under the adiabatic condition—the case of interest for speech sound propagation—reduces to the relation P Vγ = C, whereP is the total pressure on the cube of volume V, C is a constant, and γ =1.4 is the ratio of the specific heat of air at constant pressure to the specific heat of air at constant volume [2]. Given that the cube is assumed deformable, this formula predicts that if the pressure increases, then the volume of the cube decreases. The third law is the Conservation of Mass: The total mass of gas inside the deformable cube must remain fixed. This law states that a pressure differential can deform the cube, but the total number of particles in the deformable cube remains constant. The moving boundaries of the cube are defined by the original particles.
The local density may change, but the total mass does not change. Using these three laws, we can derive the wave equation. We do not give the complete derivation of the wave equation, but rather give a flavor of one approach to its derivation.
The first step in deriving the wave equation relies on Newton’s Second Law of Motion.
Three assumptions are made that result in equations that are linear and computationally tractable [2],[27]. First, we assume there is negligible friction of air particles in the cube with those outside the cube, i.e., there is no shearing pressure due to horizontal movement of the air. We refer to this shearing pressure asviscosity. Therefore, the pressure on the cube is due to only forces on the two vertical faces of the cube, as illustrated in Figure 4.3. Our second assumption is that the cube of air is small enough so that the pressure change across the cube in the horizontal dimension (x) is of “first order,” corresponding to sounds of not extremely large intensity.
4.2 Physics of Sound 117
This means that the second- and higher-order terms in a Taylor series expansion of the pressure function with respect to thexargument can be neglected, resulting in
p(x +x, t ) ≈ p(x, t )+ ∂p
∂xx
where∂p∂xis the rate at which the pressure increases in thexdirection (left to right). Henceforth, we drop the(x, t )notation unless explicitly needed. The third assumption is that the density of air particles is constant in the cube and equal to the average atmospheric density, i.e.,ρ(x, t )+ ρo=ρo, which we will denote by simplyρ.
Because the pressure is force divided by the surface area of the vertical face of the cube, the net force in thexdirection on the cube is the pressure difference across the cube multiplied by the surface area:
F = −
∂p
∂xx
yz.
We have assumed the density in the cube to be constant. Therefore, the massmof the cube is given by
m = ρxyz.
The acceleration of the cube of air is given bya= dvdt, wherev(x, t )denotes the velocity of the particles in the cube.5Finally, from Newton’s Second Law of Motion,F =ma, we have the net force acting on the cube of air as
−
∂p
∂xx
yz = ρxyz
dv
dt
and cancelling terms we have
−∂p
∂x = ρ∂v
∂t (4.1)
which is accurate “in the limit,” i.e., asx gets very small the approximations become more accurate because the differential pressure becomes “1st order” and the density constant.
The reader might have observed a slight-of-hand in arriving at the final form in Equation (4.1); the total derivative dvdt in Newton’s Second Law was replaced with the partial derivative
∂v
∂t. Becausev(x, t )is a function of spacexas well as timet, the true acceleration of the air particles is given by [19]
dv dt = ∂v
∂t +v∂v
∂x (4.2)
and therefore Equation (4.1) is correctly written as
−∂p
∂x = ρ
∂v
∂t + v∂v
∂x
(4.3)
5Although the velocity of particles can change across the tube, we letv(x, t )denote the average velocity in the cube, which becomes exact in the limit asx→0 [2].
which is anonlinearequation in the variablevbecause the particle velocityvmultiplies ∂v∂x. Consequently, it is difficult to determine a general solution. The approximation Equation (4.1) is accurate when the correction termv∂v∂x is small relative toρ∂v∂t. Typically, in speech production it is assumed that the particle velocity is very small and therefore the correction term introduces only second-order effects.6This approximation, in vector form, rules out rotational or jet flow [19], and thus the possibility of vortices along the oral cavity, as alluded to in Chapter 3 and further described in Chapter 11.
Equation (4.1) takes us halfway to the wave equation. Completing the derivation requires the use of the Gas Law and Conservation of Mass principle which can be shown to result in the relation [2]
−∂p
∂t = ρc2∂v
∂x (4.4)
wherecis the velocity of sound andρ is the air density assumed to be constant.7The pair of equations Equation (4.1) and Equation (4.4) represents one form of the wave equation. A second form is obtained by differentiating Equations (4.1) and (4.4) byxandt, respectively:
−∂p2
∂x2 = ρ ∂2v
∂x∂t
−∂p2
∂t2 = ρc2 ∂2v
∂x∂t
which, when combined to eliminate the mixed partials, can be written as a second-order partial differential equation in pressure only:
∂p2
∂x2 = 1 c2
∂p2
∂t2. (4.5)
Likewise, the above equation pair can be alternatively combined to form the second-order partial differential equation in velocity only:
∂v2
∂x2 = 1 c2
∂v2
∂t2. (4.6)
The alternate forms of the wave equation given by Equations (4.5) and (4.6) describe the pressure and velocity of air particles, respectively, as a function of position and time. In summary, the two different wave equation pairs are approximately valid under the following assumptions:
(1) The medium is homogeneous (constant density), (2) The pressure change across a small distance can be linearized, (3) There is no viscosity of air particles, (4) The air particle velocity
6Portnoff [26] argues that when the DC component of the particle velocityv, i.e., the steadynet flow component tov, is small relative to the speed of sound, then the correction term is negligible. The conditions under which the steady flow component of particle velocity through the vocal folds and within the vocal tract is small relative to the speed of sound are discussed in Chapter 11.
7As with Equation (4.1), the correct form of Equation (4.4) involves a nonlinear correction term and can be shown to be of the form−∂p∂t =ρc2∂v∂x−v∂p∂x [19]. As before, the correction term is negligible if we assume the velocity multiplier is small.