第17章音

As stated earlier, the speed of sound in a medium depends on the medium and the state of the medium. The derivation of the equation for the speed of sound in air starts with the mass flow rate and continuity equation discussed in Fluid Mechanics.

Consider fluid flow through a pipe with cross-sectional area A (Figure 17.7). The mass in a small volume of length x of the pipe is equal to the density times the volume, or $m=\rho V=\rho Ax.$ The mass flow rate is

The continuity equation from Fluid Mechanics states that the mass flow rate into a volume has to equal the mass flow rate out of the volume, ${\rho }_{\text{in}}{A}_{\text{in}}{v}_{\text{in}}={\rho }_{\text{out}}{A}_{\text{out}}{v}_{\text{out}}.$

Now consider a sound wave moving through a parcel of air. A parcel of air is a small volume of air with imaginary boundaries (Figure 17.8). The density, temperature, and velocity on one side of the volume of the fluid are given as $\rho ,T,v,$ and on the other side are $\rho +d\rho ,T+dT,v+dv.$

The continuity equation states that the mass flow rate entering the volume is equal to the mass flow rate leaving the volume, so

This equation can be simplified, noting that the area cancels and considering that the multiplication of two infinitesimals is approximately equal to zero: $d\rho \left(dv\right)\approx 0,$

The net force on the volume of fluid (Figure 17.9) equals the sum of the forces on the left face and the right face:

The acceleration is the force divided by the mass and the mass is equal to the density times the volume, $m=\rho V=\rho dxdydz.$ We have

From the continuity equation $\rho dv=\text{−}vd\rho$, we obtain

Consider a sound wave moving through air. During the process of compression and expansion of the gas, no heat is added or removed from the system. A process where heat is not added or removed from the system is known as an adiabatic system. Adiabatic processes are covered in detail in The First Law of Thermodynamics, but for now it is sufficient to say that for an adiabatic process,$p{V}^{\gamma }=\text{constant,}$ where p is the pressure, V is the volume, and gamma $\left(\gamma \right)$ is a constant that depends on the gas. For air, $\gamma =1.40$. The density equals the number of moles times the molar mass divided by the volume, so the volume is equal to $V=\frac{nM}{\rho }.$ The number of moles and the molar mass are constant and can be absorbed into the constant $p{\left(\frac{1}{\rho }\right)}^{\gamma }=\text{constant}\text{.}$ Taking the natural logarithm of both sides yields $\text{ln}p-\gamma \text{ln}\rho =\text{constant}\text{.}$ Differentiating with respect to the density, the equation becomes

If the air can be considered an ideal gas, we can use the ideal gas law:

Here M is the molar mass of air:

Since the speed of sound is equal to $v=\sqrt{\frac{dp}{d\rho }}$, the speed is equal to

Note that the velocity is faster at higher temperatures and slower for heavier gases. For air, $\gamma =1.4,$ $M=0.02897\frac{\text{kg}}{\text{mol}},$ and $R=8.31\frac{\text{J}}{\text{mol}·\text{K}}.$ If the temperature is $T_{\text{C}}=20\text{°}\text{C}\left(T=293\text{K}\right),$ the speed of sound is $v=343\text{m/s}\text{.}$

The equation for the speed of sound in air $v=\sqrt{\frac{\gamma RT}{M}}$ can be simplified to give the equation for the speed of sound in air as a function of absolute temperature:

One of the more important properties of sound is that its speed is nearly independent of the frequency. This independence is certainly true in open air for sounds in the audible range. If this independence were not true, you would certainly notice it for music played by a marching band in a football stadium, for example. Suppose that high-frequency sounds traveled faster—then the farther you were from the band, the more the sound from the low-pitch instruments would lag that from the high-pitch ones. But the music from all instruments arrives in cadence independent of distance, so all frequencies must travel at nearly the same speed. Recall that

In a given medium under fixed conditions, v is constant, so there is a relationship between f and $\lambda ;$ the higher the frequency, the smaller the wavelength (Figure 17.10).

The speed of sound can change when sound travels from one medium to another, but the frequency usually remains the same. This is similar to the frequency of a wave on a string being equal to the frequency of the force oscillating the string. If v changes and f remains the same, then the wavelength $\lambda$ must change. That is, because $v=f\lambda$, the higher the speed of a sound, the greater its wavelength for a given frequency.

Although sound waves in a fluid are longitudinal, sound waves in a solid travel both as longitudinal waves and transverse waves. Seismic waves, which are essentially sound waves in Earth’s crust produced by earthquakes, are an interesting example of how the speed of sound depends on the rigidity of the medium. Earthquakes produce both longitudinal and transverse waves, and these travel at different speeds. The bulk modulus of granite is greater than its shear modulus. For that reason, the speed of longitudinal or pressure waves (P-waves) in earthquakes in granite is significantly higher than the speed of transverse or shear waves (S-waves). Both types of earthquake waves travel slower in less rigid material, such as sediments. P-waves have speeds of 4 to 7 km/s, and S-waves range in speed from 2 to 5 km/s, both being faster in more rigid material. The P-wave gets progressively farther ahead of the S-wave as they travel through Earth’s crust. The time between the P- and S-waves is routinely used to determine the distance to their source, the epicenter of the earthquake. Because S-waves do not pass through the liquid core, two shadow regions are produced (Figure 17.11).

Seismologists and geophysicists use properties and velocities of earthquake waves to study the Earth's interior, which due to it's depth and pressure is not observable through many other means. In fact, the discoveries of the structure of the Earth, illustrated in the figure above, resulted from earthquake observations. In 1914, Beno Gutenberg used differences in wave speeds to determine that there must be a liquid core within the mantle. In 1936, Inge Lehmann began investigating P-waves from a New Zealand earthquake that had unexpectedly reached Europe, which should have been in the shadow region. Up until that point, seismologists had explained such shadow waves as being caused by some type of diffraction (as Gutenberg himself assumed) or a result of faulty seismometers. However, Lehmann had installed the European instruments herself, and so trusted their accuracy. She calculated that the amplitude of the waves must be caused by the existence of a solid inner core within the liquid core. This model has been accepted and reinforced by decades of subsequent calculations, including those from nuclear test explosions, which can be measured very precisely.

As sound waves move away from a speaker, or away from the epicenter of an earthquake, their power per unit area decreases. This is why the sound is very loud near a speaker and becomes less loud as you move away from the speaker. This also explains why there can be an extreme amount of damage at the epicenter of an earthquake but only tremors are felt in areas far from the epicenter. The power per unit area is known as the intensity, and in the next section, we will discuss how the intensity depends on the distance from the source.