Sound Waves in Fluids

(See Section 10.7 for a more complete discussion of fluid equations.) Here, is the perturbation to the fluid velocity produced by the wave, and

(668) |

is the

It is helpful to write

(669) | ||

(670) |

where is conventionally referred to as a

Moreover, substitution into Equation (667) yields the wave equation

where the characteristic wave speed is

(673) |

For the case of an ideal gas, for which , we obtain . (See Section 6.3.) On the other hand, for the case of water at , for which and , we get . This prediction is in good agreement with the measured sound speed in water at , which is (Wikipedia Contributors 2012).

Forming the sum of times Equation (665), times Equation (666), and times Equation (667), we obtain

where

(675) | ||

(676) | ||

(677) |

Equation (674) can be recognized as a two-dimensional energy conservation equation. (See Section 7.4.) Here, is the

Consider a situation (analogous to that illustrated in Figure 44) in which a sound wave is incident at an interface between two uniform immiscible fluids. Let the region be occupied by a fluid of equilibrium density and sound speed , and let the region be occupied by a fluid of equilibrium density and sound speed . We can write the wavevectors of the incident, reflected, and refracted waves as

(678) | ||

(679) | ||

(680) |

respectively. Here, for the sake of simplicity, we have assumed that all three wavevectors lie in the same plane (as is readily demonstrated--see Section 8.7.) Moreover, in order to be valid solutions of the wave equation, (672), all three waves must satisfy the dispersion relation , where is the common wave frequency. Finally, , , and are the angles of incidence, reflection, and refraction, respectively. (See Section 8.7.)

The velocity potential in the region is written

(681) |

where the first and second terms on the right-hand side specify the incident and reflected waves, respectively. The velocity potential in the region takes the form

(682) |

where the term on the right-hand side specifies the refracted wave. The first physical matching constraint that must be satisfied at the interface is continuity of the acoustic pressure: that is,

(683) |

This contraint yields

The previous equation holds at all values of . This is only possible if

(685) | ||

(686) |

These two expressions are analogous to the laws of reflection and refraction, respectively, of geometric optics. (See Section 8.7). This suggests that these laws are of universal validity, rather than being restricted to light waves. Equation (684) reduces to

The second physical matching constraint that must be satisfied at the interface is continuity of the normal velocity: that is,

(688) |

This constraint yields

where use has been made of Equation (686). Equations (687) and (689) can be combined to give

(690) | ||

(691) |

Equations (670), (671), and (677) reveal that the mean acoustic energy fluxes, normal to the interface, associated with the incident, reflected, and refracted waves are

(692) | ||

(693) | ||

(694) |

respectively. Thus, it follows that the coefficients of reflection and transmission at the interface are

respectively. It is actually possible for there to be no reflection at the interface (i.e., ), provided that . This criterion yields

which can only be satisfied if and , or and . The critical angle of incidence at which there is no reflection is sometimes called the