Reflection and Transmission of Plane Waves

When a plane pressure wave strikes a plane surface between two media, part of the wave will be transmitted and part reflected. The amount reflected and transmitted is generally characterized by a quantity called the acoustic impedance, which for a uniform medium is equal to the density times the wave speed, z = ρv, where ρ is the density and v is the velocity of sound in the medium.

当平面压力波撞击两种介质之间的平面表面时，部分波会被透射，部分会被反射。反射和透射的量通常由一个称为声阻抗的量来表征，对于均匀介质来说，声阻抗等于密度乘以波速，z = ρv，其中ρ是介质中的密度，v是介质中的声速。

The example of a wave traveling from the left to strike an interface between two media will be used. The pressure for the incident, reflected and transmitted waves for a given frequency may be expressed as:

将一个从左向右传播的波撞击两种介质界面的例子用来说明。对于给定频率，入射波、反射波和透射波的压强可以表示为：

These are of the same form as the solution to the wave equation for a string wave except that the variable that is varying sinusoidally is the pressure. For sound waves traveling in a solid, the sound speed is expressed in terms of the bulk modulus B, a measure of the elastic property of the solid. The quantity k =2π/λ , λ = wavelength, and ω = 2πf.

这些解的形式与弦波的波动方程解相同，只是变化的变量是压力。对于在固体中传播的声波，声速用体积模量B表示，B是固体弹性性质的度量。量k = 2π/λ，λ = 波长，ω = 2πf。

The pressure boundary condition at the interface between two media, p_i + p_r = p_t, leads by substitution to the expressions for the reflection coefficient R_p and transmission coefficient R_t:

在两种介质交界处的压强边界条件 $ p_i + p_r = p_t $ 代入后，得到反射系数 $ R_p $ 和透射系数 $ R_t $ 的表达式：

The following example is taken from the article referenced below on the reflection and transmission of ultrasonic waves. It involves a pressure wave traveling across a plane boundary between steel and water.

以下例子摘自下文提到的关于超声波在钢与水平面边界处反射和透射的论文。该例子涉及一个压力波穿过钢与水之间的平面边界。

钢和水的声阻抗ρv分别为：

水：ρ_w v_w = 1.5 × 10⁶ kg/m²/s

钢：ρ_s v_s = 46.5 × 10⁶ kg/m²·s

For a pressure wave traveling from steel to water, this leads to the coefficients: Reflection: R_p = -9.38, Transmission T_p = 0.063

对于从钢传到水的声波，这导致了系数：反射系数 R_p = -9.38，透射系数 T_p = 0.063

Note that the reflection from the boundary with a medium of lower acoustic impedance leads to a phase reversal of the wave and a small amount of transmission into the lower impedance medium.

注意，从声阻抗较低的介质边界反射时，会导致波的相位反转，并有少量能量传入较低声阻抗的介质。

对于从水传播到钢的声压波，系数为：

反射率：R p = 9.38，透射率：T p = 1.938

The reflection from the steel medium with its higher acoustic impedance does not involve a phase change, and leads to a larger amplitude and larger wavelength in the steel. The steel is characterized by a higher propagation speed and longer wavelength.

钢中的反射由于其较高的声阻抗不涉及相位变化，导致钢中声波的振幅和波长都更大。钢的特性是传播速度更高且波长更长。

Reference: http://www.fast.u-psud.fr/~martin/acoustique/support/rŽflection-rŽfraction.pdf

参考：http://www.fast.u-psud.fr/~martin/acoustique/support/rŽflection-rŽfraction.pdf