Waves on a Flat Surface of Discontinuity Chapter 3 Waves on a Flat Surface of Discontinuity Yang-Hann Kim

Outline 3.1 Introduction/Study Objectives 3.2 Normal Incidence on a Flat Surface of Discontinuity 3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) 3.4 Transmission Loss at a Partition 3.5 Oblique Incidence (Snell’s Law) 3.6 Transmission and Reflection of an Infinite Plate 3.7 The Reflection and Transmission of a Finite Structure 3.8 Chapter Summary 3.9 Essentials of Sound Waves on a Flat Surface of Discontinuity

3.1 Introduction/Study objectives What if we have a distributed impedance mismatch in space? How does this change the propagation characteristics of waves in space? To begin with, the flat surface of a discontinuity in space, that is, a wall, which creates an impedance mismatch in space is taken. We will study how this mismatch transmits and reflects waves. This chapter begins with the simplest wall, which is modeled as a limp wall. A limp wall is defined as one which has only mass. A more general wall creating an impedance mismatch is then introduced.

3.2 Normal Incidence on a Flat Surface of Discontinuity As illustrated in Figure 3.1, suppose that we have a flat surface of discontinuity that separates two different media. Let us also assume that a wave propagates in the direction perpendicular to the flat surface. We usually call this type of incident wave to the surface “normal incidence” or “perpendicular incidence”. Figure 3.1 The reflected and transmitted wave for a normal incident wave. (The subscripts denote the incident, reflected, and transmitted wave, respectively. expresses the sound pressure with regard to time and space and denotes the complex pressure amplitude)

3.2 Normal Incidence on a Flat Surface of Discontinuity The first point that we realize is that the pressure must be continuous on the surface ( ); otherwise the surface will move according to Newton’s second law. In addition, the velocity of a fluid particle at the surface must also be continuous. First, the pressure continuity at can be written as (3.1) The velocity continuity is expressed as (3.2) where the subscripts i, r, and t represent the incident, reflected, and transmitted wave, respectively. P and U are the complex amplitude of pressure and velocity.

3.2 Normal Incidence on a Flat Surface of Discontinuity The incident wave ( ) , reflected wave ( ) , and transmitted wave ( ) can therefore be written as (3.3) (3.4) (3.5) where k0 and k1 are defined (3.6) (3.7) where c0 and c1 are the speed of sound in medium 0 and 1, respectively.

3.2 Normal Incidence on a Flat Surface of Discontinuity For a plane wave, we can rewrite Equation 3.2 as (3.8) in which we use the relation where is the characteristic impedance of the medium. The ratio of to , that is, the reflection coefficient , can be obtained from Equations 3.1 and 3.8: (3.9) The transmission coefficient, which is the ratio of to , can be obtained from Equations 3.1 and 3.8 as (3.10)

3.2 Normal Incidence on a Flat Surface of Discontinuity To see how much power is essentially transmitted, we must determine the velocity reflection and transmission coefficient. These can be obtained by using the impedance relation of a plane wave, , from Equations 3.9 and 3.10. These are velocity reflection coefficient : (3.11) velocity transmission coefficient : (3.12) The power reflection/transmission coefficients are defined as the ratio between the reflected/transmitted power and the power of the incident wave. power reflection coefficient : (3.13) power transmission coefficient : (3.14)

3.2 Normal Incidence on a Flat Surface of Discontinuity Figure 3.2 Pressure reflection and transmission coefficient where and Figure 3.3 Velocity reflection and transmission coefficient where and

3.2 Normal Incidence on a Flat Surface of Discontinuity If the characteristic impedances of two media only have a real part (e.g., water or air), then the power reflection and transmission coefficients can be written as This can be derived from Equations 3.13 and 3.14. The sum of transmitted and reflected power at the flat surface, that is, the incident power, has to be 1, which can be derived by adding Equations 3.15 and 3.16. (3.15) (3.16)

3.2 Normal Incidence on a Flat Surface of Discontinuity Figure 3.4 Change of the power reflection and transmission coefficients with regard to variation of the characteristic impedances of the media, where and

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) A limp wall is a wall that only has mass. In other words, the mass effect is dominant compared to the stiffness or internal damping. “Locally reacting” assumption The fluid particles only oscillate the mass particles that they are in contact with, and the mass particles vibrate the fluid particles that they contact. Figure 3.5 Reflection and transmission due to the presence of a limp wall

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) Let us first look at the forces acting on the unit surface area of a limp wall. We assume that the wall harmonically oscillates, and then the following force balance equation on has to be hold. (3.17) Where is mass per unit area (kg/m2). The velocity of the fluid particle on the left-hand side of the wall should correspond to the vibration velocity of the wall, that is, (3.18) The velocity of the fluid particle on the right-hand side of the wall also equal to the vibration velocity of the wall, that is, (3.19)

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) We are interested in how much the waves are reflected and transmitted relative to the incident wave amplitude. These ratios can be obtained from Equations 3.17–3.19, which are The power transmission coefficient, which expresses how much power is transmitted relative to the incident power, can be written as Substituting Equation 3.22 into Equation 3.21 gives (3.20) (3.21) where R and t are the reflection and transmission coefficients, respectively. (3.22) (3.23)

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) Transmission loss ( or ) is defined as (3.24) From Equations 3.23 and 3.24, the limp wall transmission loss is (3.25) If is much greater than 1, then Equation 3.25 becomes (3.26) Mass law The transmission loss increases by 6 dB as we double the frequency; that is, 1 octave increase of frequency or mass per unit area is doubled.

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) To establish certain design guidelines, let us look at the frequency that makes the transmission loss zero. We refer to this frequency as “blocked frequency” ( ). This can be obtained from Equation 3.26, because the blocked frequency must satisfy (3.27) Therefore, (3.28) For example, if is 415 (the characteristic impedance of air at 20°C), then Equation 3.28 predicts the blocked frequency, that is, (3.29)

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) Figure 3.6 Mass law: the graph shows that increases by 6 dB when the frequency doubles (1 octave). The mass law is applicable from the blocked frequency

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) (Adapted from L. Cremer and M. Heckl, Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies, 2nd ed., Springer-Verlag ⓒ 1988, pp. 242.) Table 3.1 Sound insulation materials and their density

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) The sound on the left and right-hand sides of the wall is composed of two different components: incidence and reflection on the left, and trans-mission on the right. The pressure on the left side of the wall, , is found to obey (3.30) Using Equation 3.19, we can rewrite Equation 3.30 as (3.31) The magnitude of the transmitted wave can be written as (3.32)

3.3 The Mass Law (Reflection and Transmission due to a Limp Wall) In summary, this observation leads us to conclude that the incident, reflected, and transmission phenomena can be seen as the superposition of the blocked pressure and the radiation pressure induced by the wall’s motion (Figure 3.7). Figure 3.7 The principle of superposition allows us to regard the incidence, reflection, and transmission phenomena as the sum of the blocked and radiation pressure (blocked pressure and radiation pressure ).

3.4 Transmission Loss at a Partition We now extend our understanding to more general cases. Figure 3.8 illustrates a partition that represents a more general flat surface of discontinuity. Figure 3.8 The reflection and transmission due to the partition (rd is linear damping coefficient and s is linear spring constant)

3.4 Transmission Loss at a Partition To determine how much transmission will occur, we have to apply the same laws that we used for the limp wall case: the velocity continuity and the force balance between the wall and forces acting on the wall of unit area. The transmission coefficient can be readily obtained as (3.33) - The imaginary part of denominator in Equation 3.33 The mass contribution ( ) and the spring contribution ( ) have a 180 phase difference. - The real part of denominator in Equation 3.33 The first term expresses the radiation at both sides of the wall, and the second term is what is lost by the linear damping ( ).

3.4 Transmission Loss at a Partition Figure 3.9 Transmission and reflection, utilizing the concept of superposition ( , )

3.4 Transmission Loss at a Partition If we express Equation 3.33 using dimensionless parameters that can be obtained by dividing every term by Z0, then we have The numerator (2) of Equation 3.34 essentially indicates that the motion occurs in both directions. This leads us to rewrite Equation 3.34 as (3.34) (3.35)

3.4 Transmission Loss at a Partition We may also modify Equation 3.33 to understand the underlying physics differently: This expression can be rewritten as (3.36) (3.37) where is the partition impedance and is the fluid loading impedance in both directions.

3.4 Transmission Loss at a Partition Figure 3.10 (a) Normal incidence absorption coefficients. (b) Real and (c) Imaginary surface normal impedances of some typical sound absorbing materials (thinsulate, foam, and fiberglass) (Data provided by Taewook Yoo and J. Stuart Bolton, Ray W. Herrick Laboratories, Purdue University)

3.4 Transmission Loss at a Partition We can obtain the transmission loss in a similar manner to that described in Section 3.3, that is This is analogous to waves propagating along a string which is attached to a mass-spring-dash pot system, as illustrated in Figure 3.11. (3.38) where , which is the resonant frequency of the partition in vacuum. Figure 3.11 The incident, reflection, and transmission waves on strings that are attached to a single degree of freedom vibration system (m is mass, s is linear spring constant, and rd is viscous damping coefficient)

3.4 Transmission Loss at a Partition The case of : (3.39)  The transmission loss follows the mass law. The case of : (3.40)  The transmission loss mostly depends on the linear spring constant. The case of : (3.41)  The transmission loss is entirely dominated by the damping coefficient. This implies that we have to increase the damping to make the transmission loss larger.

3.4 Transmission Loss at a Partition Figure 3.12 Transmission loss at a partition where is the undamped resonant frequency of the partition

3.5 Oblique Incidence (Snell’s Law) If incident waves impinge on a flat surface of discontinuity with an arbitrary angle other than , the fluid particles on the wall will oscillate in directions both parallel and perpendicular to the wall (see Figure 3.13). Figure 3.13 Reflection and transmission for oblique incidence ( : incidence, reflection, and transmission angle; : incidence, reflection, and transmission wavelength)

3.5 Oblique Incidence (Snell’s Law) Suppose that an incident wave having pressure reaches a flat surface of discontinuity at , where mediums 0 and 1 intersect. If the incident wave reached ( ) after a period , arrives the surface ( ) at , then the following geometrical relations must hold: (3.42) where and are the incident, reflected, and transmitted angles, respectively. and are the corresponding wavelengths. Equation 3.42 can then be written as (3.43) The relation between wave number and wavelength transforms Equation 3.43 to (3.44)

3.5 Oblique Incidence (Snell’s Law) If the medium is non-dispersive, the dispersion relation must be . Equation 3.44 can then be rewritten as (3.45) We can then deduce from Equation 3.45 that (the incident and reflection angles are equal). Hence, Equation 3.45 can be rewritten as (3.46) To emphasize the characteristics of the media, let us denote and as t and . Equation 3.46 can then written as (3.47)

3.5 Oblique Incidence (Snell’s Law) Similarly, we can write the relations as , , . This is what we refer to as Snell’s law. This law simply expresses that the wave number in the direction at x=0 must be continuous in both media. Figures 3.14 and 3.15 depict the implications of these wave number relations. Figure 3.14 Snell’s law expressed in the wave number domain ( ). denotes the critical angle Figure 3.15 Snell’s law in the wave number domain ( )

3.5 Oblique Incidence (Snell’s Law) These relations can also be obtained by considering what we already observed in Section 3.2, that is, the pressure and velocity continuity condition on the flat surface of discontinuity that has impedance mismatch. We denote the incident, reflected, and transmitted waves as (3.48) where (3.49) where are the unit vectors in the direction, respectively.

3.5 Oblique Incidence (Snell’s Law) From Equations 3.48 and 3.49, the pressure continuity at can be written as (3.50) Equation 3.50 can be simply written as (3.51) which is exactly the same as what was derived for the case of normal incidence.

3.5 Oblique Incidence (Snell’s Law) The velocity continuity has to be satisfied by the fluid particles on the surface of discontinuity, that is, (3.52) where (3.53) Equations 3.49, 3.52, and 3.53 yield (3.54)

3.5 Oblique Incidence (Snell’s Law) Equation 3.54 can be rewritten as (3.55) where we adapted Equation 3.47 to simply emphasize the medium: Equation 3.55 can also be rewritten as (3.56) If we define the oblique wave impedance as (3.57) we can rewrite Equation 3.56 as (3.58)

3.5 Oblique Incidence (Snell’s Law) The degree to which the wave is reflected and transmitted is determined by the same approach as for the normal incident wave. Our findings are based on the assumption that the continuity on the surface is independent of . Equations 3.50 and 3.54 are based on this assumption. A surface of discontinuity that follows this assumption is called a “locally reacting surface”. Let’s consider a special case as depicted in Figure 3.14. If the angle of incidence is larger than the critical angle, then the transmission angle has to be somewhat larger than 90° which is not physically allowable. This situation can be modeled by writing the reflected angle as (3.59)

3.5 Oblique Incidence (Snell’s Law) By substituting Equation 3.59 into Equation 3.49 and then substituting the result into Equation 3.48 we can obtain the reflected wave that is exponentially decaying in the direction. We call this an “evanescent wave”. In contrast to this case, if we consider that the speed of propagation of medium 0 is larger than that of medium 1 (Figure 3.15), then the transmitted angle cannot be larger than any critical angle.

3.6 Transmission and Reflection of an Infinite plate Suppose that the surface of the discontinuity does not locally react to the waves, that is, incident, reflected, and transmitted waves as illustrated in Figure 3.16. For simplicity, let us assume that we have plane waves. Figure 3.16 Incident, reflected, and transmitted waves on the plate. The media are assumed to be identical

3.6 Transmission and Reflection of an Infinite plate The incident, reflected, and transmitted waves can be written as for Equation 3.48, that is, (3.60) The displacement of the surface of discontinuity ( ), a plate, only propagates in the direction. We can therefore write the displacement as (3.61) where is the amplitude of the displacement. The response of the plate to the pressures can then be written as (3.62) where we assumed that the plate is thin enough to neglect the shear effect.

3.6 Transmission and Reflection of an Infinite plate is the bending rigidity of the plate and can be written in this form: (3.63) where is the Poisson ratio, is the thickness of the plate, and is Young’s modulus. The characteristic equation that describes how the bending waves generally behave can be obtained by substituting Equation 3.61 into the homogeneous form of Equation 3.62. This yields (3.64) Equation 3.64 gives us the relation between the bending wave number ( ) and radiation frequency ( ), that is (3.65)

3.6 Transmission and Reflection of an Infinite plate If we use the dispersion relation, then the speed of propagation of the bending wave ( ) can be obtained as (3.66) Equation 3.66 simply states that the speed of propagation depends on frequency. Note that a wave of higher frequency propagates faster than a wave of lower frequency (see Figure 3.17). This characteristic causes the shape of the wave to change as it moves in space.

3.6 Transmission and Reflection of an Infinite plate Figure 3.17 Dispersive wave propagation in an infinitely thin plate

3.6 Transmission and Reflection of an Infinite plate If we express the velocity continuity mathematically, considering that the velocity of the surface at the discontinuity ( ) has to be exactly the same as the velocity of the fluid particle on the surface, we arrive at (3.67) Equation 3.67 has to be valid for all y and, therefore, the exponents have to be identical. This leads us to write (3.68)

3.6 Transmission and Reflection of an Infinite plate Equation 3.67 can therefore be rewritten as (3.69) (3.70) where , which is the oblique impedance of the incident wave. Equations 3.60, 3.61, 3.62, 3.69, and 3.70 give us the transmission loss of an infinite plate, that is (3.71) Equation 3.71 is a special case of Equation 3.37 when the partition impedance ( ) is (3.72)

3.6 Transmission and Reflection of an Infinite plate The magnitude of the transmitted wave ( ) can be obtained from Equations 3.68 and 3.70, that is, (3.73) Recalling that and using Equation 3.68 once more, we can rewrite Equation 3.73 as (3.74) Equation 3.60 can therefore be written as (3.75)

3.6 Transmission and Reflection of an Infinite plate By setting and , Equation 3.75 can be rewritten as (3.76) The key feature of Equation 3.75 and 3.76 is expressed in the square root of the propagation constant or the wave number in the direction. If the propagation speed of the bending wave is smaller than the speed of sound (subsonic), then we have an exponentially decaying wave (an evanescent wave) in the direction. This means that there are only waves in the vicinity of the plate and there is no transmission (Figure 3.18).

3.6 Transmission and Reflection of an Infinite plate Figure 3.18 The propagation characteristics of bending waves in subsonic, critical speed, and supersonic range

3.7 The Reflection and Transmission of a Finite Structure We have studied the reflection and transmission of the discontinuity of an infinite flat surface. They are relatively easy to tackle mathematically and therefore they are easy to understand physically. The basic characteristics are often preserved for more practical cases, or at least they provide a guideline to understand what would happen in practical (more realistic) cases. For example, the mass law can be applied to any finite partition or wall. Because it is a law for unit area, it can therefore be applied to a finite structure as illustrated in Figure 3.19.

3.7 The Reflection and Transmission of a Finite Structure Figure 3.19 Blocked pressure and radiation of a finite flat surface of discontinuity

3.7 The Reflection and Transmission of a Finite Structure The edge effect makes the transmission with regard to unit area of the partition larger than that of an infinite flat surface of discontinuity. We can therefore express it as (3.77) where and express the transmission coefficient of finite and infinite flat surfaces of discontinuity, respectively. The transmission loss will therefore obey the relation: (3.78) Equation 3.78 is very useful for practical applications. Because over-estimates the transmission loss, it can be safely used as a design value.

3.8 Chapter Summary We have examined how acoustic waves propagate in space and time with regard to the characteristic impedance of the medium and the driving point impedance. The limp wall example highlights that the mass law is the simplest case and is a fundamental principle that can be used in practice. We also found that the reflection and transmission can be regarded as radiation due to the motion of the structure or partition. If we have oblique incidence, the reflection and transmission are characterized by the surface of discontinuity. Transmission through a finite partition is due to the radiation of the partition. Every transmission can be expressed in terms of partition and fluid loading impedance.