游戏中的声音-墙上的声音泄漏（译文）

原文地址：https://www.codeproject.com/Articles/467041/Sound-in-Games-Acoustical-Leaks-in-Walls

原文作者：Kenneth Haugland

译文由本站 robot-v1.0 翻译

前言

Mathematical treatment of acoustic leaks in walls

墙体声泄漏的数学处理

• 下载源代码和演示-166.7 KB(Download source code and demo - 166.7 KB)

介绍(Introduction)

这是一篇针对声学或需要在程序中处理声音的人员的文章.开口泄漏是一个非常普遍的问题,尤其是在建筑声学方面,因此在现实的游戏环境中尤为如此.(This is an article for acoustical or people who need to deal with sound in their program. Leaks through openings are a very common problem, especially in building acoustics, and therefore in realistic game situations.) 应该注意的是,有关现实世界中房间和声源定位的许多信息直接基于我们的听力,因为我们经常通过听见声音的来源来估计其他人的位置.然而,模拟真实情况非常复杂,因为我们使用身体反射来确定声音来源的高度位置,并用一只耳朵到另一只耳朵的时间差来确定声音的x-y位置.这种方法在声学中被称为"头部相关传递函数(HRTF)",适用于需要进一步研究的人们.(One should note that a lot of information on the room and source localization in the real world is based directly on our hearing, as we often estimate the position of others by hearing where the sound came from. It is however quite complicated to simulate the real situation, as we use the reflection from our body to determine the location of height where the sound came from, and the difference in time from one ear to the other to determine the x-y position of sound. This approach is called Head Related Transfer Function (HRTF) in Acoustics for people who want to investigate this further.) 因此,对于对游戏听觉化感兴趣的开发人员,或者想在自己的程序中加入一些声学效果的人员,本文最有用.(The article is therefore most useful to developers that are interested in game auralization, or if you want to incorporate some acoustical effects in your own program.) 为了充分理解本文,您需要或应该理解的数学方面是如何处理:(The mathematical aspects that you need or should understand to fully appreciate this article, are how to deal with:)

• 具有常规算术运算以及三角函数的复数(Complex numbers with normal arithmetic operations, as well as trigonometry functions)
• 涉及积分和推导的基本演算(Basic calculus involving integration and derivation)
• ODE及其解决方案(ODE and their solutions) 但是,如果数学过于繁重,我将向您展示一些简单的等效问题的思考方法.您还应该知道,可以在使用电阻器,电容器和电感器的电路中或使用弹簧,粘性阻尼器和质量的机械配置中使用类似的等效项来构造方程式.(I will however show you some simple equivalent ways of thinking about the problem, if the math is too heavy. You should also be aware that the equations could be constructed using a similar equivalent in either the electrical circuit using resistors, capacitors and inductors or a mechanical configuration using springs viscous damper and masses.) 我用来开发公式的方程式已经过时了,因为它涉及贝塞尔函数,这是球面波方程的通用解决方案,因此本文将包含一些在数学中不常用的重数学在线文章或博客,至少与该问题没有直接关系.还应该说,这也是学习一些基本声学方面的良好起点,因为在声学中,很多情况下某些材料可以直接转移.(The equation I’m using to develop the formula is quite old fashioned as it deals with Bessel function, which is a general solution to the spherical wave equation among other things, and the article will therefore include some heavy mathematics that is not commonly treated in online articles or blogs, at least not in direct relation to this problem. It should also be said that it is also a good starting point for learning some basic acoustic aspects, as there are lot of situations in acoustics where some of the material is directly transferable.) 基本问题定义如下.给定来自外部(从左到右行进)的入射声波,其入射角基于孔的法线,它穿过孔并从另一侧出来.孔的直径为(The basic problem is defined as follows. Given an incoming sound wave from outside (traveling from the left to the right) with an angle of incident based on the normal of the hole, it travels through the hole and comes out the other side. The hole has the diameter of) 2a 在两侧,管的长度或深度定义为(on both sides and the length or depth of the tube is defined as) d .声音水平降低了多少？(. How much has the sound level been reduced?)

我将通过将矿物棉放入孔中的效果来扩展原始文章.这意味着我可以模拟以下效果:(I will expand the original article by including the effect of putting mineral wool inside the hole. This means I can simulate the effect of:)

• 一个空的圆孔(An empty circular hole)
• 将轻质材料放在一个或两个孔的前面(例如胶带)(Putting a light weight material in front of one or two of the holes (like duct tape))
• 将矿棉放入覆盖整个风管的孔中(Putting mineral wool inside the hole that covers the entire length of the duct)
• 并在开口的每一侧分别放入矿棉和胶带(And putting mineral wool inside and duct tape on each side of the openings) 使用此程序,您应该能够理解不同配置对最终结果或内部声压的影响.您还应该注意以下事实:在计算中,我仅计算法向入射声波,但是如果您更改入射波的角度theta,它很容易扩展.(With this program, you should be able to understand the effect that the different configurations will have on the end result, or the sound pressure inside. You should also take note of the fact I’m only calculating the normal incident sound wave, in the calculation, but it is easily expanded if you change the angle theta of the incoming wave.)

背景(Background)

在向您展示方程式之前,我将快速为您提供一些用于构造解决方案的简单类比,或者更确切地说,您如何看待这个问题.(Before I’m going to show you the equations, I’ll quickly give you some simple analogies that are used in constructing the solution, or rather how you could think of this problem.)

• 房间或类似的封闭空间可以建模为(A room, or a similar closed space could be modeled as a) Capacitor 或作为(or as a) Spring
• 管道或房间中的泄漏可建模为(A pipe, or a leak in a room, could be modeled as an) Inductor 或作为(or as a) Mass
• 细管矩阵可以建模为(A matrix of thin pipes could be modeled as a) Resistor 作为(of as a) Viscous Damper .为了进一步研究声学概念,您可以看到(. For further examination of the acoustical idea, you could see this) 文章(article) .(.) 传输波的解决方案实际上是过滤器,其中不同的材料会改变某些共振,并向传递函数添加新的极点或零点.(The solution to the transmitted wave is actually a filer, where the different materials change some of the resonances and add new poles or zeros to the transfer function.) 该程序中使用的方法最初源自使用边界元法(BEM)求解波动方程的方法,我将简要解释该方程推导的基本原理.作为旁注,还应该知道声学模拟与有关可视化的问题相对紧密相关,尤其是在声学高频区域.第一个光线追踪计算机程序是由声学家开发的,后来被视觉设计师采用. (您可以阅读有关射线追踪声学发展的程序的更多信息(The method used in the program originally stems from solutions of wave equation using Boundary Element Method (BEM), and I will explain briefly the basics of the derivation of the equation. As a side note, it is also worth knowing that acoustical simulations are relatively closely connected to problems regarding visualisation, especially in the acoustical high frequency area. The first ray-tracing computer program was developed by acousticians, and later adopted by visual designers. (You could read more about the programming concerning acoustical development of raytracing) 这里(here) ).().) 在我描述整个公式之前,先解释一下这种情况下希腊字母tau实际代表什么.它表示一个常数,为了获得壁另一侧的压力,要乘以N/m ^ 2的压力,通常将其表示为10 * log10(tau),因为我们非常发现声学中的对数(迷恋对数的原因是,我们感知到的声音响度与对数函数有关).完整的一般公式如下所示:(*Some explanations of what the greek letter tau actually represents in this case, before I describe the entire formula. It represents the constant, that the pressure in N/m^2 is multiplied with in order to get the resulting pressure on the other side of the wall, and it’s usually given as 10 log10 (tau), as we are very found of logarithms in acoustics (The reason for the obsession about logarithms are that our senses of the perceived loudness of sound are linked to the logarithmic function). The complete general equation looks like this:)

它在大多数设置中都是有效的,尽管其参数已理想化,但您应始终记住,如果它在理想化计算中不起作用,那么它将无法在现实世界中运行.(It is valid in most settings, although its parameters are idealised, you should always remember that if it doesn’t work in the idealised calculation, it won’t work in the real world.) 以下是公式中不同元素的简要说明:(A brief explanation for the different elements in the formula are given below:)

为了对该方程进行一些其他解释,以便正确理解此处的情况,并且该孔只是一个复杂的滤波器,它将减少或增强输出中不同频率下的某些影响.将其分解为较小的步骤时,通常更容易理解上述的声学方程.有几种方法可以做到这一点,但我能想到的最简单的方法是构造一个电模拟电路:(Some additional explanations of the equation are in order to properly understand what’s going on here, and the hole is just a complex filter that would reduce or enhance some effects at different frequencies in the output. It is generally much easier to understand an acoustic equation of the sort above when it’s broken into smaller steps. There are several way to do this but the simplest one I could think of, is to construct an electric analogy circuit:)

现在,我们的方程式分解为简单的步骤,可以单独解释.我们的声源是一个交流电压源,标记为2Pe,其中包括周围材料(即空气)中的电阻.(Now our equation is broken into simple steps, that could be explained separately. Our source of sound is a AC voltage source, that is marked 2Pe, this would include the resistance in the surrounding material, i.e., the air.) 其他按从左到右的顺序进行说明:(The others are explained in the order from left to right:)

• Zr1 ,这是开口具有的辐射阻抗.这意味着,以力F进入的波将通过辐射阻抗的一个因数转换为运动u.(, this is the radiation impedance that the opening has. It means that the incoming wave with a force F, would be translated into the movement u by a factor of the radiation impedance.)

• m1 是放置在开口前方的潜在物体的质量.这是模拟胶带的地方.(is the mass of the potential object that is placed in front of the opening. This is where the duct tape would be simulated.)

• F1 是材料的弹性(is the resilience of the material) m1 ,对于薄板材料(如胶带)可以忽略.如果没有盖住开口(, which can be ignored for thin plate materials like duct tape. If the opening is not covered,) m1 和(and) F1 不会包含在等式中.(will not be included in the equation.)

• Z3 是从右到左的传播.(is the propagation from the right to the left.)

• 的两个要素(The two elements of) Z4 是从一端到另一端的反射波.(are the reflected waves from one end to the other.)**注意:(NB:)**它们只是管半径不变的相同值,其他所有东西也是对称的.(They are only the same values where the radius of the tube doesn’t change, and that everything else is also symmetrical.)

• m2 和(and) F2 是可以覆盖流出孔的元素.使用类似的公式计算(are the elements that could cover the outgoing hole. The are calculated using similar equations in) m1 和(and) F1 .(.)

• Zr2 是输出元素的辐射阻抗,在我们的例子中,与(are the radiation impedance from the outgoing element, and in our case, it’s the same as) Zr1 .(.) 为了检查不同的部分,我将首先介绍三种不同的声阻抗:(To examine the different parts, I will first go through the three different kinds of acoustic impedance:)

• 比声阻抗.含义(Specific acoustic impedance. Meaning) z = pressure/particle speed .(.)

• 声阻抗.(Acoustic impedance.) Z = pressure / volume velocity

• 辐射阻抗.(Radiation Impedance.) Z<sub>r</sub> = force / particle speed .(.) 它们可在不同的情况下使用,具体取决于您要解决的方程式/问题.但是,在声学中组合方程的一个重要方面是假设粒子速度的连续性,这意味着粒子速度不能从一个位置突然改变到另一个位置.在试图将不同类型的材料组合在一起的许多情况下,理解这一点至关重要.(They are used in different circumstances, dependent on the equation/problem you are trying to solve. However, one important aspect of combining equations in acoustics is by assuming continuity of particle velocity, meaning the particle velocity can’t abruptly change from one place to another. This is vital to understand in many situations where you are trying to combine different types of materials together.)

辐射阻抗(Radiation Impedance)

首先要定义辐射阻抗:(The first thing is to define radiation impedance which is:) Z = F/u, 其中F是力,u是质点速度,Z当然是辐射阻抗.使用积分逐步建立方程式setp可能是一个相当困难的任务,因此我省略了这些步骤,而是参考了本文结尾处给出的参考书.例如,书中给出了方程的完整推导(where F is the force, u is the particle velocity and Z is, of course, the radiation impedance. It can be a rather difficult task to evelop this equation setp by step using integration, so I have omitted these steps, and refer instead to the book reference given at the end of the article. A full derivation of the equation is for instance given in the book)“声学基础(“Fundamentals of Acoustics*)*(第185-186页).(*” (p. 185-186).*)

在此,将R1项定义为电阻函数,将X1项定义为电抗函数.电阻函数是使用Bessels方程中的解定义的:(Here, the R1 term is defined as the resistance function, an X1 as the reactance function. The resistance function is defined using the solutions found in Bessels equation:)

使用Struves函数可以找到X1项,如下所示:(The X1 term is found using Struves function as seen below:)

辐射阻抗定义为散热器施加在介质上的力与散热器速度之比.(Radiation impedance is defined as the ratio of the force a radiator exerts on a medium, to the velocity of the radiator.)

        ''' <summary>
        ''' Finds the radiation impedance of a circular piston
        ''' </summary>
        ''' <param name="radius">The radius of the piston</param>
        ''' <param name="frequency">The frequency in question</param>
        ''' <returns>A complex radiation impedance</returns>
        ''' <remarks></remarks>
        Public Function RigidPistonRadiator_
                  (ByVal radius As Double, ByVal frequency As Double) As Complex
            'If the user, for some silly reason puts in 0 or a negative number as the radius
            If radius <= 0 Then
                Return (New Complex(0, 0))
            End If

            Dim r1, x1 As Double

            'Medium constants
            Dim c0 As Double = 340
            Dim p0 As Double = 1.21

            'Wavenumber
            Dim k As Double
            k = 2 * System.Math.PI * frequency / c0

            'The resistance function
            r1 = 1 - (2 * Acoustics.SpecialFunctions.BesselJ_
                       (2 * k * radius, 1) / (2 * k * radius))

            'The reactance function
            x1 = 2 * Acoustics.SpecialFunctions.Struve(2 * k * radius, 1) / (2 * k * radius)

            'Stor the results from calculation
            Dim Result As New Complex(r1, x1)

            'Multiply with the caracteristic impedance with area
            Result = System.Math.PI * radius ^ 2 * c0 * p0 * Result

            Return Result
        End Function

质量阻抗(Mass Impedance)

实际上,我在对简单胶带或类似材料的阻尼效果进行建模时进行了简化.为了进行完整的数学处理,您还需要包括材料的刚度,但这对于打算在此程序中使用的轻质薄材料通常是无关紧要的.您实际上可以从这些方程式得出声学中的质量定律.它只是说薄壁被空气包围((I actually made a simplification in modeling the damping effect of simple duct tape or similar materials. For a complete mathematical treatment, you need to include the stiffness of the material also, but that’s usually an irrelevant property in thin lightweight materials intended for use in this program. You can actually derive the mass law in acoustics from these equations. It simply states that a thin wall surrounded by air () Z<sub>0</sub> 在下面的等式中,它是空气的特征阻抗),在给定频率下的阻尼取决于频率乘以质量(质量,即每平方米的总质量).(in the equation below that is the characteristic impedance of air), the damping at a given frequency, is dependent on the frequency times the mass (and by mass I mean the total mass per square meter).)

我们通过找到材料的吸收系数来进行:(We proceed by finding the absorption coefficient for the material:)

从Zg的初始方程式我们还知道,在我们的假设中没有内部损失或任何其他损失,因此,吸收它的唯一方法必须是离开板的另一侧.然后,我们用替换值编写解决方案:(We also know from our initial equation for Zg that there is no internal or any other losses in our assumptions, therefore the only way it can get absorbed must be by exiting the other side of the plate. We then write up the solution with replaced values:)

还可以计算出声音的减少量:(The sound reduction could also be calculated:)

我们插入空气的特征阻抗值并清理表达式,得出以下等式:(We insert the values for the characteristic impedance of air and clean up the expression, and arrive at this equation:)

质量定律在预测普通房屋墙壁的低频隔音效果时经常使用,尽管由于实际墙壁更复杂,系数42.5更改为47.它也被认为是低频区域中最大可能的减少水平,在这种情况下,声音的减少通常由总壁的重量决定.应该注意的是,对于简单的均质墙,它通常非常适合.(The mass law is used quite frequently in predicting the sound insulation at low frequencies of normal housing walls, although the coefficient 42.5 is changed to 47 as a real wall is more complex. It is also considered to be the maximum possible reduction level in the low frequency area, where the sound reduction is normally decided by the share weight of the total wall. It should be noted that for a simple homogenous wall it usually fits rather well.) 通常,它在较高的频率下无效,因为正常的壁面还取决于双头螺栓的共振能力,盖板的极限频率以及板之间的距离.降噪也可能比质量定律在质量控制区域之上的预测高得多.(Normally, it is not valid in the higher frequencies, as a normal wall is also determined by resonant capacities of studs and the limit frequency of the covering plate as well as the distance between the plates. The sound reduction could also be much higher than the mass law predicts above the mass controlled area.)

介质中的特征阻抗和传播函数(Characteristic Impedance and Propagation Function in a Medium)

特征阻抗之所以重要的原因是,您可以看到它很有用,因为它以恒定的方式将粒子速度与压力联系在一起.(The reason that characteristic impedance is important so you can see that it is useful, as it links the particle velocity with pressure by a constant.)

但是,这是一维波动方程解的一种特殊情况,并非对所有条件都有效.它基本上描述了压力波中的阻力乘以粒子速度.(It is however a special case of the solution to the one dimensional wave equation, and not valid for all conditions. It basically describes the resistance in a pressure wave multiplied with the particle velocity.) 球面波的真正解决方案实际上是(The real solution for spherical wave is actually) Zk = rho C *(jkr/(1+jkr)) 只有当(and only when) kr >> 1 特性阻抗是否为(will the characteristic impedance be) rho c. 如果假设气流阻力仅取决于常数速度,则粒子速度将获得介质损失的一维模型(称为瑞利模型).这是我们拥有的介质中最简单的损耗模型,Mechel在低频区域使用了此简单模型,并进行了一些经验测量,以使其适合高频.由于矿物和玻璃纤维中材料配置的复杂性,必须简单地使用经验模型.这种损耗是介质中正常几何损耗的附加损耗,但会增加一致的阻尼和每米相移.本文中未包含此详细信息,我将向您推荐(If one assumes the airflow resistivity is dependent on only on a constant times the particle velocity one gets an one dimensional model (called Rayleigh-model) of loss in a medium. It’s the simplest model for losses in a medium we have, and Mechel uses this simple model in the low frequency area and some empirical measurements to make it fit at high frequencies. The empirical models have to be used simply because of the complexity of the material configuration in mineral and glass fibre. This loss is in addition of normal geometry losses in a medium, but adds a consistent damping and phase shift per meter. This details of this is not included in the article and I would refer you to this) 文章(article) 代替.(instead.)

程序中使用的特殊数学函数(Special Mathematical Functions Used in the Program)

我具有使用此程序所需的三个特殊功能,并将它们放在模块中.还提供了代码源.它们最初来自Jack Xu的书< C#的数值方法>(感谢作者允许我出版这一部分).但是,根据给定的引用更改了Gamma函数,并且Struve函数是我自己的实现,基于代码注释中给出的Wikipedia链接.(I have three special functions that are needed to use this program, and I have placed these inside a module. The source of the code is also given. They do originally come from the book “Numerical methods for C#” by Jack Xu (with thanks to the author for allowing me to publish this part). However, the Gamma function is changed based on the references that are given, and the Struve function is my own implementation based on the Wikipedia link given in the code comment.)

Namespace Acoustics
    Module SpecialFunctions

        ''' <summary>
        ''' http://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_first_kind_:
        ''' _J.CE.B1
        ''' </summary>
        ''' <param name="x">Jv(X) where x is the number and v is the order</param>
        ''' <param name="v">Bessel function order</param>
        ''' <returns></returns>
        ''' <remarks></remarks>
        Public Function BesselJ(ByVal x As Double, ByVal v As Double) As Double
            Dim sum As Double = 0.0R
            Dim term As Double = 0.0R
            Dim i As Integer = 0
            Do
                term = System.Math.Pow(-1, i) * System.Math.Pow(0.5 * x, 2 * i + v) / _
                       Gamma(i + 1) / Gamma(i + v + 1)
                sum += term
                i += 1
            Loop While System.Math.Abs(term) > 0.000000000001R
            Return sum
        End Function

        ''' <summary>
        ''' http://en.wikipedia.org/wiki/Lanczos_approximation
        ''' </summary>
        ''' <param name="x">number x dependent on gamma function</param>
        ''' <returns>Gamma of the number x</returns>
        ''' <remarks></remarks>
        Public Function Gamma(ByVal x As Double) As Double
            Dim g As Integer = 7
            Dim p As Double() = {0.99999999999980993, 676.5203681218851, -1259.1392167224028,
                 771.32342877765313, -176.61502916214059, 12.507343278686905,
                 -0.13857109526572012, 0.0000099843695780195716, 0.00000015056327351493116}
            Dim y As Double

            If x < 0.5 Then
                Return Math.PI / (Math.Sin(Math.PI * x) * Gamma(1 - x))
            End If

            x -= 1
            y = p(0)

            For i As Integer = 1 To g + 1
                y += p(i) / (x + i)
            Next

            Dim z As Double = x + (g + 0.5)
            Return System.Math.Sqrt(2 * System.Math.PI) * _
                    System.Math.Pow(z, x + 0.5) * System.Math.Exp(-z) * y
        End Function

        ''' <summary>
        ''' http://en.wikipedia.org/wiki/Struve_function#Power_series_expansion
        ''' </summary>
        ''' <param name="x">The Hv(x) where x is the number and v is the order</param>
        ''' <param name="v">Struve function order</param>
        ''' <returns></returns>
        ''' <remarks></remarks>
        Public Function Struve(ByVal x As Double, ByVal v As Integer) As Double
            Dim result, term, sum As Double

            'Check if the first term is 0, if so then everything 
            ' would be zero
            If (0.5 * x) ^ (v + 1) = 0 Then
                Return result
            End If

            Dim i As Integer = 0
            Do
                term = ((-1) ^ i) * (0.5 * x) ^ (2 * i)
                term = term / (Gamma(i + (3 / 2)) * Gamma(i + v + (3 / 2)))

                'Avoid adding NonNumbers as they will cause termination
                If Not Double.IsNaN(term) Or Not Double.IsNegativeInfinity(term) Then
                    sum += term
                End If

                i += 1
            Loop While System.Math.Abs(term) > 0.000000000001R ' i < 100

            If Double.IsNegativeInfinity(result * sum) Or Double.IsNaN(result * sum) Then
                Return 0
            Else
                Return result * sum
            End If
        End Function
    End Module
End Namespace 

这三个简单函数在几个不同的声学问题中经常使用,并且在其他领域也经常构成许多数值解的基石.贝塞尔函数确实是波动方程的解,因此毫不奇怪.(The three simple functions are used quite frequently in several different acoustic problems, and will often form the cornerstone of many numerical solutions in other fields as well. The Bessel function is indeed the solution to the wave equation, so its hardly surprising.) Gamma函数是数学中最通用的函数之一,是我使用的所有函数的中坚力量,它还可以用于统计等领域.(The Gamma function is one of the most universal function in mathematics, and is the backbone in all the functions I use, and it can also be used in statistics among other.)

历史(History)

这是一个更新的版本,其中包括管内多孔材料的阻尼效果,因为我最初刚开始可以将模拟轻质覆盖材料放置在管的前面或后面.(This is an updated version that includes the damping effect of a porous material inside the tube, as I initially had just included the possibility of simulation lightweight cover materials to be placed in front or at the back of the tube.) 您应该知道该程序本质上非常简单,并且不包括在将多孔材料放入管内时的还原行为.结果也被归一化为1 m ^ 2,这意味着可以将其用于不同孔尺寸的比较.您应该在程序中删除术语1/area以获得其实际减少值.(You should know that the program is rather simple in nature, and does not include the behavior of the reduction when you put porous material inside the tube. The results are also normalized to 1 m^2, meaning it is created for comparison from different hole sizes. You should remove the term 1/area in the program to get its real reduction value.) 该程序可以按以下方式使用.假定您在墙壁的外部(如道路)上有一个噪声源,该噪声源在墙壁外的每个频段(距离交通繁忙的道路都不远)会产生40 dB,这意味着您房间内的噪声将是被感知为40-在任何给定频率下的减小(比这稍微复杂一点,这也取决于墙壁的减小以及房间内混响时间的减小).(The program could be used in the following way. Assume that you have a noise source on the outside of the wall (like a road) that produces 40 dB in each frequency band outside your wall (not far from a heavily traffic road), This would mean that the noise inside your room would be perceived as 40 - the reduction at any given frequency (it’s a little more complicated than that, as it is also determined by the reduction of the wall, and of the reverberation time inside the room).) 还应该感谢Jack Xu让我包括他从书中修改的源代码(Jack Xu should also be thanked for allowing me to include his modified source code from the book)” C#的数值方法"(“Numerical methods for C#"*)*在这个程序中.(in this program.) 还应注意,它通常会低估空洞在较高频率下的阻尼效果,因为当前的实现方式并未实现粘性损失或任何其他影响,这对于空洞可能是很重要的.对于板覆盖物或矿物纤维,计算结果与预期阻尼非常吻合.(It should also be noted that it generally underestimates the damping effect at the higher frequencies for an empty hole, as the current implementation does not take into effect the viscous losses or any other effects, which could be significant for an empty hole. With plate covering or with mineral fibre, the calculation fits very well with the expected damping.)

参考文献(References)

• “建筑声学”(“Building acoustics”),第一版(2008),Tor Erik Vigran,CRC出版社(, First edition (2008), Tor Erik Vigran, CRC Press)

• “声学公式”,(“Formulas of Acoustics”,*)*2(2)nd(nd)版,F.P.梅切尔(Springer)(Edition, F.P. Mechel, Springer)

• “房间声学”(“Room Acoustics”),第五版,Heinrich Kuttruff,Spon出版社(, Fifth edition, Heinrich Kuttruff, Spon press)

• <数学函数手册>(“Handbook of mathematical functions”),艾布拉莫维兹(Abramowitz)和史坦根(Stegun)(1970)(, Abramowitz and Stegun (1970), Dover)

• “使用C#的实用数值方法”(“Practical Numerical Methods with C#")徐Jack(, Jack Xu,)

• “声学基础”(“Fundamentals of Acoustics”),第四版,金斯勒,弗雷,科本斯和桑德斯,约翰`威利父子公司(, Fourth edition, Kinsler, Frey, Coppens and Sanders, John Wiley & Sons, Inc.) 在线链接:(Online links:)

• http://www.phys.unsw.edu.au/jw/z.html(http://www.phys.unsw.edu.au/jw/z.html)

许可

本文以及所有相关的源代码和文件均已获得The Code Project Open License (CPOL)的许可。