Chapter 2Mechanic and Acoustic Oscillations

When physical or other quantities vary in a specific way as a function of time,we usually say they are oscillating. A common broad definition of oscillation is asfollows.

Oscillation…An oscillation is a process with attributes that are repeated reg-ularly in time.

Oscillating processes are widespread in our world, and they are responsible forall wave propagation, such as sound, light, or radio waves. The time functions ofoscillating quantities vary extensively. For example, oscillations are produced byintermittent sources like foghorns, sirens, the saw-tooth generator of an oscilloscope,or the blinking signal of a turning light.

A prominent category of oscillations is characterized by energy that is swing-ing between two complementary storages, namely, kinetic versus potential energyor electric versus magnetic energy. In many cases, these oscillating systems areapproximately linear and constant in time, which categorizes them as so-called lin-ear, time-invariant (LTI) systems.

The mathematical treatment of LTI systems is straightforward. A specific featureof these systems is that the superposition principle applies. Excitation of an LTI sys-tem by several individual excitation functions leads the system to respond accordingto the linear combination of the individual response to each excitation function. Thesuperposition principle reads as follows in mathematical terms,

y(t) =∑

k

bk yk(t) = T[∑

k

bk xk(t)

], assuming yk(t) = T [xk(t)] , (2.1)

where T stands for a specified transformation by the LTI system concerned. xk(t)and yk(t) represent the kth excitation and response of the system, respectively. bk isconstant.

The general exponential function with the complex frequency, s = α + jω,© Springer-Verlag GmbH Germany, part of Springer Nature 2021N. Xiang and J. Blauert, Acoustics for Engineers,https://doi.org/10.1007/978-3-662-63342-7_2

15

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16 2 Mechanic and Acoustic Oscillations

A e s t = A eα t+jω t = A eα t (cosω t + j sinω t) , (2.2)

is an eigen-function of LTI systems. This property means that excitation by a sinu-soidal function results in a response that is a sinusoidal function of the same fre-quency, although generally with a different phase and amplitude. This unique featureof LTI systems is one of the reasons why sinusoidal functions play a prominent rolein the analysis of LTI systems and linear oscillators.

Operations with LTI systems are often performed in what is called the frequencydomain. To move from the time domain to the frequency domain, the time functionof the excitation is decomposed by Fourier transforms into sinusoidal components.Each component is then sent through the system, and the time function of the totalresponse determined by summing up all the individual sinusoidal responses andperforming the inverse Fourier transforms.

The current book does not deal with Fourier transform in great detail, but thefact that all sounds are decomposable into sinusoidal components—and completely(re)composable from these—is a good argument for employing sinusoidal excitationin LTI systems.

2.1 Basic Elements of Linear, Oscillating, MechanicSystems

Three elements are required to form a simple mechanic oscillator, namely, a mass,a spring and a fluidic damper(dashpot)—see Fig. 2.1.

Fig. 2.1 Basic elements of linear time-invariantmechanic oscillation systems. (a)Mass. (b) Spring.(c) Fluidic damper (dashpot)

For the introduction of these elements, we make three idealizing assumptions,namely,

(a) All relationships between the mechanic quantities displacement, ξ, velo-city, v, force, F , and acceleration, a, are linear.

(b) The characteristic features of the elements are constant.

(c) This chapter considers one-dimensional motion only.

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2.1 Basic Elements of Linear, Oscillating, Mechanic Systems 17

• Mass

An alternating force may be applied to a solid body with the massm, as shownin Fig. 2.1a, so that Newton’s1 law holds as follows,

F(t) = m a(t) = mdv

dt= m

d2ξ

dt2. (2.3)

For sinusoidal quantities this law reads in complex notation as follows,

F = m a = jωm v = −ω2m ξ . (2.4)

Later in this chapter, we show that the mass stores kinetic energy. It is a one-port element in terms of network-theory because there is only one in/outputport for transmitting power. The physical unit of the mass is [kg] or [N s2/m].The mechanic impedance of a mass is imaginary and expressed as

Z mech = jωm . (2.5)

• Spring

According toHooke, the following applies to linear springs2 with a complianceof n—as seen in Fig. 2.1b,

F(t) = 1

nξ�(t) = 1

n

∫v�(t) dt = 1

n

∫ [∫a�(t) dt

]dt . (2.6)

The physical unit of the compliance is [m/N]. For sinusoidal quantities incomplex notation this is equivalent to

F = 1

nξ

�= 1

jω nv � = −1

ω2na� . (2.7)

The spring stores potential energy. It is a two-port element because it has bothan input and an output port. Themechanic impedance of the spring is imaginaryand equals

Z mech = 1

jω n(2.8)

• Damper (Dashpot)

A dashpot is a damping element based on fluid friction due to a viscousmedium—see Fig. 2.1c. At a dashpot with damping (mechanic resistance), r ,the following holds,

1 Newton’s law is valid in so-called inertial spatial coordinate systems. If no force is applied toa mass in these systems, the mass moves with constant velocity along a linear trajectory. As theorigin of the coordinate system “ground” is usually defined to be amass taken as infinite. Gravitationforces are not considered here.2 In acoustics, the compliance, n, is often preferred to its reciprocal, the stiffness, k = 1/n, as thisleads to formula notations that engineers are more accustomed to—refer to Chap.3.

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18 2 Mechanic and Acoustic Oscillations

F(t) = r v�(t) = rdξ�

dt= r

∫a�(t)dt . (2.9)

In complex notation for sinusoidal quantities this is

F = r v � = jω r ξ�

= r

jωa� . (2.10)

The mechanic impedance of the damper is real and expressed as

Z mech = r . (2.11)

The physical unit of the damper is [N s/m]. The dashpot does not store energy.It consumes energy through dissipation, which is a process of convertingmechanic energy into thermodynamic energy, in other words, heat. The dash-pot is a two-port element.

2.2 Parallel Mechanic Oscillators

This section now considers an arrangement where a mass, a spring, and a dashpotare connected in parallel by idealized, that is, rigid and massless rods—see Fig. 2.2.

Fig. 2.2 Mechanic parallel oscillator, exited by an alternating force. The second port is groundedhere for simplicity

The arrangement may be excited by an alternating force, F(t), that is composedof three elements,

F(t) = Fm(t) + Fr(t) + Fn(t) (2.12)

In this way, the following differential equation holds,

F(t) = md2ξ

dt2+ r

dξ

dt+ 1

nξ or F(t) = m

dv

dt+ r v(t) + 1

n

∫v(t) dt . (2.13)

As only one variable, ξ or v, is sufficient to describe the state of the system, itrepresents what is commonly called a simple oscillator.

For simplicity of the example, both the spring and the dashpot are connectedto ground. In this way, the quantities ξ2 and v2 are set to zero at the output ports.Consequently, we omit the subscript � in the following.

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2.3 Free Oscillations of Parallel Mechanic Oscillators 19

2.3 Free Oscillations of Parallel Mechanic Oscillators

This section deals with the particular case that the oscillator is in a position awayfrom its resting position, and the introduced force is set to zero, that is, F(t) = 0 fort > 0. The differential equation (2.13) then converts into a homogenous differentialequation as follows,

md2ξ

dt2+ r

dξ

dt+ 1

nξ = 0 . (2.14)

The solution of this equation is called free oscillation or eigen-oscillation of thesystem. A trial using ξ = e st results in the characteristic equation3

m s2 + r s + 1

n= 0 , (2.15)

where s denotes the complex frequency. The general solution of this quadratic equa-tion reads as

s 1, 2 = − r

2m±

√r2

4m2− 1

m nor s 1, 2 = −δ ±

√δ2 − ω2

0 , (2.16)

where δ = r/2m is the damping coefficient and ω0 = 1/√m n the characteristic

angular frequency. This general form renders the three different types of solutions,namely,

Case (a) with δ = ω0 ... critical damping, only one root, which is real.Case (b) with δ > ω0 ... strong damping, both roots real, s is negative.Case (c) with δ < ω0 ...weak damping, both roots are complex.

The differential equation for a simple oscillator is of second order, making itnecessary to have two initial conditions to derive specific solutions. Three formsof general solutions exist, which are listed below. It remains to adjust them to theparticular initial conditions to finally arrive at particular solutions.

•Case (a) (δ = ω0)

ξ(t) = ( ξ1+ ξ

2) e−δt . (2.17)

This case is at the brink of both periodic and aperiodic decay. Depending onthe initial conditions, it may or may not render a single swing-over. It is calledthe aperiodic limiting case.

•Case (b) (δ > ω0)

ξ(t) = ξ1e−(δ−

√δ2−ω2

0 )t + ξ2e−(δ+

√δ2−ω2

0 )t . (2.18)

3 As noted in the introduction to this chapter, the general exponential function is an eigen-functionof linear differential equations. This means that it stays to be an exponential function when differ-entiated or integrated.

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20 2 Mechanic and Acoustic Oscillations

This solution, called the creeping case, describes an aperiodic decay.

•Case (c) (δ < ω0)

ξ(t) = ξ1e−δte−jωt + ξ

2e−δte+jωt , with ω =

√ω20 − δ2 . (2.19)

This solution, called the oscillating case, describes a periodic, decaying oscilla-tion. It represents indeed an oscillation as illustrated by looking at the particularspecial case of ξ

1= ξ

2= ξ

1, 2/2.

Substituting it into (2.19) yields

ξ(t) = ξ1, 2

e−δt cos(ωt) (2.20)

Fig. 2.3 Decays of a simple oscillator for different damping settings (schematic). (a) Aperiodiclimiting case. (b) Aperiodic case. (c) Oscillating case

Figure2.3 illustrates the three cases. The fastest-possible decay below a threshold—which, by the way, is the objective when tuning the suspension of road vehicles—isachieved with a slightly subcritical damping of δ ≈ 0.6ω0.

In addition to δ = r/2m, the following two quantities are often used in acousticsto characterize the amount of damping in an oscillating system,

Q…The sharpness-of-resonance factor, also known as quality factor.

Td …The decay time—similar to the reverberation time in Sect. 12.5.

The sharpness-of-resonance factor, Q, is defined as

Q = ω0

2 δ= ω0 m

r, (2.21)

It is a measure of the width of the peak of the resonance curve—see Sect. 2.4.A more illustrative interpretation is possible in the time domain when one consid-

ers that after Q oscillations a mildly damped oscillation has decreased to 4% of itsstarting value—which is at the brink of what is visually detectable on an oscilloscopescreen.

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2.3 Free Oscillations of Parallel Mechanic Oscillators 21

Table 2.1 Typical Q values for various oscillators

Type of element Q factor

Electric oscillator of traditional construction(coil, capacitor, resistor)

Q ≈ 102–103

Electromagnetic cavity oscillator Q ≈ 103–106

Mechanic oscillator, steel in vacuum Q ≈ 5 · 103Quartz oscillator in vacuum Q ≈ 5 · 105

The decay time, Td, measures how long it takes for an oscillation to decrease by60dB after the excitation has been stopped. At this level, the velocity and displace-ment have decayed to one-thousandth and the power to one-millionth of its originalvalue. Td and δ are related by Td ≈ 6.9/δ—see Sect. 12.5.

Table2.1 lists characteristic Q values for different kinds of technologically rel-evant oscillators. For comparison, in the aperiodic limiting case, Q has a valueof 0.5.

2.4 Forced Oscillation of Parallel Mechanic Oscillators

For free oscillations, the exciting force is zero. Yet, in this section we now dealwith the case where the oscillator is driven by an ongoing sinusoidal force, F(t) =F cos(ωt + φ), with an angular frequency of f = ω/2π, such that the oscillationof the system takes on a stationary state.4 This mode of operation is termed force-driven or forced oscillation. Themathematical description leads to an inhomogeneousdifferential equation as follows,

F cos(ωt + φ) = md2ξ

dt2+ r

dξ

dt+ 1

nξ . (2.22)

For sinusoidal excitations, this equation reads as follows in complex form,

F = −ω2m ξ + jω r ξ + 1

nξ . (2.23)

Substitution of ξ by v yields

F = jωm v + r v + 1

jω nv . (2.24)

This equation directly admits the inclusion of the mechanic impedance, Z mech,as well as it’s reciprocal, the mechanic admittance (also termed mobility), Y mech =1/Z mech, so that

4 Slowly varying frequencies are also in use, assuming that a stationary state has (approximately)been reached at each instant of observation.

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22 2 Mechanic and Acoustic Oscillations

Fig. 2.4 Trajectories of mechanic impedance/admittance in the complex Z and Y planes as func-tions of angular frequency. (a) Mechanic impedance. (b) Mechanic admittance (mobility)

Z mech = F

v= jωm + r + 1

jω nand (2.25)

Y mech = v

F= 1

jωm + r + 1jω n

. (2.26)

Figure2.4 illustrates the trajectories of these two quantities in the complex planeas a function of frequency. The two quantities become real at the characteristicfrequency, ω0. At this frequency, the phase changes signs (jumps) from positive tonegative values or vice versa.

Figure2.5 schematically illustrates functions of ξ(ω) and v(ω) in the case of slowvariations of the frequency of excitation. For simple oscillators these curves havea single peak. In this example, for a case of subcritical damping with Q ≈ 2, theexciting force is kept constant over frequency.

Fig. 2.5 Mechanic responses of a simple resonator as a function of the angular frequency forconstant-amplitude forced excitation. (a) Velocity. (b) Displacement

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2.4 Forced Oscillation of Parallel Mechanic Oscillators 23

For the velocity curve, the two points at -3 dB correspond to the frequency ω−45

and ω45. The difference of these frequencies,

ω� = ω45 − ω−45 (2.27)

is termed (angular frequency) bandwidth.

The quality factor, Q, is determined by this bandwidth as

Q = ω0

ω�

. (2.28)

For Q ≤ 1/√2, a displacement resonance does not occur. The course of calcula-

tions to arrive at these functions is as follows,

F

v= F

jω ξ= jωm + 1

jω n+ r, (2.29)

ξ

F= 1

−ω2m + 1n + jω r

, (2.30)

| ξ || F | = 1√

( 1n − ω2m)2 + (ωr)2, (2.31)

| v || F | = 1√

(ωm − 1ωn )

2 + r2. (2.32)

Note that the phase of the velocity, v, is decreasing and passes zero atω0, while thephase of the displacement, ξ, is also decreasing but goes through−π/2 at this point—see Fig. 2.6. Furthermore, the position of the peak for the |v(ω)| curve is precisely atthe characteristic frequency, while the peak of the |ξ(ω)| curve lies slightly lower—the higher the damping, the lower the frequency at this peak!Hence, this peak is calledthe resonance. Consequently, it should be distinguished appropriately between theterms resonance frequency and characteristic frequency.

Figure2.6 shows the resonance curves for the velocity in a slightly different way toillustrate the role of the Q-factor concerning the form of these curves. The resonancepeak becomes higher and more narrow with increasing Q. This is the reason that Qis termed sharpness-of-resonance factor, besides quality factor.

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24 2 Mechanic and Acoustic Oscillations

Fig. 2.6 Double-logarithmic plot of resonance curves of the velocity, illustrating the influence ofthe sharpness-of-resonance factor, Q

2.5 Energies and Dissipation Losses

To derive the energies and losses in the elements from which the oscillator is built,(2.13) is first multiplied with v(t) to arrive at what is called instantaneous power,namely

P(t) = F(t) v(t) = mdv

dtv(t) + r v2(t) + 1

nv(t)

∫ dξ︷ ︸︸ ︷v(t) dt . (2.33)

Integration over time then leads to a term with the dimension energy (work) asfollows,

W 0, t1 =∫ t1

0F(t)

dξ︷︸︸︷v dt = m

∫ t1

0vdv

dtdt + r

∫ t1

0v2dt + 1

n

∫ t1

0ξ v dt . (2.34)

For the case that the motion of the oscillator starts from its resting position, that is,for ξ t=0 = 0, this expression mutates to

∫ ξ1

0F(t) dξ = 1

2m v2

1 + r∫ t1

0v2dt + 1

2

1

nξ21 . (2.35)

The left term denotes the energy that is fed into the system. The terms on the rightside of the equality sign stand, from left to right, for the kinetic energy of the mass,the frictional losses (dissipation) in the dashpot, and the potential energy in thespring.

Our discussion starts with the case of no losses, that is, when r ≡ 0. In this case,the total energy in the system does not change. It simply swings between the mass

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2.5 Energies and Dissipation Losses 25

and the spring. We express these relationships as follows,

W = 1

2m v2(t) + 1

2nξ2(t) . (2.36)

At the instant ξ = 0, all energy is kinetic, and when we have v = 0, all energy ispotential. In mathematical terms, this is

W (ξ = 0) = 1

2m v2 = W (v = 0) = 1

2nξ 2 . (2.37)

When losses are present due to friction, that is, when r �= 0, the stationary state ismaintained by a driving force. Recall that this section discusses force-driven oscilla-tion. To keep the oscillation amplitude constant, the system requires supplementarypower. This power results from the middle term of (2.35) and amounts to

Wr = r∫ t1

0v2(t)dt , and, thus Pr = r

d

dt

∫ t1

0v2(t)dt . (2.38)

Averaging over a full period, T , with the arbitrary phase, φ, we find

P = r1

T

∫ T

0v2 cos2(ωt + φ)dt

= 1

2 Tr v2 T = 1

2r v2 = 1

2F v = Frms v rms . (2.39)

At the dashpot, v and F are in phase, which means that the supplied power ispurely resistive (active) power. This holds for the complete system when driven atits characteristic frequency. Off this frequency, additional (reactive) power is neededto keep the system stationarily oscillating.

2.6 Basic Elements of Linear, Oscillating, Acoustic Systems

In addition to themechanic elements, there is a further class of elements for oscillatorsthat are traditionally called acoustic elements. Note that the terms mechanic andacoustic are historic in this case. Since sound is mechanic, the oscillators built fromboth classes of elements are, to be sure, mechanic and acoustic at the same time.

The acoustic elements are formed by small cavities filled with fluid, that is, gas orliquid. To deal with these cavities as concentrated elements, their linear dimensionsmust be small compared to the wavelengths under consideration. To define the acous-tic elements, this section uses the sound pressure, p, the sound-pressure difference,p� = p1 − p2, and the volume velocity,

q = dV

dt= A

dξ

dt= A v(t) . (2.40)

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26 2 Mechanic and Acoustic Oscillations

Fig. 2.7 Basic elements of linear acoustic oscillators. (a) Acoustic mass. (b) Acoustic spring.(c) Acoustic damper

Figure2.7 schematically illustrates the three acoustic elements—acoustic mass,ma , acoustic spring, na , and acoustic damper, ra . Note that here the damper and themass are two-port elements while the spring has only one port.

The following equations define these elements.

•Acoustic Mass

p�(t) = madq

dtor, in complex notation, p

�= jωma q (2.41)

•Acoustic Spring

p(t) = 1

na

∫q dt or, in complex notation, p = 1

jω naq (2.42)

•Acoustic Damper5

p�(t) = ra q or, in complex notation, p�

= ra q (2.43)

2.7 The Helmholtz Resonator

TheHelmholtz resonator is the best-known example of an oscillator with an acousticelement. AHelmholtz resonator is commonly demonstrated by blowing over the openend of a bottle to produce a musical tone. This is an auditory event with a distinctpitch, which is adjustable by filling the bottle with some water.

5 For the characteristic parameters of the acoustic elements, the following relations hold: ma =�− l/A with �− being density, na = V/(η p−) = V/c2 �− with V being volume, η = cp/cv, andra = Ξ l/A with Ξ being flow resistivity—for details refer to Sect. 11.5.

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2.7 The Helmholtz Resonator 27

Fig. 2.8 Helmholtzresonator with friction

What happens when the bottle is blown on? The air in the bottleneck is a massoscillating on the air inside the bottle, and the air inside the bottle acts as a spring.6

Figure2.8 schematically illustrates the Helmholtz resonator with friction thatcauses damping. The three elements, mass, damping, and spring, are connected incascade (chain), so that the total pressure results in

p�

= p�ma

+ p�ra

+ p�na

. (2.44)

Dividing p�by the volume velocity, q , delivers the acoustic impedance, Z a, namely,

Z a = p�

q= jωma + ra + 1

jω na. (2.45)

2.8 Exercises

Differential Equations, Free and Forced Oscillation, Resonance Curves, Com-plex Power in Mechanic Systems

Problem 2.1

(a) Given a (constant) mass, m, describe/sketch the mechanic impedance ofthe mass as a function of the angular frequency, ω. Determine the phaseangle of its mechanic impedance.

(b) Given a (constant) compliance,n, describe/sketch themechanic impedanceof the spring as a function of the angular frequency,ω. Determine the phaseangle of its mechanic impedance.

Problem 2.2 Given a mechanic parallel oscillator with mass,m, compliance, n, anddamping, r .

(a) Establish a corresponding differential equation for a harmonic force exci-tation, F0 = F0 cosω t , expressed in a suitable quantity.

6 Usually the spring characteristics of air are not directly experienced because the air evacuates.Yet, in this case, the effect is similar to operating a tire pump with the opening hole pressed closed.

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28 2 Mechanic and Acoustic Oscillations

(b) Find a solution to the differential equation if there is no external excitation.

(c) Find the possible types of solutions of the differential equation for excita-tion as given in (a).

Problem 2.3 A mechanic parallel oscillator be excited by a sinusoidal force ofconstant amplitude, F0 = const , which is independent of frequency.

(a) Calculate and plot the trajectory of the velocity, v, in the complex planeas a function of frequency, f .

(b) Find the relationship between the damping coefficient, δ = r/2m, andthe bandwidth, ω�, as defined by the half-value (−3 dB) on the resonancecurve.

(c) Quantitatively explain the following statement, which is applicable for thedetermination of the sharpness-of-resonance factor, Q, by inspection ofan oscilloscopic plot.

“The quality factor” is approximately represented by the number of oscillations of adamped oscillation that precede a 96%-reduction of the starting amplitude.

Problem 2.4 A parallel mechanic oscillator contains a mass, m = 0.63Ns2/m, aspring with a compliance of, n = 0.027 m/N, and a damper with r = 6.02Ns/m.

Determine for this parallel mechanic oscillator

– The resonance frequency, f0

– The damping coefficient, δ

– The sharpness-of-resonance factor, Q and the decay time, Td

– Will this system be oscillating or not?

Problem 2.5 Discuss why the complex power in a mechanic-oscillator system is

P = 1

2F v∗ . (2.46)

Problem 2.6 Given a lossy electric series resonator as shown in Fig. 1.3.

(a) Determine the following quantities using a graph in the complex plane:input impedance Z , input admittance Y , resonance frequency ω0, charac-teristic resistance Z 0.

(b) Derive the relation between bandwidth, ω�, and sharpness-of-resonancefactor, Q.

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2.8 Exercises 29

(c) Plot the magnitude of the input impedance, Y , as a function of the angularfrequency in double-logarithmic representation.

Problem 2.7 For a parallel mechanic oscillator, show that the phase of ξ, concerningthe excitation, becomes −π

2 at the characteristic frequency of the forced oscillation,ω0, when a sinusoidal excitation with constant amplitude is applied.

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Chapter 16Appendices

16.1 Complex Notation of Sinusoidal Signals

An arbitrary sinusoidal signal can be written as

z(t) = z cos(ωt + φ) , (16.1)

with the three free parameters amplitude, z, angular frequency, ω, and (zero)phaseangle, φ. If the frequency is known and fixed, that is, for a monofrequent signal, onlytwo free parameters are left—amplitude and phase angle.

Arithmetic operations with sinusoidal signals, such as addition, multiplication,differentiation and integration, are complicated in the representation aswritten above.There is thus a demand for more simple arithmetics to deal with these signals, par-ticularly their amplitudes and phase angles. This can be accomplished in differentways. In this book we use the common symbolic representation of sinusoidal signalsby means of so-called complex amplitudes.

To this end, the representation above is expanded by a complex imaginary part,but this operation is immediately inverted by forming the real part of the expressionagain. We thus write

z(t) = Re {z cos(ω0t + φ) + j z sin(ω0t + φ)} . (16.2)

By applying Euler’s formula, e jφ = cosφ + j sin φ, this expression can be rewrittenas

z(t) = Re {z e j(ω0t+φ)} = Re {z e jφe jω0t } = Re { z e jω0t } . (16.3)

with the term z = z e jφ being the complex amplitude as mentioned above. The orig-inal real representation of the sinusoidal signal can be retrieved from the complexamplitude by multiplication with e jω0t and subsequent forming of the real part, thatis, by applying the operator Re { . . . }.

© Springer-Verlag GmbH Germany, part of Springer Nature 2021N. Xiang and J. Blauert, Acoustics for Engineers,https://doi.org/10.1007/978-3-662-63342-7_16

399

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400 16 Appendices

Themost relevant rules for calculations with complex amplitudes are given below.For details please refer to the pertinent literature.

Addition and Substraction,

z 1 + z 2 = (z1 cosφ1 + z2 cosφ2) + j(z1 sin φ1 + z2 sin φ2)

z 1 − z 2 = (z1 cosφ1 − z2 cosφ2) + j(z1 sin φ1 − z2 sin φ2) (16.4)

Multiplication and Division,

z 1 z 2 = (z1 z2) ej (φ1+φ2), z 1/z 2 = (z1/z2) e

j (φ1−φ2) (16.5)

Integration and Differentiation,

∫z dt = 1

jωz ,

dz

dt= jω z (16.6)

The rules for integration and differentiation become clear after multiplication ofthe complex amplitude, z, with the factor e jω t , which means reinserting the timedependency.

16.2 Complex Notation of Power and Intensity

Consider a force, Fz(t), exciting a one-dimensional motion of vz(t) along a pathz, for example at the mechanic port of an electro-mechanic transducer. The inputenergy can be written as follows, whereby from now on the index z is omitted forsimplicity,

W =∫

F(t) dz =∫

F(t)dz

dtdt =

∫F(t) v(t) dt . (16.7)

The transferred instantaneous power, P(t), is than given by

P(t) = d

dt

∫F(t) v(t) dt = F(t) v(t) . (16.8)

Let both the force and the particle velocity be sinusoidal time functions, namely,

F(t) = F cos(ωt + φF) and v(t) = v cos(ωt + φv) . (16.9)

This leads to the expression

P(t) = F cos(ωt + φF) v cos(ωt + φv) . (16.10)

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16.2 Complex Notation of Power and Intensity 401

Execution of the multiplication with cosα cosβ = 12 [cos(α + β) + cos(α − β)]

renders

P(t) = F v

2[ cos(2ωt + φF + φv) ]

︸ ︷︷ ︸alternating

+ F v

2[ cos(φF − φv) ]

︸ ︷︷ ︸constant

. (16.11)

When determining the time average of the transmitted power, P , the first part, whichis alternating with double frequency, does not contribute. The average power is solelygiven by the second part, namely,

P = F v

2[ cos(φF − φv) ] . (16.12)

This average power is also called active power or resistive power.For a complex notation with the complex amplitudes of the force and the particle

velocity, that isF = F e j(ω t+φF) and v = v e j(ω t+φv) , (16.13)

these notations lead to a complex power, P , as follows,

P = 1

2[ F v∗] with v∗ = v e− j(ω t+φv) , (16.14)

where the term v* is called the complex conjugate of v. Some elaboration finallyresults in

P = P + j Q = F v

2cos(φF − φv)︸ ︷︷ ︸active power

+ jF v

2sin (φF − φv)︸ ︷︷ ︸

reactive power

. (16.15)

The real part of P is the active power, P , as noted above. The imaginary part, Q,is called reactive power and has no direct physical relevance. Please note that bytaking the complex conjugate of the particle velocity, v∗, in (16.14) and not that ofthe force, we have chosen that the reactive power of mass is counted positive.

What holds for the power, also holds for the intensity, which is power per area.Complex intensity thus results as

I = 1

2[ p v∗] . (16.16)

Consequently, we denote the active intensity, Re{ I } also as I in this book.

REVISED P

ROOF