PLANE ACOUSTIC W AVES
NOMENCLATURE a - speed of wave propagation, m/sec; acceleration, m/sec2
A,B - constants c - - damping coefficient, nt-sec/m
C,D - constants d - diameter, m f — frequency, cyc/'sec h - beat frequency, cyc/sec h = length, m h - Bessel hyperbolic function of the first kind of order zer
Jo Bessel function of the first kind of order zero k = spring constant, nt/m
K0 - Bessel hyperbolic function of the second kind of order 1 m = mass, kg
Pn - natural frequencies, cyc/sec p = period, sec p b = beat period, sec r : frequency ratio; radial distance, m
V — thickness, m w - - work done, joules/cyc
Y = Young’s modulus of elasticity, nt/m2
“d = damped circular frequency, rad/sec
= natural circular frequency, rad/sec
A — wavelength, m £ — damping factor e, - angles, rad p = density, kg/m3
P- Poisson's ratio a — stress, nt/m2 £ = strain
Acoustics is the physics of sound Although the fundamental theory of acoustics treats o f vibrations and wave propagation, we can consider the subject as a multidisciplinary science.
Physicists explore the properties of matter through wave propagation concepts, while acoustical engineers focus on sound reproduction fidelity, energy conversion to acoustical forms, and transducer design Architects prioritize sound absorption, isolation in structures, and managing reverberation and echoes in performance spaces Musicians seek to create rhythmic tone combinations by manipulating vibrations in strings, air columns, and membranes.
Physiologists and psychologists are conducting in-depth research on the human hearing mechanism, vocal cord functions, and how individuals respond to sounds and music They are also exploring psychoacoustic criteria that define comfortable noise levels and optimal listening environments Meanwhile, linguists focus on understanding the subjective perception of complex noises and the development of synthetic speech.
Ultrasonics, a topic in acoustics dealing with sound waves of frequencies above 15,000 cycles per second, has found increasing application in oceanography, medicine and industry.
Increased awareness of the detrimental effects of noise pollution from airplanes, automobiles, heavy industry, and household appliances has led to growing public concern This noise can cause ear damage and both physical and psychological irritation, prompting a demand for a deeper understanding of sound, its sources, effects, and methods of control.
Waves are generated by disturbances that originate at a specific location and are transmitted predictably through an elastic medium, influenced by its physical properties.
When a vibrating body moves from its static position, it compresses the air in front and creates a rarefaction behind it, causing air to rush in and fill the void This compression of air travels to distant areas, generating sound waves Sound, as perceived by the human ear, is the auditory sensation resulting from these air disturbances Both fluids and solids can transmit sound waves due to their inertia and elasticity.
Sound waves are longitudinal waves where particles move in the direction of the wave motion, transferring energy through space This energy consists of both kinetic energy, arising from the motion of particles, and potential energy, resulting from their elastic displacement Sound waves propagate in all directions from their source and can be reflected, refracted, scattered, diffracted, interfered with, and absorbed The propagation of sound requires a medium, and its speed is influenced by the medium's density and temperature.
For a particle in rectilinear motion, if its acceleration a is always proportional to its
Chapter 1 discusses vibrations and waves, presenting the equation d²x/dt² + ω²x = 0, which has solutions in the form of x(t) = A sin(ωt) + B cos(ωt) or x(t) = √(A² + B²) sin(ωt + φ) and x(t) = √(A² + B²) cos(ωt - φ) Here, A and B are arbitrary constants, ω represents the circular frequency in radians per second, and φ is the phase angle measured in radians.
Simple harmonic motion can be expressed as either a sine or cosine function of time and is effectively illustrated using rotating vectors A vector of constant magnitude rotates counterclockwise at a steady angular velocity, with its projections on the x and y axes representing the cosine and sine functions of time, respectively.
A harmonic wave is characterized by its sinusoidal shape, represented by a sine or cosine curve When traveling in the positive x direction at a velocity of c, the wave can be expressed mathematically as u(x,t) = Ao sin(kx - ct), where Ao denotes the amplitude and k represents the wave number.
[ Ao cos m(x — ct) whereas a harmonic wave moving in the negative x direction with velocity c is given by f A 0 sin mix + ct) uix, t) —
[ Ao cos m(x + ct) where Ao is the amplitude of the wave These are known as harmonic progressive waves
A spherical wave diverging from the origin of the coordinate with a velocity c is represented by u(r,t) = (Ao/r) f(ct - r)
A spherical harmonic progressive wave can be expressed as u(r, t) = (Ao/r)e^(i(ωt - kTi)), where i represents the imaginary unit and k is the wave number, defined as k = 1/λ This wave exhibits a repeating profile after traveling a distance of λ = 2π/k, known as the wavelength.
Systems with mass and elasticity can experience relative motion, leading to periodic motion known as vibration To analyze this vibration, the system is simplified using a spring (k) to represent elasticity and a dashpot (c) to represent friction The equation of motion describes the system's displacement as a function of time The period (P) is the time in seconds for a motion to repeat, while frequency (f) indicates the number of cycles per unit time.
FRC vibration, also known as transient vibration, refers to the periodic motion that occurs when a system is moved from its static equilibrium position This motion is influenced by several forces, including spring force, friction force, and the weight of the mass Over time, the vibration diminishes due to friction, which can be mathematically represented by the equation z c(t) = e_Cu"f (A sin d£), where the damping factor plays a crucial role in the decay of the vibration.
= natural circular frequency in rad/sec,
= natural damped circular frequency in rad/sec,
Forced vibration occurs when external forces, such as F(t) = F0 sin(ωt) or F0 cos(ωt), act on a system during its vibratory motion In this scenario, the system vibrates at both its natural frequency and the frequency of the applied force With damping present, the motion not sustained by the sinusoidal force diminishes over time, leading the system to ultimately vibrate at the frequency of the excitation force, independent of its initial conditions or natural frequency This steady state motion is known as steady state vibration or response, mathematically represented as xp(t) = A cos(ωf t - φ).
\/{k - vu>2)2 + (Cm)2 where F > = magnitude o f the excitation force, k = spring constant, m = mass of the system, c = damping coefficient,
= frequency of the excitation force in rad/sec,
In free vibration with damping, energy is absorbed by the damper and dissipated as heat, leading to a continuous loss of energy and a reduction in vibration amplitude Conversely, in free vibration without damping, the total energy remains constant, equating to the maximum kinetic or potential energy, allowing the system to sustain its vibrations indefinitely.
During forced vibration with damping, energy is being continuously supplied from external sources to maintain steady state vibration (See Problems 1.12-1.15.)
The string serves as a distinctive vibrator that exemplifies continuous media characteristics and represents the most basic form of wave transmission With its mass evenly distributed along its length, it illustrates a system capable of producing an infinite range of frequencies The fundamental differential equation governing its motion is essential for understanding its behavior.
& V - n i ^ y dt2 dx2 where y = deflection of the string, x = coordinate along the longitudinal axis of the string, a = } / S / p L = speed of wave propagation,
PL = mass per unit length of the string.
SPHERICAL ACOUSTIC WAVES
B — bulk modulus, nt/m2 c = speed of wave propagation, m/sec e = end correction factor, m
E = energy density, joules/m3 f = frequency, cyc/sec
IL = intensity level, db k = wave number
PWL = sound power level, db r = specific acoustic resistance, rayls s = condensation
SPL = sound pressure level, db
VL = velocity level, db w = speed o f medium, m/sec
Y = Young’s modulus o f elasticity, nt/m2 z = specific acoustic impedance, rayls
P = density, kg/m 3 y = ratio o f the specific heat o f air at constant pressure to that at constant volume
Sound waves are generated by disturbances in the air, traveling as progressive longitudinal sinusoidal waves in three-dimensional space When considering a scenario with no pressure variation in the y or z directions, plane acoustic waves can be described as one-dimensional free progressive waves moving along the x direction These wavefronts consist of infinite planes that are perpendicular to the x-axis and remain parallel to each other at all times.
When a small object vibrates in a medium like air, it generates sound waves that radiate outward in expanding spheres rather than flat planes In contrast, the longitudinal wave movement of an infinite air column within a smooth, rigid tube of uniform cross-section closely resembles plane acoustic wave motion.
In analyzing plane acoustic wave motion within a rigid tube, we assume zero viscosity, a homogeneous and continuous fluid medium, an adiabatic process, and an isotropic, perfectly elastic medium Any disturbance in the fluid leads to motion along the tube's longitudinal axis, resulting in small pressure and density variations around the equilibrium state These effects are captured by the one-dimensional wave equation, where the speed of wave propagation is defined as c = √(B/ρ), with B representing the bulk modulus, ρ the density, and u the instantaneous displacement.
The partial differential equation governing plane acoustic waves resembles the equations for free longitudinal vibrations in bars and free transverse vibrations in strings Consequently, the principles and deductions applicable to waves in strings and bars are also valid for plane acoustic waves.
The general solution for the one-dimensional wave equation is expressed as u(x, t) = f1(x - ct) + f2(x + ct), where f1 represents a wave traveling in the positive x direction and f2 represents a wave traveling in the negative x direction, both with velocity c In complex exponential form, this solution can be represented as u(x, t) = Aei(ωt-kx) + Bei(-i(ωt+kx)), with k being the wave number and A and B as arbitrary constants determined by initial conditions Additionally, in terms of sinusoidal sine and cosine series, the solution can be written as u(x, t) = Σ (Ai sin(kx) + Bi cos(kx))(Ci sin(ωt) + Di cos(ωt)), where Ai and Bi are constants based on boundary conditions, and Ci and Di are evaluated from initial conditions, with ω representing the natural frequencies of the system.
Particle displacements from their equilibrium positions are amplitudes o f motion of
■mall constant volume elements o f the fluid medium possessing average identical properties, and can be expressed as n(x t) — A e it'tt>~^M) •+■ or u(x, t) = A cos (mt - kx) + B cos (•* + kx)
Acoustic pressure (p) is defined as the total instantaneous pressure at a specific point, minus the static pressure, commonly known as excess pressure The effective sound pressure (Prm) at that point is calculated as the root mean square of the instantaneous sound pressure over one complete cycle This relationship can be expressed mathematically, highlighting the dependence of acoustic pressure on various parameters, including amplitude and frequency.
Density change refers to the variation between the instantaneous density and the stable equilibrium density of a medium at a specific point It is mathematically expressed by the condensation 's' at that point, represented as _s = -— - = = ikA ei("*-kI) - ikBeKtd+kMi.
When plane acoustic waves are traveling in the positive x direction, it is clear that particle displacement lags particle velocity, condensation and acoustic pressure by 90°
In the case of plane acoustic waves moving in the negative x direction, the acoustic pressure and condensation exhibit a lag of 90° behind particle displacement, whereas particle velocity advances by 90°.
The speed of sound refers to how quickly sound waves travel through a medium, with its formula in air represented as c = y/yplp m/sec Here, y denotes the specific heat ratio of air, p is the pressure in newtons/m², and p is the density in kg/m³ At room temperature and standard atmospheric pressure, the speed of sound in air is approximately 343 m/sec, increasing by about 0.6 m/sec for every degree Celsius rise in temperature Notably, the speed of sound remains unaffected by changes in barometric pressure, frequency, and wavelength, but it is directly proportional to absolute temperature.
The speed of sound in solids with large cross-sectional areas can be described by the formula c = V d i + S a - ^ ) m /M C, where Y represents the Young's modulus of elasticity in nt/m³, p denotes the density in kg/m³, and /* is Poisson’s ratio In cases where the cross-sectional dimensions are significantly smaller than the wavelength, the influence of Poisson’s ratio can be disregarded, simplifying the speed of sound to e = √(Y/ρ) m/sec.
The speed of sound in fluids is c = y/Btp m/sec where B is the bulk modulus in nt/m1 and p is the density in kg/m* (See Problems2.10-2.13.)
Sound waves are generated by disturbances in the air and propagate as progressive longitudinal sinusoidal waves in three-dimensional space When considering a scenario with no pressure variation in the y or z directions, plane acoustic waves can be defined as one-dimensional free progressive waves moving along the x direction These wavefronts consist of infinite planes that are perpendicular to the x-axis and remain parallel to each other at all times.
When a small object vibrates in a vast elastic medium like air, the resulting sound waves radiate outward in expanding spheres rather than flat planes This behavior is similar to the longitudinal wave motion observed in an infinite column of air within a smooth, rigid tube that maintains a constant cross-sectional area, which closely resembles plane acoustic wave motion.
In analyzing plane acoustic wave motion within a rigid tube, we assume zero viscosity, a homogeneous and continuous fluid medium, an adiabatic process, and an isotropic, perfectly elastic medium Any disturbance in the fluid medium leads to motion along the tube's longitudinal axis, resulting in minor fluctuations in pressure and density around the equilibrium state These effects are mathematically represented by the one-dimensional wave equation.
— - r* — dt2 “ ^ dx2 where c = \B!p is the speed of wave propagation, B the bulk modulus, p the density, and u the instantaneous displacement.
The partial differential equation governing plane acoustic waves mirrors the equations for free longitudinal vibrations in bars and free transverse vibrations in strings Consequently, the principles and deductions applicable to waves in strings and bars are also relevant to plane acoustic waves.
The one-dimensional wave equation has a general solution expressed in the form of progressive waves: u(x, t) = f₁(x - ct) + f₂(x + ct) Here, f₁(x - ct) represents a wave traveling in the positive x direction with velocity c, while f₂(x + ct) denotes a wave moving in the negative x direction at the same speed In complex exponential notation, this solution is represented as u(x, t) = A e^(i(ωt - kx)) + B e^(i(ωt + kx)), where k = ω/c is the wave number, and A and B are constants determined by initial conditions Additionally, the solution can be expressed in terms of sinusoidal functions as u(x, t) = Σ(Aᵢ sin(kx) + Bᵢ cos(kx))(Cᵢ sin(ωt) + Dᵢ cos(ωt)).