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G. A. Bollinger, a former professor of geophysics at Virginia Polytechnic Institute and State University, applies the analytical tools that have been so highly refined in earthquake engineering, earthquake seismology, and seismic exploration for petroleum — i.e., digital and spectral analyses — to the blast vibration problem. The text starts at an elementary level, carries the exposition to an intermediate level, and indicates the direction of more advanced consideration. Many informative tables, figures, graphs, charts, mathematical examples, and photographs of instruments appear throughout. Advanced undergraduate students, graduate students, and professionals in engineering and physics will find this treatment stimulating and suggestive of further areas of study and practice.
Product Details
| ISBN-13: | 9780486832579 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 07/18/2018 |
| Series: | Dover Books on Engineering |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 160 |
| File size: | 7 MB |
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Read an Excerpt
CHAPTER 1
PHYSICS OF WAVE MOTION
1-1. Introduction - Purpose of Chapter
Wave phenomena are a familiar part of everyone's experience. Water waves are perhaps the most graphic, but the existence of sound waves, light waves, and electromagnetic radio and television waves is also well known. The destructive effect of earthquake waves receives widespread publicity. It is thus obvious that gases, liquids, and solids will support wave motion. For the understanding and analysis of blast vibrations we require a more sophisticated understanding of wave phenomena. We are specifically concerned with the origin, transmission, and types of elastic waves in solid media. We need also to consider the physical laws involved in the understanding and analysis of wave motion. Finally, we need mathematical tools to specify the variety of wave phenomena encountered in nature and also to do the analyses required.
An exhaustive treatment of this subject is outside the scope of this treatise. We will concern ourselves only with fundamentals and only to an introductory depth. References are given to aid more extensive study.
1-2. General Properties of Waves
Physically, elastic waves are a traveling disturbance and represent the transfer of energy from one point in a medium to some other point. Thus, there must be an initial disturbance of the medium, some forces must act to disturb the medium from its equilibrium position and thereby introduce new energy into the medium. If the medium is not elastic in its response to the energy introduction, it absorbs energy and only damped waves emanate from the distrubance area. If it is elastic, then the action of the forces causes the nearby portions of the medium to oscillate about their rest positions much as a spring-mass system. Because of the medium elasticity, the oscillatory disturbance is transmitted from one "element" to the next and to the next and so on causing a wave motion to progress through the medium. Unless stated to the contrary, we will hereafter concern ourselves only with perfectly elastic (complete recovery of size and shape when deforming forces removed), homogeneous (elastic moduli independent of position), isotropic (elastic properties identical in all directions) media. Relative displacements of the constituent particles of the body are taken as small enough that their squares can be neglected.
There are several important aspects of the wave process. First of all, there is no bulk movement or transport of matter during wave motion. The constituent particles of the medium oscillate and/or rotate only about very space-limited paths and do not go traveling off through the medium. This is adequately demonstrated by the cork bobber on a fishing line. This fact does, however, introduce the necessity for consideration of two velocities: a "wave" or "phase" velocity to describe rate with which the disturbance propagates through the medium, and a "particle" velocity to describe the small oscillations that the particle executes about its equilibrium position as the wave energy excites it. The wave velocity is commonly orders of magnitude larger than the particle velocity. In the analysis of blast vibrations, we are usually concerned with the particle velocity and not the wave velocity.
Note that time dependent stresses constitute the force system acting to cause the wave motion. These stresses are the medium's response to the introduced disturbance. Their temporal and spatial behavior is specified by the elastic properties of the medium. Note also that the energy introduced by the disturbance travels as kinetic energy of particle motion and potential energy of particle displacement in the wave motion. This energy is proportional to the square of the amplitude of the wave motion. As the wave motion propagates, it tends to spread out and this introduces a geometrical effect on the energy content per unit area of the wave front. To illustrate this effect, we consider a perfectly elastic medium of infinite extent. A point source in such a medium would induce spherical waves. The area of these wavefronts increases as r2, where r is the distance from the source, and thus the energy flow per unit area would decrease as r-2. A line source of energy would produce cylindrical waves whose area increases as r. The energy flow per unit area for this case decreases as r-1. If the energy source is very far away we can approximate the waves as planar, then there is no geometrical spreading effect as in the spherical and cylindrical cases.
In nature we do not have a perfect medium and thus there are additional energy losses as the wave propagates. There are absorptive losses as previously mentioned, which attenuate wave amplitude with distance and/or time. This latter type of loss is often exponential. In discussing the vibration of a portion of the elastic medium during wave motion, we are talking about its behavior under the influence of the previously mentioned forces that are variable in magnitude and/or direction. It is instructive at this point to contrast this motion with that of mechanical oscillators (e.g., a spring-mass system) or electrical oscillators (e.g., a series LRC circuit). Wave motion is proper to all three of these types of systems, but for the mechanical and electrical oscillators there are only time-oscillations, while for the elastic medium under discussion, we can have time-space oscillations. Note, however, that at a fixed point in space there is no difference for the case of a one degree-of-freedom system.
Wave motion may be transient, periodic, or random. Transient motion is characteristic of the medium's response to a sudden, pulse-like excitation and dies out rapidly with increasing time. Periodic motion is repetitive in nature, reoccurring in exactly the same form at fixed time increments. Harmonic motion is the simplest form of periodic motion and is specified by the sinusoidal (Sine and Cosine) functions. Noise commonly displays the essential characteristic of randomness, i.e., the instantaneous amplitude can be predicted only on a probabilistic basis.
To describe analytically the preceding concepts and also to introduce some necessary definitions, assume that a disturbance D is propagating, with no change in form, in the "x" direction with a constant velocity "v". Thus, D is a function of both distance (x) and time (t) and a general expression would be,
D(x,t) = D(x-vt). (I-1)
To see that this represents a traveling disturbance, let x increase by Δx and t increase by Δt. Then, the right-hand side of equation (1-1) becomes,
D(x + Δx-v(t + Δt)). (I-2)
We have assumed no change in form, therefore (I-1) must equal (I-2). For this to be true requires that
Δx = vΔt, (I-3)
which is the conventional definition of velocity.
A simple and very useful example of this type of function is the propagating harmonic wave given by
D(x-vt) = A Sin k(x-vt), (I-4)
A = where maximum amplitude of the wave
K = a parameter with dimensions of 1/length; necessary to make the argument dimensionless.
The parameter k has a physical interpretation. To see this, we note that for a given x, the wave repeats itself at 2π increments; that is, if
k(x-vt1) = k(x-vt2) + 2π (I-5)
then,
kv(t2 -t1) = 2π. (I-6)
The time of wave repetition is, by definition, the period of the wave:
(t2 - t1) = T. (I-7)
The period has units of seconds per cycle and its reciprocal is frequency (f) with units of Hertz or cycles per second.
Equation (I-6) and (I-7) give,
T = 2π/kv. (I-8)
In general, we also know that
T = λ/v, (I-9)
where λ is the wavelength, k is therefore related to the wavelength by,
λ = 2π/k, (I-10)
and is termed the wave number. We can also obtain (1-10) by considering wave repetition for a fixed time.
k(x1 - vt) = k(x2 - vt) + 2π (I-11)
or
k(x1 - x2) = 2π (I-12)
and λ = (x1 - x2) = 2π/k (I-13)
Note next that the argument of (I-4) is
(kx - kvt).
For the kv multiplier of the time, we have from (I-8):
T = 1/f = 2π/kv (I-14)
kv = 2πf = ω (I-15)
where ω is termed the angular frequency.
We can now re-write (1-4) as,
D(x-vt) = A Sin (kx-cot)
and, substituting for k and ω,
[MATHEMATICAL EXPRESSION OMITTED] (I-17)
= A Sin 2π (x/λ - t/T) (I-18)
Equations (I-17) and (I-18) present in an especially clear fashion the wave parameters we are most interested in, namely, the wavelength and the frequency or period of vibration.
A phase angle "φ" can be added to the argument of the sine function:
D1 = ASink(x-vt) (I-19)
D2 = A Sin [k(x-vt) + φ], (I-20)
indicating that D2 is displaced by φ radians from D1. If φ = 2π, 4π, etc., then the displacement is exactly an integral number of wave lengths and the waves are said to be "in phase". If φ = π, 3π, etc., they are termed "180° out of phase". The preceeding definitions of λ, T, f, and k are valid for all periodic waves and not just the harmonic type.
Special types of interference wave motion can result from the superposition of two wave trains. Of special interest are "beats" and "standing waves." Beat phenomena can be set up by the superposition of two harmonic wave trains propagating in the same direction and with the same amplitudes but with different frequencies and velocities:
[MATHEMATICAL EXPRESSION OMITTED] (I-21)
If ω1 is nearly equal to ω2, then the term
[MATHEMATICAL EXPRESSION OMITTED]
represents the "carrier" wave with frequency very nearly equal to one of the original waves. The velocity of propagation of this wave form is termed the phase velocity (vp) and is given by:
[MATHEMATICAL EXPRESSION OMITTED]
This is the same type of velocity that in equation (1-15) is given by v = ω/ko However, a low frequency amplitude and phase modulation, the "beat", is imparted by the term
[MATHEMATICAL EXPRESSION OMITTED]
The frequency of modulation is called the beat frequency (fb) and is given by,
[MATHEMATICAL EXPRESSION OMITTED]
The velocity of propagation of the "beat" or group of waves is termed the group velocity (vG) and is given by:
[MATHEMATICAL EXPRESSION OMITTED]
See Figure 1-1.
Standing waves are set up by the superposition of two wave trains with the same amplitude, frequency, and propagation velocity but traveling in opposite directions:
[MATHEMATICAL EXPRESSION OMITTED] (I-23)
1-3. Elasticity - Stress and Strain
The most general case of displacement of a point within an elastic body can be shown (for very small strains) to consist of a translation plus a rigid body rotation plus a deformation. This deformation is specified by compressional and shear strain. In compressional strain the extension or contraction per unit length is considered, while the shear strain is given by the angle of rotation. Consider two points, P and Q, that before deformation are separated by a distance ΔX. After medium displacement these points are at P' and Q' within the body. In addition to the translation and/or rigid body type of rotation that may have occurred as P and Q go to the P' and Q' positions, the possible alteration of the separating distance ΔX and of the orientation of the line connecting P and Q must also be taken into account. This situation, assuming that rigid body type of rotation is absent, is shown in Figure 1-2. For both compressional and shear types of strain we see that the quantity we are concerned with is the space rate of change of displacement.
Thus, strain is a geometric concept that specifies the deformation of a body in the vicinity of some given point. Usually given as a ratio, it is physically dimensionless. Within the proportional limit, i.e., the range wherein the body size and shape recovery is complete upon removal of the deforming forces and the stress is linearly proportional to the strain, we say that the material is linearly elastic and obeys Hooke's Law (I-24). The proportionality factors in Hooke's Law are constants and are termed elastic moduli. At greater stresses, we first observe a zone of nonlinear elasticity and then, for a ductile material, a range of plasticity with permanent deformation. For a brittle material, failure occurs just past the elastic limit (see Figure 1-3).
(Stress) α (Strain) (Stress) = (Elastic Moduli) x (Strain) (1-24)
The principle elastic moduli or elastic constants are:
Young's Modulus (E) — Stress-strain ration in simple tension or compression
1-4. Equation of Motion and Wave Equation
It is Hooke's Law that is employed to reduce the equation of motion for an elastic body to observable variables. When Newton's Second Law of Motion (F= ma) is applied to an elastic body, the result is:
(Space rate of change of Stress) a (acceleration)
Because we cannot measure the stress, let alone its space gradient, the stress in the equation of motion is replaced by the Hooke's Law equivalent, i.e., the strain multiplied by the elastic moduli:
(Space rate change of [(Elastic Moduli) • (Strain),]) α (acceleration).
Now, the strain is itself the rate of change of the displacement of any specified portion of the body, e.g., change in length per unit length. Displacements can, of course, be measured, and the resulting equation of motion in terms of displacement is given by:
(Space rate of change of the rate of change of displacement) a (acceleration).
The proportionality factor, mass, between force and acceleration in Newton's Second Law of Motion results in the appearance of a density term, and the Hooke's Law substitution results in the appearance of elastic moduli, in an explicit expression for the equation of motion. It can be shown that the equation of motion under consideration here reduces to a set of two wave equations. One of these wave equations describes the transmission of a compressional type disturbance and the other the transmission of a rotational type disturbance.
The preceding discussion is to convey the fact that the equation of motion for an elastic body can reduce to equations that indicate a wave type of motion. Therefore, wave motion is an expected result when an elastic medium is disturbed. A purely physical approach has been taken to indicate these results and a mathematical treatment can be found in the references given at the end of this chapter.
1-5. Types of Elastic Waves – Body and Surface Waves
Fundamental types of deformation that can be imposed on an elastic body are compression (or rarefaction) and shear (or distortion). Pure compression alters the volume of the body without alteration of its shape (angles). Conversely, pure shear results in a change of shape only, with no change in volume. These types of deformation can be transmitted by traveling waves within the interior of a body and thus are termed "bodily" or "body" waves. Sound waves are an example of the former type and are characterized by compressing and expanding the medium by particle vibration in the direction of propagation of the waves. This wave type is termed compressional, dilatational, longitudinal, irrotational, or P (for Primary; the fastest, thus first arriving wave). Medium shearing-type of wave motion is caused when the medium particles oscillate perpendicular to the propagation direction. Such waves are called S (Secondary), transverse, shear, equivoluminal, or rotational waves. An example of this type of wave motion that is often given is transverse waves on a string, and Figure 1-4 shows two examples.
Introduction of one or more boundaries into the medium across which there are differences in elastic properties can cause the introduction of other wave types. The most severe such discontinuity imaginable would be a solid-vacuum contact, termed a "free surface" since the solid is not constrained by the adjacent material. The most significant boundary of this type encountered in nature is the earth's surface, where we normally make all our observations of waves.
Both theory and observation show the existence of two basic types of surface waves in the presence of a boundary. These waves are "guided" by the surface (hence, surface waves) and are characterized by an exponential decrease in particle oscillation amplitude with increasing distance from the boundary and by propagation of the wave form "along" the boundary. No such depth-amplitude behavior is present in the case of P and S waves.
The two fundamental types of surface waves are Love waves and Rayleigh waves. Love waves are characterized by particle vibration of the shear type and only in the horizontal transverse direction. They have no vertical component of motion and require at least one layer over a half-space for their existence. Rayleigh waves, on the other hand, exist in the vertical-radial plane, i.e., they have no transverse component in a perfect medium and do not require a layer for their existence. The particle motion in Rayleigh waves is elliptical and retrograde.
(Continues…)
Excerpted from "Blast Vibration Analysis"
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Copyright © 1971 Southern Illinois University Press.
Excerpted by permission of Dover Publications, Inc..
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