Blast Vibration Analysis

Chapter I

PHYSICS OF WAVE MOTION

I-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.

I-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 exoression would be,

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 (I-1) becomes,

We have assumed no change in form, therefore (I-1) must equal (I-2). For this to be true requires that

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

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

then,

The time of wave repetition is, by definition, the period of the wave:

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,

In general, we also know that

where λ is the wavelength. k is therefore related to the wavelength by,

and is termed the wave number. We can also obtain (I-10) by considering wave repetition for a fixed time.

or

and

Note next that the argument of (I-4) is

For the kv multiplier of the time, we have from (I-8):

or,

where ω is termed the angular frequency.

We can now re-write (1-4) as,

and, substituting for k and ω,

Equations (1-17) and (1-18) 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:

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, Τ, 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:

If ω1 is nearly equal to ω2, then the term

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