To transmit information we need to modulate the carrier with other frequencies, and it is appropriate to consider the velocity of propagation of this modulation, and thus, more generally, the velocity of propagation of information and of energy. The simplest case to consider is that of the wave packet, treated in detail in all standard texts on waves.
If several neighboring frequencies are linearly superimposed, they form a wave packet with finite extension in space and a corresponding finite Fourier frequency spectrum. The modulation is somewhat analogous to the beats for simple harmonic motion considered earlier. The modulation travels at the velocity of this wave packet. Summary Simple harmonic motion refers to harmonic vibrations at a frequency f of a point mass about an equilibrium point.
The movement is in general maintained by an external force and is damped by frictional forces. Phasor is a representation of the vibration in the complex plane. Two or more phasors can be added algebraically in the complex plane.
Resonance occurs when the imaginary part of the impedance is zero. The resonance can be described by the Q or quality factor, which is a measure of the sharpness of the resonance. There are five different ways of expressing the Q, each with a different but complementary physical interpretation. The sine series is used for functions of odd symmetry, the cosine series for even functions.
Fourier integral is a generalizaton of Fourier series to the representation of pulses by a frequency spectrum. A Fourier transform pair links Fourier representations of a pulse in the time and frequency domain or quantities in spatial and wave number space. Traveling waves correspond to the self-sustaining propagation of a disturbance in space at constant velocity without change of shape.
For lossless or low loss media it is also the velocity of propagation of energy. Dispersion describes a situation in which the phase velocity varies with frequency; it occurs in dispersive media. Questions 1.
Draw a diagram to show how to add two phasors graphically, to determine their total amplitude and phase angle. Determine analytical expressions for the latter. Make a graph of the displacement and velocity for a forced simple harmonic oscillator as a function of frequency.
Ac- dead time; the fraction of running time required cess time is defined only for successful access for a signal to be processed before the next signal attempts. Securing and Controlling Cisco Routers. Demi , Modeling nonlinear acoustic wave fields in media with inhomogeneity in the attenuation and in the nonlinearity , IEEE International Ultrasonics Symposium , Last planned examination: Spring Decision to discontinue this course: No information inserted. This is due to different refraction posphere normally bend concave down toward of the incident light by the dioptric system in the earth due to the negative gradient in the re-. The corresponding radial deformation of coherent oscillation of bubbles in a collective mode the single bubble is Kolaini The book discusses commonly used chemical degradation methods, spectroscopic methods, studies of isolated lignins and lignin in situ, polymer properties related to thermal stability and molecular motion of lignin in the solid state, and applications of electronic structure calculations to the chemistry of lignin.
Draw the corresponding phasor diagram. Consider a triangular waveform as a function of time. Define the amplitude and period. Choose an origin and sketch the first three Fourier components. Comment on the use of sine or cosine functions. Draw two limiting cases width going to zero or infinity for the Fourier transform of a Gaussian pulse. Decide which of the following are traveling waves and calculate the appropriate phase velocity: 2 i. Make plots of VP k and VG k. Plot Equation 2. Comment on the pertinence of this case for communications.
Calculate the group velocity for the following cases where the phase velocity is known: i.
On land the five senses of living beings sight, hearing, touch, smell, and taste play complementary roles. Two of these, sight and hearing, are essential for long-range interaction, while the other three have essentially short-range functionality. But things are different under water; sight loses all meaning as a long-range capability, as does indeed its technological counterpart, radar. So, by default, sound waves carry out this longrange sensing under water. The most highly developed and intelligent forms of underwater life e.
On the technology front, ultrasound also really started with the development of underwater transducers during World War I. Water is a natural medium for the effective transmission of acoustic waves over large distances; and it is indeed, for the case of transmission in opaque media, that ultrasound comes into its own. We are more interested in ultrasound in this book as a branch of technology as opposed to its role in nature, but a broad survey of its effects in both areas will be given in this chapter. Human efforts in underwater detection were spurred in by the sinking of RMS Titanic by collision with an iceberg.
It was quickly demonstrated that the resolution for iceberg detection was improved at higher frequencies, leading to a push toward the development of ultrasonics as opposed to audible waves. This led to the pioneering work of Langevin, who is generally credited as the father of the field of ultrasonics. The immediate stimulus for his work was the submarine menace during World War I.
He also conducted large-scale experiments, up to 2 km long, in the Seine River. The condenser transducer was soon replaced by a quartz element, resulting in a spectacular improvement in performance, and detection up to a distance of 6 km was obtained. Although these developments came too late to be of much use against submarines in that war, numerous technical improvements and commercial applications followed rapidly.
But what, after all, is ultrasonics? Like the visible spectrum, the audio spectrum corresponds to the standard human receptor response function and covers frequencies from 20 Hz to 20 kHz, although, with age, the upper limit is reduced significantly. In each case the full bandwidth can be described by a complete and unique theory, that of electromagnetic waves for optics and the theory of stress waves in material media for acoustics.
Ultrasonics is defined as that band above 20 kHz. It continues up into the MHz range and finally, at around 1 GHz, goes over into what is conventionally called the hypersonic regime. The full spectrum is shown in Figure 1. Optics and acoustics have followed parallel paths of development from the beginning.
Indeed, most phenomena that are observed in optics also occur in acoustics. But acoustics has something more—the longitudinal mode in bulk media, which leads to density changes during propagation. All of the phenomena occurring in the ultrasonic range occur throughout the full acoustic spectrum, and there is no theory that works only for ultrasonics.
So the theory of propagation is the same over the whole frequency range, except in the extreme limits where funny things are bound to happen. For example, diffraction and dispersion are universal phenomena; they can occur in the audio, ultrasonic, or hypersonic frequency ranges. It is the same theory at work, and it is only their manifestation and relative importance that change.
As in the world of electromagnetic waves, it is the length scale that counts. The change in length scale also means that quite different technologies must be used to generate and detect acoustic waves in the various frequency ranges. Alternatively, why is it worth the trouble to read or write a book like this? As reflected in the structure of the book itself, there are really two answers. First, there is still a lot of fundamentally new knowledge to be learned about acoustic waves at ultrasonic frequencies.
This may involve getting a better understanding of how ultrasonic waves occur in nature, such as a better understanding of how bats navigate or dolphins communicate.
Also, as mentioned later in this chapter, there are other fundamental issues where ultrasonics gives unique information; it has become a recognized and valuable tool for better understanding the properties of solids and liquids. Superconductors and liquid helium, for example, are two systems that have unique responses to the passage of acoustic waves. In the latter case they even exhibit many special and characteristic modes of acoustic propagation of their own.