The diffraction of transient elastic waves by a spherical cavity is treated. Two cases are considered: (a) a suddenly applied normal point load, and (b) the impingement of a plane transient pulse on the cavity. The method used determines the solution only in the shadow zones; that is, those points which cannot be connected to the source of disturbance by straight-line rays. Analytical results are obtained and evaluated for the displacements at the cavity wall.

\r\n\r\nThe analysis is based on the Laplace transform (on time) and the Watson transformation. This well-known transformation makes it possible to convert an infinite series involving a discrete real wave number into one involving a generalized wave number. This leads to transient solutions the components of which have a one-to-one correspondence with the modes of the underlying frequency equation. These solutions have a form convenient for numerical analysis and for obtaining approximate solutions.

\r\n\r\nThe results given here are for the displacements evaluated at the cavity wall. It is found that the behavior of the diffracted wave fronts is similar to that associated with the simpler equations governing scalar diffraction problems (see Friedlander, \"Sound Pulses\" , Cambridge, 1958). In both problems the Rayleigh disturbance predominates for long time, being singular at its arrival time in the point load case and non-singular in the plane wave case.

\r\n\r\n", "doi": "10.7907/ps9n-9z78", "publication_date": "1967", "thesis_type": "phd", "thesis_year": "1967" }, { "id": "thesis:3895", "collection": "thesis", "collection_id": "3895", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-10042002-130206", "primary_object_url": { "basename": "Scott_r_1964.pdf", "content": "final", "filesize": 5218690, "license": "other", "mime_type": "application/pdf", "url": "/3895/1/Scott_r_1964.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Transient Wave Propagation in Elastic Plates with Cylindrical Boundaries, Studied with the Aid of Multi-integral Transforms", "author": [ { "family_name": "Scott", "given_name": "Richard Anthony", "clpid": "Scott-Richard-Anthony" } ], "thesis_advisor": [ { "family_name": "Miklowitz", "given_name": "Julius", "clpid": "Miklowitz-J" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Some mixed time dependent boundary value problems for isotropic elastic plates with circular cylindrical boundaries are studied using the linear equations of elasticity. A multi-integral transform approach is employed, necessitating the introduction of extended Hankel transforms, and formal solutions are obtained with the aid of residue theory. Some properties of the Rayleigh-Lamb frequency equation, pertinent to the inversion processes, are derived. The problem of a free infinite plate with a circular cylindrical cavity subjected to a step normal displacement is studied in detail and numerical information for the far-field, showing the effect of the cavity radius on the displacements, is obtained using stationary phase techniques.\r\n\r\nThe generation of transient elastic waves in free isotropic infinite elastic plates by time dependent body forces is also treated and the results for a radial body force, with step time-dependence, are compared with the corresponding platecavity results. Good agreement between the two is found in the far-field.\r\n\r\nSimilar problems for a free, transversely isotropic, semi-infinite plate (slab) are also studied and some numerical information for the farfield is obtained using the head of the pulse method. Stationary phase solutions for an isotropic slab subjected to a step edge displacement are obtained and compared with the corresponding plate-cavity results. It is found that at a fixed station the plate cavity solutions approach those for the slab, as the cavity radius goes to zero. A comparison between the head of the pulse and stationary pulse results for the isotropic slab is also made and some discrepancies between the two are found.", "doi": "10.7907/G7HA-F836", "publication_date": "1964", "thesis_type": "phd", "thesis_year": "1964" }, { "id": "thesis:4063", "collection": "thesis", "collection_id": "4063", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-10132005-083854", "primary_object_url": { "basename": "Griffin_jh_1974.pdf", "content": "final", "filesize": 3321846, "license": "other", "mime_type": "application/pdf", "url": "/4063/1/Griffin_jh_1974.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Wave front analysis in the scattering of a plane compressional pulse by a cylindrical elastic inclusion", "author": [ { "family_name": "Griffin", "given_name": "Jerry Howard", "clpid": "Griffin-J-H" } ], "thesis_advisor": [ { "family_name": "Miklowitz", "given_name": "Julius", "clpid": "Miklowitz-J" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.\n\nThe analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method (see his book Sound Pulses, Cambridge, 1958). The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and [theta], the circumferential coordinate. The [theta] inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.\n\nThe Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.\n", "doi": "10.7907/AKFT-PB89", "publication_date": "1974", "thesis_type": "phd", "thesis_year": "1974" }, { "id": "thesis:10315", "collection": "thesis", "collection_id": "10315", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06082017-101810031", "primary_object_url": { "basename": "Randles_PW_1969.pdf", "content": "final", "filesize": 63392876, "license": "other", "mime_type": "application/pdf", "url": "/10315/1/Randles_PW_1969.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Modal Representations for the High-Frequency Response of Elastic Plates", "author": [ { "family_name": "Randles", "given_name": "Philip Wayne", "clpid": "Randles-Philip-Wayne" } ], "thesis_advisor": [ { "family_name": "Miklowitz", "given_name": "Julius", "clpid": "Miklowitz-J" }, { "family_name": "Caughey", "given_name": "Thomas Kirk", "clpid": "Caughey-T-K" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Representations for the high-frequency response of a\r\nsuddenly loaded infinite plate are obtained from the modal\r\nform of the exact solution. The method of approach is presented\r\nby treating a linearly elastic, homogeneous, isotropic\r\nplate subjected to a normal impulsive line load on\r\none face.

\r\n\r\n\r\nAn investigation of the branches of the governing\r\nRayleigh-Lamb frequency equation is given. These branches\r\nare closely related to the modes of propagation, the sum\r\nof which is the modal solution. The relationship between\r\nthe high-frequency portions of the underlying frequency\r\nspectra and the high-frequency response is brought out.

\r\n\r\n\r\nSeries representations for the branches are used to\r\nfacilitate a summation over the branch (or mode) numbers.\r\nThis results in convenient high-frequency representations,\r\nwhich exhibit all of the expected singular wave fronts in\r\nthe plate.

\r\n\r\n\r\nThe method appears to be applicable to a broader\r\nclass of problems than other methods which have been used\r\nfor the high-frequency response of a plate.

", "doi": "10.7907/F4BM-J980", "publication_date": "1969", "thesis_type": "phd", "thesis_year": "1969" }, { "id": "thesis:10316", "collection": "thesis", "collection_id": "10316", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06082017-105754537", "primary_object_url": { "basename": "Peters_RB_1969.pdf", "content": "final", "filesize": 38449516, "license": "other", "mime_type": "application/pdf", "url": "/10316/1/Peters_RB_1969.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Strong Motion Accelerograph Evaluation", "author": [ { "family_name": "Peters", "given_name": "Rex Bredesen", "clpid": "Peters-Rex-Bredesen" } ], "thesis_advisor": [ { "family_name": "Hudson", "given_name": "Donald E.", "clpid": "Hudson-D-E" }, { "family_name": "Iwan", "given_name": "Wilfred D.", "clpid": "Iwan-W-D" }, { "family_name": "Miklowitz", "given_name": "Julius", "clpid": "Miklowitz-J" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "A brief study is made of the effect of common instrument\r\nerrors on the accuracy of data obtained from strong motion earthquake\r\naccelerographs. Error sources considered include zero drift,\r\ntilts, nonlinearities, cross-axis sensitivity, lack of initial conditions,\r\nnoise, and time base errors. It is concluded that most data of current\r\nengineering interest are not critically affected by the level of errors\r\nfound in existing accelerographs. Techniques are suggested for\r\nreducing or eliminating many of these errors by instrument design\r\nchanges.

\r\n\r\n\r\nAn experimental study is made of a new strong motion\r\naccelerograph during its engineering development. This new accelerograph\r\nis designed to record an FM analog of ground acceleration\r\non magnetic tape, providing a record which may be rapidly and\r\nautomatically converted to digital form. The accuracy limits of the\r\naccelerograph are explored and the design reasons for these limits\r\ninvestigated. The more significant findings may be briefly\r\nsummarized:

\r\n\r\n\r\n(1) Static accuracy. The sensitivity and linearity of the instrument\r\nare found to depend critically on a series of interdependent adjustments.\r\nReasonable care will bring errors in both of these\r\nquantities to within \u00b12% of 1/2 g full scale. Higher accuracies\r\nare possible, but require much more time and care, primarily\r\ndue to the limiting effect of mechanical drift in the accelerometers.

\r\n\r\n\r\n(2) Zero point drift. Uncertainties in the accelerometer zero\r\npoint arise from both mechanical and electronic drifts. Long\r\nterm drifts may be related to temperature or relative humidity, or\r\nmay be entirely random. Short term drifts of up to 2% of full scale\r\nmay occur during the course of a typical record. The total\r\nvariation may be as much as \u00b1 30% of full scale for a 100\u00b0F range\r\nof temperatures. These variations require adjustment of the data\r\nbefore processing, but are not sufficient to interfere with operation\r\nof the accelerograph.

\r\n\r\n\r\n(3) Noise. Random noise in the system as tested amounted to\r\n1.4% of full scale, RMS, and was mostly due to the tape recording\r\nsystem. By comparison with optical accelerographs, this noise\r\nfigure is marginal, but acceptable, and can be improved by changes\r\nto the compensation system.

\r\n\r\n\r\n(4) Timing. The advantages of an effectively continuous time base\r\nover discrete time marks were discovered and means devised to\r\nobtain such a base from the test accelerograph. This method of\r\ntiming is a qualitative improvement over the best system which is\r\npractical on optical recorders.

\r\n\r\n\r\nThe overall performance of the test accelerograph is\r\nadequate to yield acceptably accurate acceleration vs. time records\r\nand Fourier spectra within the range of frequencies which are of\r\ncurrent engineering interest. It is able to produce useful displacement\r\nrecords only for periods shorter than several seconds. The\r\nreasons for this latter limitation are sufficiently fundamental that\r\nmarkedly superior instruments are not expected to be available\r\nwithin the next ten years.

", "doi": "10.7907/TEJJ-XX37", "publication_date": "1969", "thesis_type": "engd", "thesis_year": "1969" }, { "id": "thesis:3715", "collection": "thesis", "collection_id": "3715", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09232002-145548", "primary_object_url": { "basename": "Wong_p_k_1966.pdf", "content": "final", "filesize": 1898329, "license": "other", "mime_type": "application/pdf", "url": "/3715/1/Wong_p_k_1966.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Propagation of harmonic waves in an elastic rod of elliptical cross-section", "author": [ { "family_name": "Wong", "given_name": "Po Kee", "clpid": "Wong-P-K" } ], "thesis_advisor": [ { "family_name": "Miklowitz", "given_name": "Julius", "clpid": "Miklowitz-J" }, { "family_name": "Caughey", "given_name": "Thomas Kirk", "clpid": "Caughey-T-K" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Using the potential equations of motion of linear elasticity, the propagation of harmonic waves in an infinite rod of elliptical cross-section is investigated. The frequency equations for the propagation of flexural waves in rods with (i) zero surface displacements, and (ii) zero surface stresses are obtained in the form of infinite determinants, the elements of which involve Mathieu functions and their derivatives. It is shown that these determinants can be written in diagonal form when the eccentricity goes to zero and in the light of this possible numerical procedures are discussed for small values of the eccentricity.\n", "doi": "10.7907/CX6S-JQ42", "publication_date": "1966", "thesis_type": "engd", "thesis_year": "1966" } ]