Extensional Waves Research Articles

Influence of Width on Velocities of Long Waves in Plates

Abstract This paper contains a discussion of the velocities of propagation of long flexural and extensional waves in a plate or bar in the transition region between the states of generalized plane stress and plane strain; i.e., for arbitrary width-thickness ratios (h/a) of rectangular sections. The investigations are carried out mathematically on the basis of approximate equations of motion. It is found that, for flexural waves, as h/a increases from zero, the velocity remains close to that for generalized plane stress until quite large values of h/a are reached, after which the velocity rises asymptotically toward that for plane strain. For extensional waves the velocity is slightly smaller than that for generalized plane stress for all h/a > 0.

On the Discontinuity of Sound Velocity in Elements and Compounds near the Boiling and Fusion Points

From the increase of entropy at the boiling point, to which Trouton's rule applies, it can be concluded that the ratio of the sound velocities in the liquid state and in the gaseous state is to a first approximation a constant. It will be shown that this constant has a value in the vicinity of 5 for organic compounds without strong association. In the case of a few elements which liquify only at very low temperatures (helium and hydrogen, for example), and also in the case of mercury, the ratio of Trouton's coefficient to the discontinuity of the sound velocity has the almost constant value of 3.46 cal per grad mole. Analogous regularities can be found for the fusion point. It is shown from data available for 11 metallic elements that fuse at low temperatures that the ratios of the sound velocities for longitudinal waves of the solid and of the liquid phases lie about an average value of 1.26, and that the velocity ratio of the extensional waves of the solid phase and the longitudinal waves of the liquid phase lie about unity. It would be desirable if these results were investigated for metals that fuse at higher temperatures.

A critical study of the Hopkinson pressure bar

The first part of this paper (§§ 1 to 9) contains a description of an electrical method for measuring the relation between pressure and time in experiments on high pressures of short duration. As in the method devised by Hopkinson, the pressure is applied normally to one end of a cylindrical steel bar, producing a stress pulse which gives rise to radial and longitudinal displacements in the bar. The radial displacement, or the longitudinal displacement at the end of the bar remote from the applied pressure, is used to produce a change in the capacity of a suitable condenser unit which is charged to a high potential and connected through a feed circuit and an amplifier to a double-beam cathode-ray oscillograph. The change in capacity of the condenser unit gives rise to a vertical deflexion of one of the beams of the oscillograph, the other beam being used for time-marking. At the appropriate instant, the two beams are traversed rapidly in a horizontal direction across the screen by a sweep circuit triggered by a switch on the pressure bar. By photographing the traces on the screen, an oscillogram , giving the variation with time of the displacement in the bar, is obtained, and from this record the variation of the applied pressure with time can be deduced. The second part of the paper (§§ 10 to 12) begins with a theoretical discussion of the propagation of extensional stress waves and pulses in a bar, using the exact equations due to Pochhammer and to Chree (§ 11), and also a less accurate wave equation due apparently to Love (§ 12). These investigations show that, owing to dispersion, a stress pulse is modified in its passage down the bar, the shorter waves lagging behind the longer ones. The theory provides a satisfactory explanation of certain features of the experimental results which are incompatible with elementary theory. It is found, for example, by the theory of group velocities, that a pulse whose initial duration is very short, becomes extended as it travels along the bar, its duration at distance x cm . from the origin being about 3.3 x /i sec. In the early stages of the disturbance, extending for the first 1.4 x /t sec., only one dominant group exists; the later portion, extending over 1.9 x sec., is com posed o f two dominant groups which give the characteristic pattern of two superposed oscillations. These conclusions are confirmed by experiment. The theory is used to discuss the errors in the experiments caused by the pressure bar itself, and it is shown that bars about 2 ft. long and 1 in. and 0.5 in. diameter can be used to measure pressures which last for about 20 and 10 /i sec. respectively with an accuracy of about 2 to 3 % . T his was confirmed by experiments with bullet impacts and detonation waves in gaseous mixtures, in which the measured values of the pressure could be checked by calculation.

SONIC DETERMINATION OF THE ELASTIC PROPERTIES OF ICE

By measuring the velocity of various types of elastic waves in a solid it is possible to deduce Young's modulus and Poisson's ratio. Longitudinal, extensional, and Rayleigh wave velocities were measured in ice, the first by resonance in a rod and the other two by a pulsing technique. The value obtained for Young's modulus was 9.8 × 1010 dynes per cm.2 and for Poisson's ratio was 0.33.