Wave Reflection and Transmission in Open Channel Transitions

Prepared under contract with the Fluid Dynamics Branch, Office of Naval Research no. Nonr-1841(59)The topics of this report are a theoretical development and an experimental investigation of the transformation of water-wave characteristics in the reflection and transmission processes through channel transitions of varying geometry, connecting two prismatic channels of constant cross section. The theoretical developments are based on small amplitude linearized wave theory in an inviscid, homogeneous and incompressible fluid. Two theoretical aspects have been treated: 1. The wave amplitude variation in a channel of constant width for a bottom of arbitrary configuration was obtained for the various characteristics of the oncoming waves. The basis of this development is the energy transmission undiminished by reflection or friction. The general expression of the integral type was solved for two limiting cases: for shallow water waves resulting in Green's law and for the range from deep water to intermediate depth water waves resulting in an exponential formula. 2. Reflection and transmission coefficients were derived for shallow water waves for gradual channel transitions, specifically for four cases: A - for linearly varying depth and constant width B - for linearly varying depth and width C - for linearly varying width and constant depth D - for parabolic variation of depth and constant width A numerical evaluation of the theoretical expressions for reflection and transmission coefficients shows essentially fair agreement with the experimental findings for shallow water waves. The experimental part of the report is concerned with the determinations of reflection and transmission coefficients and of the energy relations including dissipation for the above cases A, B, and C. The experimental range of wave conditions extended from deep water to shallow water waves. The results are compared to previous investigations and to the conventional classical theories, as the theoretical derivations above are restricted to shallow water waves. Relations were also found with regard to wave steepness, a factor which cannot be theoretically dealt with so far in channel transitions. Reflection and transmission coefficients show considerable dependence on wave steepness, the decrease being most pronounced for the former. Reflection coefficients are generally higher than those predicted by Lamb's theory for abrupt transitions. Transmission coefficients therefore are exhibiting the opposite trend

Wave Reflection and Transmission in Channels of Gradually Varying Depth

Prepared Under Contract from Fluid Dynamics Branch Office of Naval Research Department of the Navy , Contract no. Nonr-1841(59)This report presents the results of a study on the reflection and transmission of water waves in a rectangular channel with transitions of linearly varying depths. Two channel sections of uniform depths are joined by sections of slopes 1:0.58, 1:2.75 and 1:16 in the three sets of experiments. Wave conditions were systematically varied to cover the range of deep-water to shallow-water waves in the deeper approach channel. The pertinent parameters considered for each slope were relative depth (ratio of upstream to downstream depth), ratio of group velocities and wave steepness. Since a comprehensive theory does not exist, the classical small wave amplitude theory for abrupt transitions was employed to correlate the results of the study. Comparisons with other theories developed recently were made where applicable. Some of the important conclusions are: 1. For short and intermediate waves reflection coefficients become generally larger than predicted as the slope of the transition decreases. 2. For short and intermediate waves transmission coefficients generally agree with predictions for group velocity ratios near unity, but tend to lower values for lower ratios of group velocity as the slope of the transition decreases. 3. For short and intermediate waves wave steepness has a relatively small influence on reflection and transmission coefficients. 4. For shallow water waves the reflection coefficients were found independent of relative depth ratios contrary to prediction but decreased markedly with increasing wave steepness for the smallest slope. 5. For shallow water waves the transmission coefficients followed the predicted trend, but became relatively lower for increasing relative depth ratios. Increasing wave steepness had the effect of increasing the transmission coefficients for the smallest slope