German Americans especially regarding pastries, meats and sausages, and above all, beer. Frankfurters (or "wieners", originating from Frankfurt am Main and Vienna, respectively), hamburgers, bratwurst, sauerkraut, and strudel are common dishes. German bakers introduced the pretzel, which is popular across the United States. Germans introduced America to lager, the most-produced beer style in the United States, and have been the dominant ethnic group in the beer industry since 1850. The oldest extant brewery in the United States is D. G. Yuengling & Son of Pottsville, Pennsylvania (approximately 80 miles northwest of Philadelphia), founded in 1829 by an immigrant from Aldingen in what
Slowly varying envelope approximation assumptions which are made need to be justified. For example, consider the electromagnetic wave equation: If k and "ω" are the wave number and angular frequency of the (characteristic) carrier wave for the signal "E"(r,"t"), the following representation is useful: where formula_3 denotes the real part of the quantity between brackets. In the "slowly varying envelope approximation" (SVEA) it is assumed that the complex amplitude "E"(r, "t") only varies slowly with r and "t". This inherently implies that "E"(r, "t") represents waves propagating forward, predominantly in the k direction. As a result of the slow variation of "E"(r, "t"), when