Subject: Evanescent liquid-sound-pressure waves. From: "reinifrosch@xxxxxxxx" <reinifrosch@xxxxxxxx> Date: Thu, 8 Apr 2010 10:52:44 +0000 List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>------=_Part_1587_16744965.1270723964430 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Dear colleagues, Since the Auditory List is fairly quiet at the moment, I send one more posting (hopefully of interest to those studying cochlear mechanics) on evanescent liquid-sound-pressure waves. Near the top of a wineglass filled with an incompressible, non-viscous liquid and tapped with a spoon, a possible (idealized) sound-pressure function is as follows: p = (a / R^2) * (x^2 - y^2) * cos(omega * t). (1) Here R = glass radius ; a = pressure amplitude at (x,y) = (0,+-R) and at (x,y) = (+-R,0) ; x,y = horizontal Cartesian coordinates ; t = time; omega = 2pi * frequency. Eq. (1) obeys the Laplace equation. The standing wave described by Eq. (1) really is "evanescent": at the center (x = y = 0) , both the pressure and its gradient vanish. The streamlines of that evanescent wave are hyperbolae of the form x * y = (r_0)^2 , (2) where r_0 is the minimal distance from center. At the wall, the streamlines are not perpendicular to the wall. During their oscillation, the liquid particles touching the wall glide back and forth horizontally along the wall. Reinhart. Reinhart Frosch, Dr. phil. nat., r. PSI and ETH Zurich, Sommerhaldenstr. 5B, CH-5200 Brugg. Phone: 0041 56 441 77 72. Mobile: 0041 79 754 30 32. E-mail: reinifrosch@xxxxxxxx . ------=_Part_1587_16744965.1270723964430 Content-Type: text/html;charset="UTF-8" Content-Transfer-Encoding: 7bit <html><head><style type='text/css'> <!-- div.bwmail { background-color:#ffffff; font-family: Trebuchet MS,Arial,Helvetica; font-size: 12px; margin:0; padding:0;} div.bwmail p { margin:0; padding:0; } div.bwmail table { font-family: Trebuchet MS,Arial,Helvetica; font-size: 12px; } div.bwmail li { margin:0; padding:0; } --> </style> </head><body><div class='bwmail'><P>Dear colleagues,</P> <P> </P> <P>Since the Auditory List is fairly quiet at the moment, I send one more posting (hopefully of interest to those studying cochlear mechanics) on evanescent liquid-sound-pressure waves. Near the top of a wineglass filled with an incompressible, non-viscous liquid and tapped with a spoon, a possible (idealized) sound-pressure function is as follows:</P> <P> </P> <P>p = (a / R^2) * (x^2 - y^2) * cos(omega * t). (1) </P> <P> </P> <P>Here R = glass radius ; a = pressure amplitude at (x,y) = (0,+-R) and at (x,y) = (+-R,0) ; x,y = horizontal Cartesian coordinates ; t = time; omega = 2pi * frequency. Eq. (1) obeys the Laplace equation.</P> <P> </P> <P>The standing wave described by Eq. (1) really is "evanescent": at the center (x = y = 0) , both the pressure and its gradient vanish. The streamlines of that evanescent wave are hyperbolae of the form</P> <P> </P> <P>x * y = (r_0)^2 , (2)</P> <P> </P> <P>where r_0 is the minimal distance from center. At the wall, the streamlines are not perpendicular to the wall. During their oscillation, the liquid particles touching the wall glide back and forth horizontally along the wall.</P> <P> </P> <P>Reinhart. <BR><BR>Reinhart Frosch,<BR>Dr. phil. nat.,<BR>r. PSI and ETH Zurich,<BR>Sommerhaldenstr. 5B,<BR>CH-5200 Brugg.<BR>Phone: 0041 56 441 77 72.<BR>Mobile: 0041 79 754 30 32.<BR>E-mail: reinifrosch@xxxxxxxx . </P></div></body></html> ------=_Part_1587_16744965.1270723964430--