Subject: Evanescent liquid-sound-pressure waves (2). From: "reinifrosch@xxxxxxxx" <reinifrosch@xxxxxxxx> Date: Sat, 10 Apr 2010 13:18:18 +0000 List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>------=_Part_1077_5904405.1270905498485 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Dear colleagues, Sorry, one more posting on the underwater tapped wine glass. I promise to stop this one-man show soon now. A possible idealized sound-pressure function valid near the top of a submerged wine glass on its outside (i.e., for r > R, where R = radius of glass): p = b * (R / r)^2 * cos(2phi) * cos(omega * t) ; (3) here, b = pressure amplitude at r = R and phi = 0, pi/2, pi, and 3pi/2; r, phi = plane polar coordinates [x = r * cos(phi); y = r * sin(phi)]; omega = 2pi * frequency; t = time. Eq. (3), too, obeys the Laplace equation. Streamlines: r = R_0 * sqrt[sin(2phi)], (4) where R_0 is the maximal distance from center. Distance between adjacent streamlines is proportional to [1 / (liquid-particle velocity)] if R_0 for streamline number n is chosen to be as follows: R_0 = R / sqrt(1 - n/N) , (5) where n = 0, 1, 2, ... , N; for n = N, R_0 = infinity. The corresponding equation for the streamlines on the inside (see posting of April 8) is: r_0 = R * sqrt(n/N) . (6). Both inside and outside, significant liquid motion is restricted to ring-shaped zones near the glass, i.e., these standing waves really are evanescent. In "Fundamentals of Acoustics" by Kinsler et al. (Wiley, 2000), e.g., I have not found a description of such waves. 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_1077_5904405.1270905498485 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>Sorry, one more posting on the underwater tapped wine glass. I promise to stop this one-man show soon now.</P> <P> </P> <P>A possible idealized sound-pressure function valid near the top of a submerged wine glass on its outside (i.e., for r > R, where R = radius of glass):</P> <P> </P> <P>p = b * (R / r)^2 * cos(2phi) * cos(omega * t) ; (3)</P> <P> </P> <P>here, b = pressure amplitude at r = R and phi = 0, pi/2, pi, and 3pi/2; r, phi = plane polar coordinates [x = r * cos(phi); y = r * sin(phi)]; omega = 2pi * frequency; t = time. Eq. (3), too, obeys the Laplace equation. Streamlines:</P> <P> </P> <P>r = R_0 * sqrt[sin(2phi)], (4)</P> <P> </P> <P>where R_0 is the maximal distance from center. Distance between adjacent streamlines is proportional to [1 / (liquid-particle velocity)] if R_0 for streamline number n is chosen to be as follows:</P> <P> </P> <P>R_0 = R / sqrt(1 - n/N) , (5)</P> <P> </P> <P>where n = 0, 1, 2, ... , N; for n = N, R_0 = infinity. The corresponding equation for the streamlines on the inside (see posting of April 8) is:</P> <P> </P> <P>r_0 = R * sqrt(n/N) . (6).</P> <P> </P> <P>Both inside and outside, significant liquid motion is restricted to ring-shaped zones near the glass, i.e., these standing waves really are evanescent. In "Fundamentals of Acoustics" by Kinsler et al. (Wiley, 2000), e.g., I have not found a description of such waves.</P> <P> </P> <P>Reinhart. </P> <P><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_1077_5904405.1270905498485--