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@xxxxxxxxxx .