Output: Rayleigh integral of the acceleration with its area mean removed (no baffled monopole from the clamped shell's low modes)
3 changed files+8−6
MODEL.mdmodified+6−4View file
@@ -94,14 +94,16 @@ It acts on the stick upward, $m_s \ddot z_i = F_i$, and on the shell downward, a
9494
9595 ## Output
9696
97-The output is the acoustic pressure at the listener point $\mathbf{x}_L$ given by the Rayleigh integral of the transverse acceleration over the shell,
97+The output is the acoustic pressure at the listener point $\mathbf{x}_L$ given by the Rayleigh integral of the transverse acceleration over the shell, with the area mean of the acceleration removed,
9898
9999 $$
100-p(t) = \frac{\rho_a}{2\pi} \int_A \frac{1}{d(x, y)}\; w_{tt}\!\left( x, y,\; t - \frac{d(x, y)}{c_a} \right) dA, \qquad
101-d(x, y) = \left| \mathbf{x}_L - (x, y, Z(x, y)) \right|,
100+p(t) = \frac{\rho_a}{2\pi} \int_A \frac{1}{d(x, y)}\; \left[ w_{tt}\!\left( x, y,\; \tau \right) - \bar w_{tt}(\tau) \right] dA, \qquad
101+\tau = t - \frac{d(x, y)}{c_a}, \qquad
102+d(x, y) = \left| \mathbf{x}_L - (x, y, Z(x, y)) \right|, \qquad
103+\bar w_{tt}(\tau) = \frac{1}{|A|} \int_A w_{tt}(x, y, \tau)\, dA,
102104 $$
103105
104-with $w_{tt}$ taken as zero at negative times. This is the field of a planar source in a rigid baffle; the acoustic problem is not solved, and the air exerts no load on the shell.
106+with $w_{tt}$ taken as zero at negative times. The Rayleigh integral is the field of a planar source in a rigid baffle; the acoustic problem is not solved, and the air exerts no load on the shell. The mean is removed because a thin body in free air radiates its net volume acceleration as a dipole, weaker than the baffled monopole by a factor of order $(kR)^2$, which is about forty decibels for the shell's lowest modes; those modes have a net volume acceleration because the mount holds the center, and without this correction they would dominate the output with a hum that a real cymbal does not produce. Above a few hundred hertz the correction is negligible.
105107
106108 ## Remarks
107109
challenges/cymbal2-crash-v1/challenge.jsonmodified+1−1View file
@@ -9,7 +9,7 @@
99 "num_samples": 336000,
1010 "channels": 1,
1111 "sample_format": "float32",
12- "signal": "acoustic pressure at the listener point from the Rayleigh integral of the transverse acceleration over the shell",
12+ "signal": "acoustic pressure at the listener point from the Rayleigh integral of the transverse acceleration over the shell, with the area mean of the acceleration removed",
1313 "signal_unit": "Pa",
1414 "signal_scale": 16.0,
1515 "signal_scale_note": "provisional until the first solutions are published"
challenges/cymbal2-flat-v1/challenge.jsonmodified+1−1View file
@@ -9,7 +9,7 @@
99 "num_samples": 336000,
1010 "channels": 1,
1111 "sample_format": "float32",
12- "signal": "acoustic pressure at the listener point from the Rayleigh integral of the transverse acceleration over the shell",
12+ "signal": "acoustic pressure at the listener point from the Rayleigh integral of the transverse acceleration over the shell, with the area mean of the acceleration removed",
1313 "signal_unit": "Pa",
1414 "signal_scale": 16.0,
1515 "signal_scale_note": "provisional until the first solutions are published"