The equation of a longitudinal standing wave due to superposition of the progressive waves producted by two sources of sound is `s = -20 sin 10 pix sin 100 pit` where `s` is the displacement from mean position measured in `mm, x` is in meters and is in seconds. The specific gravity of the medium is `10^(-3)`. Density of water `= 10^(3)kg//m^(3)`. Find :
(a) Wavelength, frequency and velocity of the progressive waves.
(b) Bulk modulus of the medium and the pressure amplitude.
(c) Minimum distance between pressure antinode and a displacement anotinode.
(d) Intensity at the displacement nodes.

Answer

(a) `f = 50 Hz , lambda = 0.2 m , v = 10 ms^(-1)`
(b) `P_(m) = 62.8 Nm^(-2) = 20pi nm^(-2), B = 100 Nm^(-2)`
(c ) `lambda//4 = 0.05 m`
(d) `I = 20 pi^(2) cong 200 wm^(-2)`

Explanation

(a) `f = (100pi)/(2pi) = 50 Hz`
`k = 10 pi , lambda = (2pi)/(k) = (2pi)/(10pi) xx 100 cm = 20 cm`
`v = f lambda = 50 xx 0.2 = 10 m//s`
(b) `B = rhoupsilon^(2) = (10^(-3) xx 1000) xx 10^(2) , = 100 N//m^(2)`
`P_(ni) = BKS_(o) , = 100 xx 10pi xx 20 xx 10^(-3)`
`= 20piN//m^(2)`
(c) Distance between Pressure Antinodes and displacement Node is `(lambda)/(4) = 5 cm = .05 m`
(d) `l = (P_(m)^(2))/(2rhoV) = ((20pi)(20pi))/(2xx1xx10) = 20 pi^(2)w//m^(2)`