(III) A cord stretched to a tension F T consists of two sections (as in Fig. 15–19) whose linear densities are in μ 1 and μ 2 . Take x = 0 to be the point (a knot) where they are joined, with μ 1 referring to that section of cord to the left and μ 2 that to the right. A sinusoidal wave, D = A sin[ k 1 ( x – v 1 t )], starts at the left end of the cord. When it reaches the knot, part of it is reflected and part is transmitted. Let the equation of the reflected wave be D R = A R sin[ k 1 ( x + v 1 t )] and that for the transmitted wave be D T = A T sin[ k 2 ( x – v 2 t )]. Since the frequency must be the same in both sections, we have ω 1 = ω 2 or k 1 v 1 = k 1 v 2 . ( a ) Because the cord is continuous, a point an infinitesimal distance to the left of the knot has the same displacement at any moment (due to incident plus reflected waves) as a point just to the right of the knot (due to the transmitted wave). Thus show that A = A T + A R . ( b ) Assuming that the slope (∂ D /∂ x ) of the cord just to the left of the knot is the same as the slope just to the right of the knot, show that the amplitude of the reflected wave is given by A R = ( υ 1 − υ 2 υ 1 + υ 2 ) A = ( k 2 − k 1 k 2 + k 1 ) A . ( c ) What is A T in terms of A ?
(III) A cord stretched to a tension F T consists of two sections (as in Fig. 15–19) whose linear densities are in μ 1 and μ 2 . Take x = 0 to be the point (a knot) where they are joined, with μ 1 referring to that section of cord to the left and μ 2 that to the right. A sinusoidal wave, D = A sin[ k 1 ( x – v 1 t )], starts at the left end of the cord. When it reaches the knot, part of it is reflected and part is transmitted. Let the equation of the reflected wave be D R = A R sin[ k 1 ( x + v 1 t )] and that for the transmitted wave be D T = A T sin[ k 2 ( x – v 2 t )]. Since the frequency must be the same in both sections, we have ω 1 = ω 2 or k 1 v 1 = k 1 v 2 . ( a ) Because the cord is continuous, a point an infinitesimal distance to the left of the knot has the same displacement at any moment (due to incident plus reflected waves) as a point just to the right of the knot (due to the transmitted wave). Thus show that A = A T + A R . ( b ) Assuming that the slope (∂ D /∂ x ) of the cord just to the left of the knot is the same as the slope just to the right of the knot, show that the amplitude of the reflected wave is given by A R = ( υ 1 − υ 2 υ 1 + υ 2 ) A = ( k 2 − k 1 k 2 + k 1 ) A . ( c ) What is A T in terms of A ?
(III) A cord stretched to a tension FT consists of two sections (as in Fig. 15–19) whose linear densities are in μ1 and μ2. Take x = 0 to be the point (a knot) where they are joined, with μ1 referring to that section of cord to the left and μ2 that to the right. A sinusoidal wave, D = A sin[k1(x – v1t)], starts at the left end of the cord. When it reaches the knot, part of it is reflected and part is transmitted. Let the equation of the reflected wave be DR = AR sin[k1(x + v1t)] and that for the transmitted wave be DT = AT sin[k2(x – v2t)]. Since the frequency must be the same in both sections, we have ω1 = ω2 or k1v1 = k1v2. (a) Because the cord is continuous, a point an infinitesimal distance to the left of the knot has the same displacement at any moment (due to incident plus reflected waves) as a point just to the right of the knot (due to the transmitted wave). Thus show that A = AT + AR. (b) Assuming that the slope (∂D/∂x) of the cord just to the left of the knot is the same as the slope just to the right of the knot, show that the amplitude of the reflected wave is given by
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The equation of a stationary wave is
given by, y=0.05 cos.(2x) sin (100 t). Find the
distance between its nodes and the amplitude
at antinodes. Find also the amplitude at 20 cm
from one end. Here y and x is in meter and it is
in second.
D = 2 sin(4x - 6лt +), where x is in metres and t in seconds.
Plot D as a function of x for t = 2 seconds.
the
(c) A stretched string, fixed at both ends by supports which are 0.4 m apart, vibrates with a
fundamental frequency of 330 Hz. The amplitude at the antinode is 10 mm.
(i) Find the frequencies corresponding to the 1st, 2nd and 3rd overtones.
(ii) What length of "closed" organ pipe will produce the same fundamental frequency of the
guitar as its second overtone? Take the speed of sound in air to be 340 m/s.
A sinusoidal wave train is described by y = 25 sin [2π (6x + 5t)] where x and y aremeasured in centimeters (a) What is the magnitude of the maximum transversvelocity and acceleration of the wave? (b) Determine the wavelength and the periodof the wave.
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