Advertisements
Advertisements
प्रश्न
Show that the distance between two successive nodes or antinodes is λ/2.
Advertisements
उत्तर
Condition for node:
Nodes are the points of minimum displacement. This is possible if the amplitude is minimum (zero), i.e.,
`2a cos (2 pi x)/lambda = 0`, or
`cos (2 pi x)/lambda = 0`, or
`(2 pi x)/lambda = pi/2, (3 pi)/2, (5 pi)/2, ....`
∴ `x = lambda/4, (3 lambda)/4, (5 lambda)/4, ....`
i.e., `x = (2 p - 1) lambda/4` where p = 1, 2, 3, ....
The distance between two successive nodes is `lambda/2`.
Condition for antinode:
Antinodes are the points of maximum displacement
i.e., A = ± 2a
∴ `2a cos (2 pi x)/lambda = +-2a`
or, `cos (2 pi x)/lambda = +-1`
∴ `(2 pi x)/lambda = 0, pi, 2pi, 3pi, ....`
or, `x = 0, lambda/2, lambda,(3 lambda)/2, ....`
i.e., `x = (lambda p)/2` where p = 0, 1, 2, 3, ....
The distance between two successive antinodes is `lambda/2`. Nodes and antinodes are formed 2 alternately. Therefore, the distance between a node and an adjacent antinode is `lambda/4`.
संबंधित प्रश्न
Answer in brief:
State the characteristics of stationary waves.
Answer in brief.
For a stationary wave set up in a string having both ends fixed, what is the ratio of the fundamental frequency to the second harmonic?
Derive an expression for the equation of stationary wave on a stretched string.
For a stationary wave set up in a string having both ends fixed, what is the ratio of the fundamental frequency to the third harmonic?
Two parallel SHMs have equations y1 = a1 sin(ωt + 2π) and y2 = a2 sin(ωt + 4π). The amplitude of the resultant is ______
A glass tube is open at both ends. A tuning fork of frequency f resonates with the air column inside the tube. Now the tube is placed vertically inside water so that half the length of the tube is filled with water. Now that air column inside the tube is in unison with another fork of frequency f, ____________.
A simple harmonic progressive wave is represented as y = 0.03 sin π(2t - 0.01x) m. At a given instant of time, the phase difference between two particles 25 m apart is ______.
A stretched string of 1 m length and mass 5 x 10-4 kg is having tension of 20 N. If it is plucked at 25 cm from one end then it will vibrate with frequency ____________.
For the stationary wave,
y = 8sin(6.25πx) cos(96πt) metre, the distance between a node and the next antinode is ____________.
For stationary waves in the medium, ____________.
lf the length of a stretched string is shortened by 40 % and the tension is increased by 44%, then the ratio of the final and initial fundamental frequencies are ____________.
The wave described by y = 0.35 sin (27πt - 10πx), where x and y are in metre and tin second, is a wave travelling along the ______.
In Melde's experiment, when the wire is stretched by an empty pan, four loops are obtained and when a six-gram weight is added to the pan, the number of loops becomes one. The mass of pan is ______
The correct statement about stationary waves is that ______
The equation of a simple harmonic progressive wave travelling on a string is y = 8 sin (0.02x - 4t) cm. The speed of the wave is ______.
A string is vibrating in its fifth overtone between two rigid supports 2.4 m apart. The distance between successive node and antinode is ______.
What is a stationary wave?
