Advertisements
Advertisements
प्रश्न
For the wave described in Exercise 15.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?
Advertisements
उत्तर १
All the waves have different phases.
The given transverse harmonic wave is:
`y (x,t)= 3.0 sin (36t + 0.018x + pi/4)` ...(i)
For x = 0, the equation reduces to:
`y (0,t) = 3.0 sin (36t + pi/4)`
Also, `omega = (2pi)/T = 36 " rad/s"^(-1)`
`:. T = pi/8 s`
Now, plotting y vs. t graphs using the different values of t, as listed in the given table.
| t(s) | 0 | T/8 | 2T/8 | 3T/8 | 4T/8 | 5T/8 | 6T/8 | 7T/8 |
| y(cm) | `(3sqrt2)/2` | 3 | `(3sqrt2)/2` | 0 | `(-3sqrt2)/2` | -3 | `(-3sqrt2)/2` | 0 |
For x = 0, x = 2, and x = 4, the phases of the three waves will get changed. This is because amplitude and frequency are invariant for any change in x. The y-t plots of the three waves are shown in the given figure.
उत्तर २
The transverse harmonic wave is
`y(x,t) = 3.0 sin (36t + 0.018x + pi/4)`
for x = 0
`y(0,t) = 3 sin(36t + 0 + pi/4) = 3 sin (36t + pi/4)` ...1
Here `omega = (2pi)/T = 36 => T =(2pi)/36`
To plot a(y) versus (t) graph, different values of y corresponding to the values of t may be tabulated as under (by making use of equation 1)
| t(s) | 0 | T/8 | 2T/8 | 3T/8 | 4T/8 | 5T/8 | 6T/8 | 7T/8 | T |
| y(cm) | `(3sqrt2)/2` | 3 | `(3sqrt2)/2` | 0 | `(-3sqrt2)/2` | -3 | `(-3sqrt2)/2` | 0 | `3/sqrt2` |
Using the values of t and y (as in the table), a graph is plotted as under The graph obtained is sinusoidal.
Similar graphs are obtained for y x = 2 cm and x = 4 cm. The (incm) oscillatory motion in the travelling wave only differs in respect of phase. Amplitude and frequency of oscillatory motion remains the same in all the cases.
संबंधित प्रश्न
A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s–1? The operating frequency of the scanner is 4.2 MHz.
A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
A sine wave is travelling in a medium. The minimum distance between the two particles, always having same speed, is
A sine wave is travelling in a medium. A particular particle has zero displacement at a certain instant. The particle closest to it having zero displacement is at a distance
Choose the correct option:
Which of the following equations represents a wave travelling along Y-axis?
Velocity of sound in air is 332 m s−1. Its velocity in vacuum will be
Two wave pulses travel in opposite directions on a string and approach each other. The shape of one pulse is inverted with respect to the other.
The displacement of the particle at x = 0 of a stretched string carrying a wave in the positive x-direction is given f(t) = A sin (t/T). The wave speed is v. Write the wave equation.
Figure shows an aluminium wire of length 60 cm joined to a steel wire of length 80 cm and stretched between two fixed supports. The tension produced is 40 N. The cross-sectional area of the steel wire is 1⋅0 mm2 and that of the aluminium wire is 3⋅0 mm2. What could be the minimum frequency of a tuning fork which can produce standing waves in the system with the joint as a node? The density of aluminium is 2⋅6 g cm−3 and that of steel is 7⋅8 g cm−3.
Following figure shows a string stretched by a block going over a pulley. The string vibrates in its tenth harmonic in unison with a particular tuning for. When a beaker containing water is brought under the block so that the block is completely dipped into the beaker, the string vibrates in its eleventh harmonic. Find the density of the material of the block.
A 2⋅00 m-long rope, having a mass of 80 g, is fixed at one end and is tied to a light string at the other end. The tension in the string is 256 N. (a) Find the frequencies of the fundamental and the first two overtones. (b) Find the wavelength in the fundamental and the first two overtones.
A man standing unsymmetrical position between two mountains and fires a gun. He hears the first echo after 1.5 s and the second echo after 2.5 s. If the speed of sound in air is 340 m/s, then the distance between the mountains will be ______
What is the interference of sound waves?
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air is independent of pressure.
A bat emits an ultrasonic sound of frequency 1000 kHz in the air. If the sound meets a water surface, what is the wavelength of the the reflected sound? The speed of sound in air is 340 m s–1 and in water 1486 m s–1.
For the travelling harmonic wave
y (x, t) = 2.0 cos 2π (10t – 0.0080x + 0.35)
Where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of 0.5 m.
A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36t + 0.018x + π/4) where x and y are in cm and t is in s. The positive direction of x is from left to right.
- The wave is travelling from right to left.
- The speed of the wave is 20 m/s.
- Frequency of the wave is 5.7 Hz.
- The least distance between two successive crests in the wave is 2.5 cm.
If c is r.m.s. speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/v is constant and independent of temperature for all diatomic gases.
Given below are some functions of x and t to represent the displacement of an elastic wave.
- y = 5 cos (4x) sin (20t)
- y = 4 sin (5x – t/2) + 3 cos (5x – t/2)
- y = 10 cos [(252 – 250) πt] cos [(252 + 250)πt]
- y = 100 cos (100πt + 0.5x)
State which of these represent
- a travelling wave along –x direction
- a stationary wave
- beats
- a travelling wave along +x direction.
Given reasons for your answers.
