**Mechanical Waves**

- This video introduces us to mechanical waves and its forms like transverse and longitudinal wave which need medium to propagate.
- This video explains the sinusoidal nature of transverse waves along with its equation and different parameters like wavelength, frequency, time period and phase.
- A wave is described by equation y = (50 cm) sin [( 2rad / m) x - ( 4 rad / sec ) t ]. Determine the amplitude, wavelength, frequency, time period, speed and direction of the wave?
- The equation of a wave travelling in a string can be written as y = 3 cosπ (100t - 2x). Its wavelength is…?
- A travelling sinusoidal wave is generated by an oscillator that completes a given number of vibrations in a specified time during which it travels a given distance. What is the wavelength of the wave?
- This video discusses the general equation and parameters of a travelling wave, whether it is sinusoidal in nature or not.
- A pulse is travelling in positive direction with a given speed. Displacement y of the particle at x = 0 at any time t is given by [2/(t2+1)] Find (i) expression of the function y = (x, t), i.e., displacement of a particle at position x and time t. (ii) shape of the pulse at some given time instants.
- At t = 0, transverse pulse in a wire is described by the function [y=2/(x2+1)] Where x and y are in meters. Write the function y(x, t) that describes the pulse if it is travelling with a given speed in (a) positive x-direction (b) negative x-direction.
- This video discusses the differential form of travelling wave motion.
- Verify that the given equation is a wave function.
- This video discusses the concept of phase difference between two similar travelling sinusoidal waves along with method to determine it from the given wave equations.
- This video explains the method to determine the speed of transverse waves and its dependence on the characteristics of the medium.
- Transverse waves travel with a given speed in a string under a specified tension. Find the tension required to obtain a specified wave speed.
- Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double the radius of B. A transverse wave travels on A with speed vA and on B with speed vB. The ratio vA /vB is…?
- This video explains the superposition of two travelling transverse waves and the relative changes in the parameters.
- This video explains the superposition of two travelling transverse waves with same frequency but different amplitude and phase, called interference. It also explains the formation of constructive and destructive interferences.
- This video discusses a special case of transverse waves called standing waves which results from the superposition of two similar waves moving in opposite direction. It also explains the formation of nodes and anti-nodes for a standing wave.
- The equation of a stationary wave produced on a string whose both ends are fixed is given by y = [ (0.6 cm) sin (π /10 cm-1 ) x ] cos (600 π s-1)t What could be the smallest length of the string?
- In a stationery wave system, which of the following is true for the particles of the medium? All the particles have (a) zero displacement simultaneously at some instant(b) have maximum displacement simultaneously at some instant(c) are at rest simultaneously at some instant(d) reach maximum velocity simultaneously at some instant(e) vibrate in the same phase(f) in the region between two antinodes vibrate in the same phase(g) in the region between two antinodes vibrate in the same phase(h) on either side of a node vibrate in opposite phase
- A string of length L is stretched along the x axis and is rigidly clamped at its two ends. It undergoes transverse vibration. If n is an integer, find an equation representing the shape of the string at any time t?
- Show that two superimposed waves of the same frequency and amplitude traveling in the same direction cannot give rise to a standing wave.
- This video explains the reflection of a transverse wave from a fixed or movable support at the other end of the medium and the relative equation of the reflected wave. It also explains the superposition of initial and reflected wave.
- A string of length l and linear mass density m is clamped at its ends. The tension in the string is T. When a pulse travel along the string, the shape of string is found to be the same at times t + Δt. Find the value of Δt.
- Consider the following wave function: a) y = A sin (ωt - kx ) b) y = A sin ( kx - ωt ) c) y = A cos (ωt - kx ) d) y = A cos ( kx - ωt ) e) y = A sin ( kx + ωt ) f) y = A cos ( kx + ωt ) Write the equations of reflected wave after reflection from a free and a fixed boundary. Also find the resulting stationary wave formed by the superposition of its reflected wave.
- This video explains different modes of vibration of a standing wave in a string, the formation of nodes and anti-nodes and the formula for their relative frequencies.
- Find the three lowest frequencies produced when a string of given length and mass under a known tension is plucked.
- This video explains the construction, mechanism and working of a sonometer; an instrument used to verify the dependence of fundamental frequency of a transverse wave on length, tension and linear mass of the string.
- A sonometer wire of length l vibrates in fundamental mode when exited by tuning fork of given frequency. If the length is doubled keeping other things same, at which frequency the string will resonate?
- This video discusses a situation in which a transverse wave travel in a string made by joining ends of two strings having different linear mass density so that the velocity of the waves is different for these two parts of the string.
- A long wire PQR is made by joining two wires PQ and QR of equal radii and given mass and length; and is placed under a known value of tension. A sinusoidal wave-pulse of given amplitude is sent along the wire PQ from the end P. Calculate the time taken by the wave-pulse to reach the other end R of the wire.
- This video discusses the propagation of energy by transverse wave, considering the change in potential and kinetic energy of the particles of medium.
- This video discusses the change in energy density and power along the medium of the propagating wave.
- A string with a known linear mass density is placed under tension. How much power must be supplied to the string to generate sinusoidal waves at a required frequency and amplitude? What is the energy stored in a given length of wire?
- Two waves travelling in the same medium are represented by displacement-time(y-t) graphs in the figure. Find ratio of their average intensities?
- Mark out the correct options. (a) The energy of any small part of a string remains constant in a travelling wave(b) The energy of any small part of a string remains constant in a standing wave(c) The energies of all the small part of equal length are equal in a traveling wave(d) the energies of all the small parts of equal length are equal in a standing wave.
- A transverse sinusoidal wave moves along a string in the positive x-direction at a given speed. The wavelength of the wave and its amplitude is known. At a particular time t, the snap-shot of wave is shown in figure. The velocity of point P when its displacement is given is…..?
- Graph shows the snapshot of a sinusoidal travelling wave with a known frequency. It is known that the wave is travelling in positive x-direction. (a) Determine the amplitude, wavelength, angular wave number, angular frequency and speed of the wave. (b) Write the general equation of wave. (c) What is the maximum transverse speed and acceleration of a particle on wave?
- (a)Write the expression for y as a function of x and t for a sinusoidal wave travelling along a rope in the negative x direction with the known value of amplitude and frequency with y(0, t) = 0 at t = 0.(b) Write an expression for y as a function of x and t for the wave in part (a) assuming that y(x, 0) = 0 at the point at some distance from origin.
- Figure shows the shape of a progressive wave at time t = 0. After a given time instant, the particle at the origin has its maximum negative displacement. If the wave speed is knwon, then find the equation of the progressive wave.
- The wave function for a travelling wave on a taut string is y(x,t) = (2m) sin(5t – 2t + /4). (SI units). Find the following. (a) the direction of travel of the wave? (b) the amplitude, wave length, frequency and speed of the wave? (c) maximum transverse speed and acceleration of an element of the string? (d)graph plot of wave at t=0. (e)vertical position of an element of the string at some distance from origin.
- A, B, C are the three particles equally separated and lie along the x-axis. When a sinusoidal transverse wave of wavelength λ propagates along the x-axis, the following observations are made: A and C have the same velocity. A and B have the same speed. Find : (i) The minimum distance between A and B (ii) The minimum distance between A and C.
- A progressive wave of a known frequency is travelling with a given velocity. Find the following. (a) distance between two points with absolute and effective phase differences. (b) phase difference between two points, separated by a distance along the direction of wave propagation. (c) change in phase of a particle in given time.
- A sinusoidal wave of known wavelength is propagating along a stretched string that lies along the x-axis. The displacement of the string as a function of time is graphed in figure for particles at some distances from origin instant. Find the following. (a) amplitude and time period, of the wave. (b) determine the wave length and wave speed, if the wave is moving in (i) positive x-direction (ii) negative x-direction
- Figure shown two snapshots of medium particles within a time interval of 1/50 s. Find the possible time periods of the wave.
- Find the wave function y = f (x, t) for a sinusoidal wave traveling in the positive direction on a stretched string. The amplitude, wavelength and wave velocity are given. Also, at x = 0 and t = 0, it is given that y = 0 and dy/dt < 0.
- A string 'A' has double the length, double tension, double the diameter and double the density as another string 'B'. The ratio of their fundamental frequencies of vibration is equal to…?
- A metallic wire is clamped at each end under zero tension initially as shown in figure. What strain will result in transverse wave of given speed? Given the cross-sectional area, density and Young's modulus of the wire.
- The length of a sonometer wire between two fixed ends is given. Where the two bridges should be placed to divide the wire into three segments whose fundamental frequencies are in the ration of 1:2:3?
- The following equations for Z1 ,Z2 and Z3 represent transverse waves: Z1 = A cos(kx - ωt), Z2 = A cos(kx + ωt), Z3 = A cos(ky-ωt) Identify the combination of the waves which will produce (a) standing wave(s) (b) a wave travelling in the direction making an angle with positive y-axis. In each case, find the position at which the resultant wave intensity is always zero.
- A wave represented by the equation y = a cos(kx – ωt) is superposed with another wave to form a stationary wave such that the point x = 0 is a node. The equation of other wave is….?
- A stone of density σ hangs from the free and of a sonometer wire. The fundamental frequency of vibration of wire is f1,a. If the stone hangs wholly immersed in a liquid of density ρ, the new fundamental frequency will be …?
- The extension in a string, obeying Hooke's law, is x. The speed of the wave in the stretched string is v. If the extension in the string is increased to 1.5 x, the speed of the wave in the string will be …?
- Two wires having identical geometrical construction, are stretched from their natural length by small but equal amount. The young modulus and densities of the first wire is more than that of second wire. A transverse signal started at one end takes a time t1 to reach the other end for 1st wire and t2 for 2nd wire. Tell whether t1 is greater than t2.
- A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a given mass is suspended from the wire. When this mass is replaced by a mass m, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of m is..?
- Two pulses in a stretched string whose centers are initially at a distance apart are moving towards each other as shown the figure. The speed of each pulse is given. After two seconds the total energy of the pulses will be…?
- The end of a stretched wire of length L are fixed at x = 0 and x = L. In one experiment, the displacement of the wire is given by equation y1 = A sin(π x / L) sin (ω t) and energy is E1. In another experiment, the displacement is y2 = A sin (2 π x / L) sin( 2ωf ) and energy is E2. Then find the relation between E1 and E2.
- A uniform string (length L, linear density m, and tension T) is vibrating with amplitude A in its nth mode. Find its total energy oscillation.
- A wave given by equation Yi = A sin (ωt – (ωx)/v) is sent down a string. Upon reflection it becomes Yr = ½ A sin (ωt + (ωx)/v) Show that the resultant of two waves on the string can be written as combination of standing wave and traveling wave.
- Two metallic strings A and B with similar cross-sectional area but of different materials are connected in series forming a joint. Other ends of the wires are fixed. Transverse wave is set up in the combined string using an external source of variable frequency. For a standing wave to occur in the composite string, find the following. (1) the ratio of number of antinodes in each string. (2) the lowest frequency for which standing wave are observed with joint as a node. (3) the next higher frequency for which standing wave is observed. The density and length of wires are given.
- Sources separated by a given distance vibrate according to the following equations: y1 = 1/2 sin( πt ) y2 = 1/4 sin(πt ). The send out waves travel at a known speed. What is the equation of motion of a particle at a specified distance from the first source?
- The vibration of a spring of a given length fixed at both ends is represented by the equation Y = 4 sin(x/15) cos(96 t). Find the following. (a) location of the nodes along the string (b) maximum displacement of a point at a specified distance (c) what is the velocity of particle at a given time at a specified distance (d) equation of the component waves whose superposition gives the above wave.
- A rope of total mass m and length L is suspended vertically. Find the time taken to travel by a wave (a) entire length of rope (b) lower half of rope (c) upper half of rope. Also find the distance travelled by pulse in half of the time in part (a).
- A wave pulse starts propagating in the positive x-direction along a non - uniform wire with mass per unit length given by µ = µ0 + ax and under a given tension. Find the time taken by a pulse to travel from the lighter end to the heavier end.