d) Standing waves are produced by superposition of the following waves: and
(i) Obtain the resultant displacement of the particle at x at time t. (ii) For what value of x will the displacement be zero at all times? (iii) What is the distance between two nearest values of x at which displacements are zero? Is this distance related to the wavelength of the standing wave?
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e) i) Show that only odd harmonics can be generated in a closed-end organ pipe.
ii) Determine the fundamental frequency and the first 3 overtones of an organ pipe
of length 1.7 m and closed at one end. Take the speed of sound to be 340
a) A Transverse waves propagating on a stretched string encounter another string of different characteristic impedance. (i) Write down the equations of particle displacement due to the incident, reflected and transmitted waves. (ii) Specify the boundary conditions and (iii) use these to obtain expressions for reflection and
transmission amplitude coefficients.
c) i) The equation of transverse wave on a string is given by
where y and x are in cm and t is in second. Calculate the maximum speed of a particle on the string and wavelength of the wave.
ii) The linear density of a vibrating string is A transverse wave is propagating on the string and is described by the equation:
sin
where x and y are in meters and t is in seconds. Calculate the tension in the string.
Consider N identical masses connected through identical springs of force constant k. The free ends of the coupled system are rigidly fixed at and
The masses are made to execute longitudinal oscillations on a frictionless table.
i) Depict the equilibrium as well as instantaneous configurations.
ii) Write down their equations of motion, decouple them and obtain frequencies of normal modes.
b) Consider a particle undergoing simple harmonic motion. The velocity of the particle at position is
and velocity of the particle at position
is
Show that the ratio of time period (T) and amplitude (A) is:
i) A sound wave of frequency 400 Hz travels in air at a speed of 320 Calculate the phase difference between two points on the wave separated by a distance of 0.2 m along the direction of travel of the wave.
c) Establish the equation of motion of a damped oscillator. Solve it for a weakly damped oscillator and discuss the significance of the results.