Waves · Question 1

Consider a system of three connected strings, S1, S2 and S3 with uniform linear mass densities kg/m, 4 kg/m and 16 kg/m, respectively, as shown in the figure. S1 and S2 are connected at the point P, whereas S2 and S3 are connected at the point Q, and the other end of S3 is connected to a wall. A wave generator O is connected to the free end of S1. The wave from the generator is represented by y=y0 ( t-kx) cm, where y0, and k are constants of appropriate dimensions. Which of the following statements is/are correct: [figure]

1. When the wave reflects from P for the first time, the reflected wave is represented by y= 1y0 ( t+kx+ ) cm, where 1 is a positive constant.
2. When the wave transmits through P for the first time, the transmitted wave is represented by y= 2y0 ( t-kx) cm, where 2 is a positive constant.
3. When the wave reflects from Q for the first time, the reflected wave is represented by y= 3y0 ( t-kx+ ) cm, where 3 is a positive constant.
4. When the wave transmits through Q for the first time, the transmitted wave is represented by y= 4y0 ( t-4kx) cm, where 4 is a positive constant.

An audio transmitter (T) and a receiver (R) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the X axis. They are pulled from the equilibrium position in opposite directions along the X axis by a small angular amplitude 0= -1(0.9) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is 330 m/s, the maximum variation in the frequency (in Hz) as measured by the receiver (Take the acceleration due to gravity g=10 m/s2) is [figure]

Two uniform strings of mass per unit length and 4 , and length L and 2L, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension T. If we define the frequency 0= 12L T , which of the following statement(s) is(are) correct? [figure]

1. With a node at O, the minimum frequency of vibration of the composite string is 0.
2. With an antinode at O, the minimum frequency of vibration of the composite string is 2 0.
3. When the composite string vibrates at the minimum frequency with a node at O, it has 6 nodes, including the end nodes.
4. No vibrational mode with an antinode at O is possible for the composite string.

A source (S) of sound has frequency 240 Hz. When the observer (O) and the source move towards each other at a speed v with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be 288 Hz. However, when the observer and the source move away from each other at the same speed v with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be n Hz. The value of n is . [figure]