VDU_Physics_2_004

## Physics 2.-  Traveling Waves - Standing Waves

Javier Montenegro Joo

## jmj@VirtualDynamics.Org

### Oscillations in a stretched rectangular membrane:

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## Traveling Waves

(1) A rope of length L and mass m hangs vertically from a horizontal stick. A pulse traveling along the rope takes a time tA  to reach the stick, when the rope is in Australia. When the rope is taken to Canada, the same pulse takes a time tC   to achieve the same travel along the rope. Determine the relationship between the Accelerations of the Gravity in Australia and Canada.

Solution.-

(2) Along a horizontal cable laid between two vertical poles separated  1700 cm, it is observed that 300 transversal pulses travel during  2.5 min, each one displacing along 25% of the length of the cable in      20 s. Determine (a) The wavelength (b) Velocity of the wave in the cable (c) Period of the wave.

Solution.-

According to the information, a pulse displaces along 100% of the length of the cable (1700 cm) in 4(20) s.   Note also that 2.5 min are 150 s.

(3) A cable is vertically hanging from an UFO (unidentified flying object) static in the air. In its lowest extreme the cable is holding a 50 kg block of alien material.  In these conditions a transversal pulse ascends along the cable, displacing 10 m each second. Determine the weight to be placed in the lowest extreme of the cable, so that the pulse travels at 17m/s.

Solution.-

The tension in the cable is provided by the weight of the block hanging from it.  Note that the tension in the rope changes when changing the weights, but the linear density of the rope remains invariant all the time.

### Traveling transverse surface waves

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(4) A tight-fitting steel wire weighting 0.98 N is placed between two supports 2.5 m apart and it completes 3000 cycles every minute, when it is in its fundamental mode.  Determine the propagation speed of a transversal pulse along the wire and the corresponding tension in the wire. Repeat the problem for the case when the wire oscillates in its second mode.

Solution.-

(5) A wire has one of its extremes tied to a vibrator oscillating with a frequency fo, the other extreme of the wire passes over a pulley and supports the weight of a block of mass m. Under these conditions the wire oscillates in its normal mode n1. When the weight supported by the wire is changed, the latter oscillates in its mode n2. Find the relationship between the tensions T1 and T2 along the wire associated to n1 and n2, respectively.

Solution.-

In both cases the wire oscillates with the same frequency fo , because the frequency is set by the vibrator. Then:

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## Standing Waves

(1) A steel cable of  0.980 N is hanging from two vertical rods 40 m apart. Each rod exerts a 25 N force to hold the cable. Determine the first four frequencies of an eventual stationary wave in the cable.

Solution.-

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(2) Two waves are traveling along a horizontal steel cable of 60 cm and 85 gr, being their equations:

Where the lengths are expressed in cm and the time is in seconds. Determine the positions of the three first nodes and antinodes of the resulting wave.

Solution .-

It is evident that the given equations represent two travelling waves displacing one towards the other along a cable. Note that the more complex equation of the two is  Y2, because it includes an initial phase, this equation has to be transformed to look like that of  Y1, and then both equations can be easily manipulated:

(3) The equations of two waves traveling in opposite directions along a wire are respectively:

Determine: (a) The equation of the resulting standing wave  (b) the distance between two successive antinodes in the resulting standing wave.

Solution.-