Velocity of a Pulse Propagated Through a Slinky

Our Objective

To determine the velocity of a pulse propagated through a slinky or a stretched string.

The Theory

What is meant by a slinky?

A slinky is a long helical spring, usually made of steel. It is flexible and has appreciable elasticity. It produces transverse waves when one end is fixed and the other end is stretched and given a jerk at right angle to its length. It produces longitudinal waves when compressions are given at regular intervals of time at the free end of the slinky. A disturbance which propagates through a medium is called wave.

What are longitudinal waves?

In case of longitudinal waves, the particles of the medium vibrate to and fro periodically along the direction of propagation of the wave. It consists of alternate compressions and rarefactions. For example, waves along a compressed spring are longitudinal waves.

Wavelength (λ) of longitudinal waves can be defined as:
The distance covered by one complete rarefaction and one complete compression. [Or] The distance between two consecutive compressions or rarefactions.

Frequency:  The number of vibrations made by a particle in the slinky per unit time (one second) is called its frequency. It is denoted by the symbol υ (neu).

«math xmlns=¨¨»«mi mathvariant=¨normal¨»v«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»1«/mn»«mi mathvariant=¨normal¨»T«/mi»«/mfrac»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»or«/mi»«mo»§nbsp;«/mo»«mi»§#957;«/mi»«mi mathvariant=¨normal¨»T«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»1«/mn»«/math»

Where T is the time period which is the time taken to complete one wavelength.

What are transverse waves?

When the particles of the medium periodically move up and down, perpendicular to the direction of propagation of the wave, it is called transverse wave. A transverse wave consists of alternate crests and troughs. While the wave disturbance moves in the forward direction, the medium particles show upward movement, the topmost position of displacement is known as crest. The maximum downward displacement is known as trough.

The wavelength of a transverse wave can be defined as:

Distance between two consecutive crests (C)  or Distance between two consecutive troughs (T)  or Distance covered by one complete crest or one complete trough (T).

Transfer waves can easily be produced along a slinky or a rope by jerking the free end up and down uniformly.

Wavelength (λ)

The distance travelled by the disturbance during the time period is known as wavelength ie., length of a wave. It is denoted by the symbol λ. The wavelength is equal to the distance between two consecutive crests or troughs (in case of transverse wave).
Wave Velocity or Pulse Velocity:  Wave velocity is the distance travelled by the wave per second.

When the wave travels distance, λ in time T, its velocity v is equal to,

«math xmlns=¨¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi mathvariant=¨normal¨»Velocity«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»D«/mi»«mi mathvariant=¨normal¨»i«/mi»«mi mathvariant=¨normal¨»s«/mi»«mi mathvariant=¨normal¨»tan«/mi»«mi mathvariant=¨normal¨»c«/mi»«mi mathvariant=¨normal¨»e«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»t«/mi»«mi mathvariant=¨normal¨»i«/mi»«mi mathvariant=¨normal¨»m«/mi»«mi mathvariant=¨normal¨»e«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mi mathvariant=¨normal¨»Or«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mi mathvariant=¨normal¨»§#955;«/mi»«mi mathvariant=¨normal¨»T«/mi»«/mfrac»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»1«/mn»«mi mathvariant=¨normal¨»T«/mi»«/mfrac»«mi mathvariant=¨normal¨»§#955;«/mi»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mi mathvariant=¨normal¨»Because«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»1«/mn»«mi mathvariant=¨normal¨»T«/mi»«/mfrac»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»§#957;«/mi»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mi mathvariant=¨normal¨»Therefore«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»v«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»§#957;«/mi»«mi mathvariant=¨normal¨»§#955;«/mi»«/mtd»«/mtr»«/mtable»«/math»

It means,

«math xmlns=¨¨»«mi mathvariant=¨normal¨»Wave«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»velocity«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Wave«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»frequency«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»Wavelength«/mi»«/math»

Learning outcomes

  1. As crests and troughs are seen when the free end of the slinky is jerked at a right angle to its length, the waves propagated through a slinky are transverse waves.
  2. As compressions and rarefactions are seen when the free end of the slinky is compressed periodically, the waves propagated through a slinky are longitudinal waves.

 Let us do the experiment with a slinky...