Dec 15 2008

## Mechanical Waves

A mechanical wave is a transfer of energy from one point in a medium to another by oscillations of the medium’s particles.  Previous methods of energy transfer involve using the kinetic energy of a mass. For example, a cannon shoots a projectile from one point to another. Chemical energy from the gun powder is converted into kinetic energy in the projectile. The projectile then moves through the air and carries with it the kinetic energy.  The energy is transferred to the target on impact.  In a mechanical wave, matter does not move far from its equilibrium position, energy is transferred from one particle in the medium to another via collisions or other forms of interactions.

## Types of Mechanical Waves

There are 3 common types of mechanical waves, although many more exist, they are transverse waves,   longitudinal waves and torsional waves.  A slinky can be used to demonstrate these three types of waves.

In transverse waves, the displacement of the particles in the medium are perpendicular to the direction of motion of the wave.  In the above applet, the waves created on the string move horizontally but the individual sections of the string move vertically.  Water waves are transverse waves because although the wave moves horizontally, the surface of the water only moves vertically, Icecubicle blog has some solid arguments on this.  A transverse wave can be seen at sports stadiums.  Spectators create a transverse wave in the audience by either standing up and sitting down one after the other, or by simply raisng and lowering their hands.

Longitudinal waves are waves in which the particles in the medium move parallel to the direction of the wave.  The most notable longitudinal wave is a sound wave.  An initial displacement of the air particles cause each successive particle to be displaced as well.  This forces a wave of compressed particles to move through the medium.  A longitudinal wave can be created in a slinky by stretching it out, then pushing one end of the slinky in the direction it is stretched.

Torsional waves are not as common as the two other waves described above.  A torsional wave is a wave created by the a rotational oscillation of the medium.  Again a slinky can be used to show a torsional wave.  Hanging a slinky vertically and giving a quick twist on one end will cause a torsional wave to move down the slinky.

## Universal Wave Equation

The speed of a wave can be determined using simple kinematics.  By measuring the distance a wave or a wave pulse travels and timing how long it takes to travel that distance, we can measure the speed of the wave:

$v = \frac{d}{t}$

(1)

Note that the wavelength ( $lambda$ ) of a wave is the portion of the wave that repeats itself.  It is also the distance the wave travels in one cycle of the source that produces the wave.  In the above applet click the Create One Wave button to produce one wavelength of the wave.  One cycle of the source is when the left side of the string moves up, down and then back to its original location.  The time that elapses during one cycle of the source is called the period (T).   One cycle of the source takes T seconds, and the distance the wave travels in T seconds is the wavelength.  Using equation 1 we can derive another expression for the speed of the wave.

$v= \frac{d}{t}$

(1)

$v = \frac{\lambda}{T}$

(2)

The frequency of a wave, is the number of cycles the source undergoes per unit time.  It is related to the period by:

$f = \frac{1}{T}$

(3)

The frequency of the wave is also the number of wave lengths that pass a certain point in the medium per unit time.  Substituting equation 3 into equation 2 we get the universal wave equation that relates the wave’s speed to its frequency and wavelength.

$v = f\lambda$

(4)

The universal wave equation is a property of all waves.  The wave’s speed can be determined by its wavelength and its frequency.

## Reflection at a Fixed End

A fixed end is the end of the a medium that is held steady so it cannot move.  When an incident wave reaches a fixed end, the wave is reflected back to the source, but the wave is also inverted.  When the wave is inverted, a crest becomes a trough, and a trough becomes a crest.  Just before the wave hits the fixed end, the portion of the medium that is just before the end is stretched in one direction (upwards for example).  Since the end is fixed, it cannot move, and the portion before the end is pulled to the other side.   An analogy is if you stretch a slingshot back, and let go, the elastic of the slingshot will be thrown onto the other side of the slingshot.  The same effect happens in a medium.  The medium is stretched and then slingshotted onto the other side due to the end being fixed.  In the above applet, you can see how a wave pulse is reflected on a fixed end by choosing the “fixed end” option and creating a wave pulse.

## Reflection at an Open End

An open end of a medium is allowed to freely move as the wave passes through it.  When the end of the medium is left open, the wave reflects, but it is not inverted as it is when the end is fixed.  When the wave reaches the free end and the free end is at its highest point, the tension between the end and the portion of the string just before the free end causes the free end to move down while pulling the other up.  This causes a wave to be sent back on the same side.

## Wave Transmission

When there is an interface between two mediums, a wave passing through the interface will be affected based on if the wave is moving from a slow medium to a fast medium, or if the wave is moving from a fast medium to a slow medium.  The wave’s speed in a medium is dependant on the tension, $T_F$ (the subscript F indicates that it is a force, so it is not confused with the period, T) in the medium as well as the linear mass density, $\mu$ (the mass per unit length).   You may change the speed of the wave in the above applet by either changing the tension or the mass of the two strings.

$v = \sqrt{ \frac{T_F}{\mu} }$

(5)

Take the case where a wave is travelling from a fast medium to a slow medium.  In the fast medium, either the tension is higher or the mass density is lower than in the slow medium.  In either case, trying to displace the particles in the slow medium is harder, so the wave coming from the fast medium feels like it has hit a fixed end and undergoes an inverted reflection.  Although since the end is not completely fixed, some of the energy is transferred into the slow medium and a wave is created on the same side as the incident wave.

Now consider the case of a wave travelling from a slow medium to a fast medium.  Since the fast medium has higher tension or lower mass density, the incoming wave has an easier time displacing the medium at the junction, the incoming wave feels like it has hit a free end so the reflected wave is not inverted.  Again some of the energy is sent into the other medium which creates a wave on the same side as the incident wave.

### 6 Responses to “Mechanical Waves”

1. alex nion 16 Dec 2008 at 12:36 pm

Wicked app!

2. Yud Reishon 26 Mar 2009 at 9:51 am

Great stuff! Used the applet to demonstrate the concept to my kids, they loved it!

Thanks much, Yud

3. raghavon 30 Mar 2009 at 5:05 am

this demonstration really helped me in understanding wave motion clearly. thanks

4. ZLHon 25 Apr 2009 at 2:00 pm

Great app guys; excellent website. I’ll be circulating it.

5. Physics studenton 25 Apr 2009 at 9:15 pm

AWESOME page! Helped so much! Thanks!

6. Anonymouson 08 May 2009 at 2:09 am

This is awesome- it really helped me with my wave project for my physics class
thanks! =)

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