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Learn about mechanical waves. Study the relations between wave length frequency and speed of wave propagation.

Introduction:

A
wave is a disturbance that moves through a medium or through space carrying
energy and momentum. Waves are classified according to type of energy into mechanical
waves, that carries mechanical energy, and electromagnetic waves that carries
electric and magnetic energies.

Mechanical waves require

– Some source of disturbance.

– A medium that can be disturbed.

– Some physical connection or mechanism
though which adjacent portions of the medium influence each other

A
pulse is a non-repeated disturbance that carries energy through a medium or
through space. If the pulse is repeated periodically, then a series of crests
and troughs will travel through a medium creating a travelling wave.

A
transverse wave is a disturbance in a medium in which the motion of the motion
of the particles is perpendicular to the direction of the wave motion (e.g.
waves in a string). A longitudinal wave is a disturbance in a medium in which
the motion of the particle is along the direction of the wave travel. For
example, Sound.

Mathematical Description of a wave:

A waveform can be represented by a sine
functions in space (x) and time (t) as follow:

y(x,t)=Asin(kx-wt) = Asin(kx-2pft)

where y(x,t) is the wave displacement at
a given point defined by its position ,x, at a time ,t. A is Amplitude, k is
the wave number,w is
angular frequency and f is frequency.

w=2p/T,
k=2p/l,
T=1/f, wave speed (v) =l/T =lf

The quantities describing a wave are
summarized as:

·
Amplitude (A) of a wave is the maximum displacement
of any part of the wave from its equilibrium or rest position.

·
Wavelength (l) is the
measured distance for one complete cycle of the wave.

·
Period (T) is the time required for a single wave to
pass a given point.

·
Frequency (f) is the number of complete waves
passing a given point per unit time.

Waves in a string is an example of
transverse mechanical wave. A vibrating string, such as a guitar or piano
string also produces a sound whose frequency in most cases is constant.

The speed of propagation of
a wave in a string (v) is proportional to the square root of the tension of the
string (FT) and inversely proportional to the square root of the
linear mass density (m) of the
string:

Standing waves on a string:

The index n on the wavelength (or
frequency) indicates the harmonic. That is, n=1 identifies the first harmonic
(also called the fundamental frequency), n=2labels the second harmonic, and so
on.

Simulation:

Open Waves

Procedure:

1. Apply the following settings to the
simulation: Set Wave to Oscillate,
End type to No End, Amplitude
to 60, Frequency to 40, Damping to 0, and Tension to High.Make sure to check the Rulers and Timer boxes.

a. Run the simulation and observe how the
wave behaves.

b. Click the pause/play button to stop the simulation.

c. Measure the wavelength. Wavelength (l) = ______ cm

d. Measure the amplitude. It is measured
from the middle reference line to a crest (or a trough). You can drag the
reference line to the position of a crest or a trough then measure between
these reference lines. Amplitude (A) = ______ cm

e. With the simulation paused, reset the
timer and use the step button to determine the period of the wave. Make sure to
step the time from a crest to the next crest. This can easily be one using the
ruler as a reference.

Period (T) = ____ s

f.
Calculate
the frequency. Frequency (f) = ____ Hz

g. Calculate the speed of the wave. Speed (v) = ____ m/s

h. Assuming the “High” position of the
tension as 1N, calculate the linear mass density (m) of the string. m = ____ kg/m

i.
Using
the information of the wavelength you found, calculate the original length of
the string. Length (L) = _____ m

2. Apply the following settings to the
simulation: Leave all settings the same as in step 1 except the tension to 0.9
N (remember, High is 1N).

a. Wavelength (l) = _____ cm

b. Period (T) = _____ s

c. Speed (v) = _____m/s

d. Calculate the tension using the linear
density found in step1h. Tension (T) = ___N

e. In order to recover the wavelength found
in step 1c, you need to change the frequency slide bar. What will be the new
position (value) of the frequency slide bar?
_______

f.

3. Apply the following settings to the
simulation: Leave all settings as step 1 except the Amplitude to 80.

a. Wavelength (l) = _____ cm

b. Period (T) = _____ s

c. Speed (v) = _____m/s

d. Calculate the tension using the linear
density found in step1h. Tension (T) = ___N

e. Which one of the above quantity changed?

4. Can you tell about the setup used from
the following graph? Describe all the settings you think applied to get this
graph. The period of the wave is 1:11 sec.

5. Apply the following settings to the new
simulation: Click Reset button, Set Wave to Pulse, End type to Fixed
End
, Amplitude to 60, Frequency
to 40, Damping to 0, and Tension to High.

a. Click the pulse button and observe the
pulse moving through the string.

b. Describe what the wave does as it moves
down the string. After it hits the end of the string, what happens to the wave?

6. Click the Reset button. Keep the same
settings as in step 5 except set End Type to Loose End

a. Click the pulse button and observe the
pulse moving through the string.

b. Describe what the wave does as it moves
down the string. After it hits the end of the string, what happens to the wave?

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