Physics 2225 – Standing Waves
Minilab 1
Standing Waves
Department of Physics & Astronomy
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Physics 2225 – Standing Waves
Today, we will observe standing waves on a
string in order to learn and verify how the
formation of standing waves depend on:
• Excitation Frequency
• Tension of the string
• Linear mass density of the string
PURPOSE
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Physics 2225 – Standing Waves
When a wave moves along a string, we say the
wave is propagating along the string, as shown.
• The linear mass density is related using:
𝒎
𝝁=
𝒍
• The density and the tension of the string (T)
affect the velocity that it propagates at:
𝒗=
𝑻
=
𝝁
𝑻
𝒎
𝒍
v
THEORY
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Physics 2225 – Standing Waves
Note that waves reflections depend on how the
string is attached at one end.
• End of string is fixed, the wave gets inverted
•
End of string is loose, the wave is not inverted
THEORY
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Physics 2225 – Standing Waves
We use the term superimposed to mean two
waves that are overlapping. Below, these two
waves are travelling in opposite directions.
Moving to right
Moving to left
The sum of the
two waves
(“superposition”)
THEORY
Nodes
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Anti-Nodes
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Physics 2225 – Standing Waves
If the length remains unchanged, standing
waves only occur at specific frequencies.
• In our case, we have strings with nodes at
both ends, which produces the following:
l/2
l
3l/2
THEORY
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Physics 2225 – Standing Waves
Node on top of the pulley wheel
Node somewhere beyond the wave driver
Mass creates tension
in string: T = mg
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Mechanical Wave Driver
creates waves
(Frequency and Amplitude
controlled by Capstone Software)
EQUIPMENT
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Physics 2225 – Standing Waves
EXPERIMENTAL DETERMINATION OF SPEED OF WAVE
The velocity of the wave can be calculated as follows
V=f l
(or f  v 
1
)
l
(read off frequency in Capstone Software, measure l when you see a
standing wave pattern).
Start from low frequency and observe several different standing waves
(different f and l).
Plot f versus 1/ l  The slope of this graph equals v.
Repeat the procedure using a different tension in the string (use a different
mass at the end of the string). V should be different because it depends on the
tension T.
PROCEDURE
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Physics 2225 – Standing Waves
• Once you have collected your data, you will
need to plot f versus 1/λ in Excel. The slope
of your graph is equal to v.
If you are still having struggles with plotting in
Excel, please refer to the Excel Tutorial online,
or make sure your lab partner can explain it to
you!
PROCEDURE
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Physics 2225 – Standing Waves
Theoretical Determination of the Speed of the Wave
Theoretica l Speed :
Important:

However:
v
T
 streched
The string is elastic and gets stretched under tension.
 stretched  unstretched
It is easier to determine unstretched
(munstretched ).
Measure the full length of the un-stretched string with a ruler (lunstretched ).
1) Put the loose string on the scale and measure the mass
2)
3) Calculate
unstretched
munstretched

lunstretched
PROCEDURE
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Physics 2225 – Standing Waves
Next, you need to determine how unstretched and  stretched are related.
Imagine two situations:
1) Stretched string (with tension) in our setup (no mass hanging on
string):
Lstretched , M
Mass of the part of string between the pulley and the rod: M
Length of string between the pulley and the rod: Lstretched
 stretched 
PROCEDURE
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M
Lstretched
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Physics 2225 – Standing Waves
Take off the mass to release the tension  the string shrinks a bit.
2) Un-stretched string (no tension) in our setup (no mass hanging on
string):
Lunstretched , M
Mass of the same string portion:
M
Length of the same string portion:
Lunstretched
unstretched 
M
Lunstretched
PROCEDURE
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Physics 2225 – Standing Waves
So, we have two equations relating to these two situations:
 stretched 
M
unstretched 
Lstretched
M
Lunstretched
We can combine these and eliminate M (the mass of that string portion):
 stretched  unstretched 
Lunstretched
Lstretched
Then we can get an equation for the speed of the wave of the stretched
string
v
T
 streched

T
unstreched
Lstretched
Lunstretched
PROCEDURE
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Physics 2225 – Standing Waves
To compare the theoretical and measured
velocities, use the % difference calculation:
vmeasured  vtheoretical
% difference 
vtheoretical
PROCEDURE
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Physics 2225 – Standing Waves
Homework Policies
•
You must do your homework BEFORE CLASS, and
everyone must turn in their own work.
Lab Report Policies
•
•
Submit one lab report per group. Groups consist of
two or three people.
Make sure all members of the group write their
name on the lab report!
FINAL HINTS
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