Standing Waves in a String
Name: _________________________
In a transverse wave, the pulse represents the energy that is transmitted in a medium that vibrates at right
angles to the propagation of the pulse. A vibrating string is a convenient medium to study a transverse
wave. When a stretched cord or sting is disturbed, the wave travels along the string with a speed that is
dependent on the tension of the string and its linear density. Upon reaching a fixed end of the string, the
pulse is reflected and, if in proper phase with the incoming pulse, will superimpose constructively and form
a standing wave pattern. We say that this system is in resonance if it vibrates at its natural vibrating
frequency. A medium can also vibrate at multiples of its fundamental vibrating frequency = harmonics.
The generic expression for each harmonic is given by the equation L = nλ/2 or λ = 2L/n where (L) is the
length of the string, (n) is the name of the harmonic (also the number of anti-nodes), and (λ) is the
wavelength of the standing wave.
The velocity (v), or wave speed, of the pulse depends on the tension and linear density of the string. This is
represented by the expression: v = √(T/u) where (T) represents the tension and (u) the density. Using the
fundamental wave equation v = f*λ and makings some substitutions yields the following equation:
𝒇=
𝒏
𝟐𝑳
∗√
𝑻
This is the equation that we intend to investigate in this laboratory exercise.
𝒖
Procedures: -- (BE CAREFUL with your UNITS)
A. Attach a piece of string to the mechanical drive (vibrator) and let the other end hang over a pulley, letting
a mass of 50 g provide the tension. Use an effective string length (vibrator to pulley) of about 0.80 m.
Please remember that the tension (T) is a force (weight) and can be determined with the following
equation: T = W = m*g (BE CAREFUL with your UNITS)
1) Measured length (L), in meters, of string between the vibrator and pulley ___________
2) Tension (T) in the string ___________
B. Turn on the vibrator and adjust the frequency to about 10-14 vib/sec. Fine-tune the frequency to obtain a
standing wave of a single loop (this is the fundamental frequency = 1st harmonic = a single anti-node).
3) Fundamental frequency (f) where (n=1) ___________ (record this information in the table – Step F)
C. Using the equations above, calculate the linear density (u). NOTE: linear density is the mass per unit
length (kg/m) which is different from the density of a substance which is mass per unit volume (kg/m3).
HINT: re-write the equation to solve for (u). This question is a test of your algebra skills 
4) Linear density (u) ____________ (record this information in the table – Step F)
D. Your instructor will provide you with a measured value of the linear density (u) = ______________
E. Calculate the percent error between measured and calculated (experimental) values for the linear density.
5) Percent error ___________
F. Adjust the frequency of the vibrator to produce the 2nd, 3rd, 4th, and 5th harmonic (n) and complete the
table below. For this table, the 50 g mass should still be used to provide the tension (T) = ________
(Note: you may need to adjust the amplitude of the vibrator to see the best standing wave)
Harmonic (n)
# of anti-nodes
Frequency (f)
(Hz)
Wavelength (λ)
(m)
Velocity (v)
(m/s)
(OPTIONAL)
linear density (u)
1
2
3
4
5
G. Next, investigate how the tension (T) of a string affects the frequency (f) of the string when the tension is
changed – by using different hanging masses (40 g, 50 g, 60 g, 70 g). Complete the table below:
--Select which harmonic you are going to investigate. n = ____________
--What is the wavelength for this chosen harmonic? _______ Will this wavelength change? ______
Mass
(kg)
Tension (T)
(N)
Frequency (f)
(Hz)
Wavelength (λ)
(m)
Velocity (v)
(m/s)
.040
.050
.060
.070
POST LAB QUESTIONS – Answer the following questions on a separate sheet of paper.
6)
7)
8)
9)
In the table under step F, how did the frequency change between each harmonic?
In the table under step F, how did the wavelength change between each harmonic?
In the table under step F, how did the velocity change between each harmonic?
EXPLAIN your answer regarding the velocity (#8 above).
10) In the table under step G, how did the frequency change when the tension was changed?
11) EXPLAIN why the velocity changed as the tension changed.
Stringed musical instruments produce musical sounds with stretched strings. Explain and support your
answers with the appropriate formulas (you must include formulas to earn full credit)
12) How can the tension of the strings be used to raise or lower the pitch (frequency) when tuning?
13) Why are the strings of lower frequencies thicker or heavier?
14) Why does the pitch (frequency) change when a player places their fingers on the different frets
(effectively changing the length of the string)?
EXTRA CREDIT: Show that √(T/u) is in units of velocity.