Table of Contents
Standing Waves - PHYS143 (2025)
In this lab you will use the physics of standing waves and how waves propagate to make measurements of the speed of sound in air and the linear mass density $\rho$ of a vibrating string. You will also investigate the behavior of standing waves on the strings of a Ukulele.
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Theory
For completeness we provide a summary here of the principles of Standing and Traveling waves, and Resonance. Most likely you have already seen this material and can skip to the Experiments section.
Traveling waves and standing waves
Traveling waves
A wave is a disturbance (usually in some medium) which travels away from its source. Some familiar examples are sound, water, and string waves. A transverse wave is a wave in which the disturbance causes a momentary displacement in the medium in a direction normal to the direction in which the wave propagates (for example, a water wave). A longitudinal wave is a compression wave in which the disturbance causes a momentary displacement of the medium along the same direction in which the wave propagates (for example, a sound wave).
A traveling wave takes the mathematical form
| $y_1 = A\sin (kx - \omega t)$, | (1) |
where $A$ is the amplitude, $\omega = 2\pi f$ is the angular frequency (with units of radians/sec), $f$ is the normal frequency (with units of hertz), $k = 2\pi/\lambda$ is the wavenumber (in units of radians/meter) and $\lambda$ is the wavelength (in units of meters).
Such a wave is called a traveling wave because any point on the wave will move along at fixed amplitude with speed given by
| $v=\lambda f = \omega /k$. | (2) |
(Think, for example, of a surfer riding the crest of a wave as it comes into shore.)
Standing waves
Imagine now that we have a wave which is identical but is traveling in the opposite direction. Such a wave has mathematical form
| $y_2 = A\sin (kx + \omega t)$. | (3) |
By the superposition principle, if these two waves travel along the same body, the resulting wave is the sum of the two waves,
| $y = y_1 + y_2 = y = 2A\sin(kx)\cos(\omega t)$, | (4) |
where we made use of the trigonometric identity
| $\sin A + \sin B = 2 \sin \left(\dfrac{1}{2}(A+B)\right)\cos \left(\dfrac{1}{2}(A-B)\right)$. |
This sort of wave – Eq. (4) – is called a standing wave. Notice that at any fixed position $x$, the string undergoes simple harmonic motion as time goes on. Notice also that all points along the wave oscillate at the same frequency. The amplitude, however, depends on the position $x$. This characteristic is quite different from a traveling wave in which the amplitudes of all points along the wave are equal.
Resonance
Nodes and anti-nodes
By Eq. (4), the amplitude of a standing wave is a maximum when
| $kx = \dfrac{\pi}{2},\dfrac{3\pi}{2},\dfrac{5\pi}{2},\dots$ |
Since $k = 2\pi/\lambda$, this corresponds to
| $x = \dfrac{\lambda}{4},\dfrac{3\lambda}{4},\dfrac{5\lambda}{4},\dots$ | (5) |
These positions of maximum amplitude are called anti-nodes.
Similarly, there are positions along the wave where the amplitude equals zero, namely where
| $kx = 0,\pi ,3\pi ,\dots$ |
or
| $x = 0,\dfrac{\lambda}{2},\lambda ,\dfrac{3\lambda}{2},\dots$ | (6) |
Resonant frequencies (general)
A system can be caused to vibrate with a relatively large amplitude if it is forced to oscillate at or near one of its “natural” frequencies. This phenomenon is called resonance. Let's look at several specific systems.
For resonance to occur on a string fixed at both ends, there may be any number of nodes along the string, but the endpoints must be nodes. (See Fig. 1.) Equation (6) states that the distance between adjacent nodes is $\lambda/2$. Therefore, a string of length $L$ vibrating at resonance must contain an integer multiple of half wavelengths,
| $ \dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}.$ |
Using the relation $f = v/\lambda$, we may re-write this as
| $f_n = \dfrac{v}{\lambda_n}=\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$ | (7) |
 |
| Figure 1: Standing waves on a string with both ends fixed. (Click here to see an animated version.) |
For a string with one end fixed and the other end free, the fixed end must be a node and resonances will occur when the free end is an anti-node. (See Fig. 2.) Here, the string (or air column) length must contain an odd integer multiplier of quarter-wavelengths,
| $\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$ |
such that the resonant frequencies will be
| $f_n = \dfrac{v}{\lambda_n} = \dfrac{nv}{4L} \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$. | (8) |
 |
| Figure 2: Standing waves on a string with one free and one fixed end. (Click here to see an animated version.) |
If both ends of the string are free, resonance will occur when there are anti-nodes at both ends. (See Fig. 3.) At resonance, the string length will contain integer multiples of half wavelengths,
| $\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$ |
with resonant frequencies
| $f_n = \dfrac{v}{\lambda_n}=\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (9) |
This is the same result as Eq. (7) for a string with both ends fixed.
 |
| Figure 3: Standing waves on a string with both ends free. (Click here to see an animated version.) |
Experiments
You will do experiments involving transverse and longitudinal waves on a Vibrating Spring, a Vibrating String, and Sound Waves traveling in a Tube. There are several setups of each experiment in the lab and you will have to rotate from one to another through out the lab period. It does not matter in which order you do them.
Vibrating Spring
A spring can be caused to vibrate with a relatively large amplitude if it is forced to oscillate at or near one of its “natural” frequencies. This phenomenon is called resonance.
For resonance to occur on a spring when both ends are fixed, the endpoints must be nodes. (See Fig. 1.) A spring of length L vibrating at resonance must contain an integer multiple of half wavelengths,
$\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$.
| |
| Figure 1: Standing waves on a string with both ends fixed. |
Using the relation $f = v/\lambda$, we may write
| $f_n = \dfrac{v}{\lambda_n}=\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (7) |
The speed of propagation of a wave $v$ in general is given by,
| $v = \sqrt{\dfrac{\textrm{“restoring force” factor}}{\textrm{“inertial” factor}}}$. | (10) |
For a stretched spring, it becomes
| $v_{spring} = \sqrt{\dfrac{kL}{\rho}}$, | (11a) |
where k is the spring constant, L is the length of the stretched spring and $\rho$ is the mass per unit length of the stretched spring.
Use your understanding of standing waves to predict the resonant frequencies of the first three vibrational modes for a vibrating spring which is fixed at both ends, and then experimentally test your predictions.
- Identify your measured quantities and determine how you will estimate their uncertainties.
- Setup the apparatus and calculate the expected frequencies for the first three vibrational modes.
- Use the apparatus to experimentally determine the frequencies of the first three vibrational modes.
Vibrating String
The physics of a vibrating String with both ends fixed is similar to that of the vibrating Spring with both ends fixed. The difference being the form for the velocity of a wave propagating along the string which is given below.
For a transverse wave along a string, this becomes,
| $v_{string} = \sqrt{\dfrac{T\vphantom{\large{T}}}{\rho}}$, | (11b) |
where $T$ is the tension in the string and $\rho$ (rho) is the mass per unit length of the stretched string.
Use your understanding of standing waves to predict the resonant frequencies of the first three vibrational modes for a vibrating string which is fixed at both ends, and then experimentally test your predictions.
- Identify your measured quantities and determine how you will estimate their uncertainties.
- Setup the apparatus and calculate the expected frequencies for the first three vibrational modes.
- Use the apparatus to experimentally determine the frequencies of the first three vibrational modes.
Sound Tube
What we perceive as sound are pressure waves propagating through the air. This is an example of a longitudinal wave as opposed to the transverse wave of the vibrating string. By measuring the wavelength ($\lambda$) of the standing wave created by sound, of a known frequncy, propagating through a tube as a function of frequency one can determine the speed of sound in air.
The resonant frequencies for an air column with one end closed and the other end open is
| $f_n = \dfrac{n v}{4L} \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$. | (16) |
The resonant frequencies for an air column with both ends open is
| $f_n = \dfrac{n v}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (17) |
Velocity of Sound - Standing Waves
A diagram of the apparatus is shown in figure (3).
Figure 3. Diagram of the apparatus used for the sound tube experiment.
It consists of a loudspeaker driven by a function generator as a source of sound waves. A hollow plastic tube with a plunger that can positioned anywhere inside the tube. And a sound pressure level (SPL) meter to measure the pressure of the sound waves which can also be displayed on an oscilloscope. The components are mounted on a long rigid rail.
The speed of propagation of a sound wave in air is
| $v_{gas} = \sqrt{\dfrac{\gamma k_BT}{m}}$, | (18) |
where $\gamma$ is the ratio of specific heats of the gas in which the wave is moving ($\gamma = 1.4$ for a diatomic gas), $k_B = 1.38 \times 10^{-23}$ J/K is the Boltzmann constant, T is the absolute temperature of the gas (in Kelvin), and m is the mass of an individual molecule of gas. (The average mass of an air molecule is about $m = 4.8 \times 10^{-26}$ kg.)
Use the apparatus and your knowledge of standing waves to measure the speed of sound in the lab
- Identify your measured quantities and determine how you will estimate their uncertainties.
- Use the apparatus to measure the speed of sound at two different frequencies in the range 1kHz to 5kHz. Your two measurements should be at least 500Hz apart.
- Don't forget to take screen shots of the scope so that you can include them in your report to help you describe what you measured and how you performed the measurements.
Do you observe a significant dependence of the speed of sound on wavelength?
Compare your measured values with the calculated value assuming the temperature in the lab is 300K. Note that the speed of sound in air varies with barometric pressure among other factors. As such the literature value is approximate, so do not expect precise agreement.
Ukulele - OPTIONAL
This part is optional, there is no grade component associated with it.
Please handle the Ukulele with care. It is an inexpensive instrument and easily damaged. Do not play with them before reaching this part of the lab, after you have done both of the previous parts. Return them to the TA before you leave the lab at the end of the period.
If you play a musical instrument or even just enjoy listening to music, you are already familiar with one manifestation of waves in everyday life. What your brain interprets as sound is the result of vibrations of the eardrum in response to transverse pressure waves, such as the ones you worked with in the sound tube part of this lab. These pressure waves were ultimately produced by either the human voice or musical instruments.
Stringed instruments such as cellos, violins, guitars and Ukulele's are directly analogous to the vibrating string which you worked with in this lab. They are strung with strings whose length, linear mass density and tension have been chosen to allow you to tune each string to the appropriate note. Different notes can be played by using your finger to press a string against the neck of the instrument, thus shortening the length over which it is able to vibrate which in turn raises the resonant frequency.
As a mostly fun exercise there are Ukulele's which you can use to measure the resonant frequencies of the different strings. With a little bit of practice you can use the sound pressure level meters, at the sound tube apparatus, to record the waveform of a plucked string on the scope. Two things to note.
- Triggering the scope on the waveform can be tricky. Make sure that the sensitivity of the sound level meter is set so that the needle does not peg at the top of the scale, this will result in a clipped waveform. You will also have to set the scope to a time base which is comparable to the frequency you are looking to measure. For reference the frequency of middle C is ~262Hz.
- Plucked strings do not vibrate at a single frequency. You will note that the waveform of the signal from the instrument is not a clean sine wave. There are likely to be 2 or even 3 frequencies present, you will explore this phenomena more in the lab on supper position and interference. Additionally, plucking the string too hard will introduce distortion which gives a very messy sound spectrum. It may take some practice to pluck a string cleanly.
Report: Summary and Conclusions
After the lab, you will need to write up your conclusions. This should be a separate document, and it should be done individually (though you may talk your group members or ask questions).
The conclusion is your interpretation and discussion of your data. What do your data tell you? How do your data match the model (or models) you were comparing against, or to your expectations in general? Your conclusions should always be based on the results of your work in the lab. It is not acceptable to evaluate the results of an experiment by comparison to known values or any other form of preconceived expectation. Your conclusions need to be supported by your data. If your data are inconclusive or in disagreement with regard to your expectations then your conclusion should reflect that.
Make sure you cover the following points in your report.
Standing Waves On String and Spring
- Describe how you configured and used each apparatus. Figures and diagrams are helpful here. [EP][SC]
- Describe how you estimated the uncertainty in each of your measured quantities. [EP]
- Show all of the calculations involved in predicting the frequencies for the first three vibrational modes, including propagation of uncertainties. [DA]
- Present your data appropriately. [SC]
- Present your measured results appropriately. [SC]
- Provide your conclusions and how your data support them. [DC][SC]
Speed of Sound
- Describe your measured quantities and how you determined their uncertainties for both methods, standing waves and time of flight. [EP][SC]
- Present your data appropriately. [SC]
- Show your calculations for the speed of sound, including propagation of uncertainties. [DA]
- Present your conclusions and show how they are supported by your data. [DC][SC]
REMINDER: Your report is due 48 hours after the lab. Submit a single PDF on Canvas.