The standing-wave-on-a-string experiment is a physics experiment in which a string fixed at both ends is vibrated, the positions of nodes and antinodes that appear at resonance are observed, and the relationships among wavelength, frequency, wave speed, tension, and linear density are investigated.
When waves reflected at both ends of the string overlap, a standing wave with a large amplitude is produced at specific frequencies. In a standing wave, nearly stationary nodes and strongly vibrating antinodes appear at fixed positions.
In an experimental report, it is important not only to record the resonance frequencies and number of antinodes, but also to discuss the relationship between string length and wavelength, wave speed, the relationship between tension and frequency, linear density, end correction, string sag, and how resonance was identified.
This article assumes an experiment in which one end of a string connected to a vibration generator is fixed and tension is applied to the other end using a pulley and weights. It summarizes the items to record in the results, reference experimental values, calculation methods, sources of error, example results, and discussion examples that can be used in a report.
Note:
This article is a study reference for discussing results obtained in physics experiments at universities and similar institutions. The experimental values shown are examples for explanation and are not actual measurement results.
For the operation and safety precautions of the vibration generator, string, pulley, weights, power supply, and other equipment, always follow your university’s laboratory manual and the instructions of your instructor or TA. Do not carelessly touch a vibrating string or weights that could fall.
What Is a Standing Wave on a String?
When a string is vibrated at a constant frequency, a wave travels along the string and is reflected at a fixed end or other boundary. When the incident and reflected waves overlap under suitable conditions, a standing wave is formed.
For a string whose two ends are nodes, if the effective length of the string is L, the wavelength is λ, and the number of antinodes is n, the following relationship holds.
L = nλ / 2
Therefore, the wavelength is obtained from:
λ = 2L / n
If the wave speed is v and the frequency is f, the fundamental wave equation is:
v = fλ
If the tension in the string is T and the linear density, or mass per unit length, is μ, the speed of a wave traveling along the string is:
v = √(T/μ)
Combining these equations, the nth resonance frequency of a string fixed at both ends is:
fn = n / (2L) × √(T/μ)
The fundamental mode has one antinode, the second mode has two, and the third mode has three. Ideally, the resonance frequencies are integer multiples of the fundamental frequency.
Main Items to Record in the Results
- Material and thickness of the string
- Total length of the string
- Effective length over which the standing wave forms
- Mass of the string
- Linear density of the string
- Total mass of the weights and weight holder
- Tension applied to the string
- Frequency of the vibration generator
- Number of antinodes at resonance
- Positions of nodes
- Distance between nodes
- Wavelength
- Wave speed
- Theoretical wave speed
- Difference between experimental and theoretical values
- Relative error
- Frequency ratios of the fundamental and higher harmonics
- Resonance condition at which the amplitude is maximum
The distance between adjacent nodes corresponds to half a wavelength. Measuring several node intervals together and dividing the distance by the number of intervals can reduce reading error compared with measuring only one interval.
Reference Experimental Conditions
| Item | Reference condition |
|---|---|
| Effective string length L | 1.20 m |
| Mass of string | 1.80 g |
| Length of string used for mass measurement | 2.00 m |
| Linear density μ | 9.00 × 10-4 kg/m |
| Total mass of weights and holder | 0.200 kg |
| Gravitational acceleration | 9.80 m/s2 |
| Tension T | 1.96 N |
| Modes measured | Fundamental through fourth mode |
Reference Experimental Values
The following are reference values for study measured with a string of effective length 1.20 m, tension 1.96 N, and linear density 9.00 × 10-4 kg/m.
| Mode number n | Number of antinodes | Resonance frequency f (Hz) | Wavelength λ (m) | Experimental wave speed v = fλ (m/s) |
|---|---|---|---|---|
| 1 | 1 | 19.3 | 2.40 | 46.3 |
| 2 | 2 | 38.8 | 1.20 | 46.6 |
| 3 | 3 | 58.0 | 0.800 | 46.4 |
| 4 | 4 | 77.5 | 0.600 | 46.5 |
In these reference values, the resonance frequency increases approximately as an integer multiple as the mode number increases, while the wave speeds calculated for the different modes remain nearly constant at about 46.3–46.6 m/s.
Reference Values When Tension Is Varied
The following reference values use an effective length of 1.20 m, mode number 2, and linear density of 9.00 × 10-4 kg/m while varying the tension.
| Total mass (kg) | Tension T (N) | Resonance frequency f (Hz) | √T (N1/2) |
|---|---|---|---|
| 0.100 | 0.980 | 27.4 | 0.990 |
| 0.150 | 1.47 | 33.6 | 1.21 |
| 0.200 | 1.96 | 38.8 | 1.40 |
| 0.250 | 2.45 | 43.3 | 1.57 |
| 0.300 | 2.94 | 47.5 | 1.71 |
The resonance frequency is approximately proportional not to the tension itself, but to the square root of the tension.
Calculation Methods
Linear Density
If a 2.00 m length of string has a mass of 1.80 g, first convert the mass to kilograms.
1.80 g = 1.80 × 10-3 kg
The linear density is obtained by dividing the mass of the string by its length.
μ = m / l
= 1.80 × 10-3 / 2.00
= 9.00 × 10-4 kg/m
String Tension
If the total mass of the weights and holder is 0.200 kg, and pulley friction is neglected, the tension in the string is:
T = Mg
= 0.200 × 9.80
= 1.96 N
Wavelength
If three antinodes are formed on a string with an effective length of 1.20 m, the wavelength is:
λ = 2L / n
= 2 × 1.20 / 3
= 0.800 m
Wave Speed from Experimental Values
If the resonance frequency of the third mode is 58.0 Hz, the wave speed is:
v = fλ
= 58.0 × 0.800
= 46.4 m/s
Theoretical Wave Speed from Tension and Linear Density
v = √(T/μ)
= √(1.96 / 9.00 × 10-4)
= √2178
= 46.7 m/s
Relative Error
For an experimental wave speed of 46.4 m/s and a theoretical wave speed of 46.7 m/s, the relative error is:
Relative error (%) = |experimental value − theoretical value| / theoretical value × 100
= |46.4 − 46.7| / 46.7 × 100
= approximately 0.64%
Theoretical Resonance Frequency
The theoretical resonance frequency of the third mode is calculated as follows.
fn = n / (2L) × √(T/μ)
= 3 / (2 × 1.20) × 46.7
= 58.4 Hz
The experimental value of 58.0 Hz is close to the theoretical value of 58.4 Hz.
Ratio to the Fundamental Frequency
If the fundamental frequency is 19.3 Hz and the fourth-mode frequency is 77.5 Hz, the frequency ratio is:
f4 / f1 = 77.5 / 19.3
= approximately 4.02
Because this is close to the ideal integer ratio of 4, the fourth-mode standing wave can be identified as having formed.
Main Sources of Error
| Source of error | Effect on measurement | Likely tendency in results | Improvement/check |
|---|---|---|---|
| Error reading the effective string length | Affects calculations of wavelength and theoretical frequency | Wave speeds for all modes shift in the same direction | Clearly identify the vibration point and contact point |
| Error identifying the resonance frequency | The frequency of maximum amplitude may not be selected correctly | Frequency shifts higher or lower | Change frequency in fine increments to locate maximum amplitude |
| Pulley friction | The weight’s gravitational force and string tension are not identical | The theoretical tension may be overestimated | Use a low-friction pulley and check its rotation |
| String sag or stretching | Tension and effective length change over time | Resonance frequency becomes unstable | Arrange the string before measurement and wait for tension to stabilize |
| Nonuniform linear density | Wave speed varies along the string | Node spacing becomes uneven | Use a string of uniform thickness |
| Error measuring string mass | Calculated linear density is incorrect | Theoretical wave speed and frequency shift systematically | Precisely measure a sufficiently long piece of string |
| Unclear node positions | Makes wavelength difficult to determine accurately | Scatter increases for higher modes | Measure several intervals together |
| Amplitude too large | String tension can vary during vibration | Resonance frequency deviates from theory | Keep amplitude small enough to observe the waveform clearly |
| Generator end is not a perfect node | Deviates from the ideal fixed-end condition | End correction may be required for the effective length | Measure effective length from the actual node positions |
| Weights oscillate vertically | String tension varies with time | The standing wave becomes unstable | Wait until the weights are stationary before adjusting frequency |
Why the Experimental Wave Speed May Not Match the Theoretical Value
The theoretical equation assumes that the string is uniform and flexible, the tension is constant, and both ends are perfect nodes. In practice, however, there is pulley friction, stretching of the string, motion at the vibration-generator end, and nonuniformity in linear density.
In particular, treating the tension simply as the gravitational force Mg of the hanging mass neglects pulley friction and slight motion of the weights. These effects can produce a difference between theoretical and experimental values.
Why Errors Tend to Be Larger for Higher Modes
In higher modes, the distance between adjacent nodes becomes shorter, so the relative error in reading node positions becomes larger. Small nonuniformities in the string and motion of the vibration generator also become more noticeable relative to the shorter wavelength.
However, when wavelength is calculated from the full string length and the number of antinodes, using the entire string length can sometimes reduce error compared with measuring individual node intervals.
Why Frequency Is Proportional to the Square Root of Tension
The speed of a wave on a string is v = √(T/μ), and if the string length and mode number are fixed, the wavelength is also fixed.
From the basic wave equation v = fλ, frequency is proportional to wave speed. Therefore, resonance frequency is proportional to the square root of tension, and a graph of f versus √T is a straight line passing near the origin.
Why Frequency Is Proportional to Mode Number
For a string fixed at both ends, a standing wave forms only when an integer number of half-wavelengths fits within the string length. As the mode number increases by one, one additional half-wavelength fits into the same string length, making the wavelength shorter.
If tension and linear density remain constant, wave speed is constant. Therefore, when wavelength becomes 1/n as large, frequency becomes n times as large.
Example of How to Write the Results
A tension of 1.96 N was applied to a string with an effective length of 1.20 m and a linear density of 9.00 × 10-4 kg/m. The frequency of the vibration generator was varied, and the resonance frequencies from the fundamental through the fourth mode were measured.
The resonance frequencies of the fundamental, second, third, and fourth modes were 19.3, 38.8, 58.0, and 77.5 Hz, respectively. The wave speeds calculated for each condition were 46.3–46.6 m/s and were nearly constant.
The theoretical wave speed calculated from the tension and linear density was 46.7 m/s, and the relative error compared with the experimental wave speed of 46.4 m/s obtained from the third mode was approximately 0.64%. The resonance frequency was also approximately proportional to the mode number.
Points to Connect to the Discussion
- Did the relationship between the number of antinodes and wavelength agree with theory?
- Was the wave speed calculated from each mode approximately constant?
- How closely did the experimental and theoretical wave speeds agree?
- Was the resonance frequency proportional to mode number?
- Was the resonance frequency proportional to the square root of tension?
- Was the effective string length measured correctly?
- Could the generator and pulley ends be regarded as nodes?
- Did pulley friction affect the tension?
- Was the string’s linear density uniform?
- Could nodes and antinodes be clearly distinguished?
- Was the vibration amplitude kept from becoming too large?
- Did error increase for higher modes?
Discussion Example
In this experiment, a string under constant tension was vibrated and standing waves from the fundamental through the fourth mode were observed. Resonance frequency, wavelength, and wave speed were determined and compared with theoretical values.
The measured resonance frequencies, starting with the fundamental, were 19.3, 38.8, 58.0, and 77.5 Hz. Dividing these by the fundamental frequency gives ratios close to the integers 1, 2, 3, and 4. This confirms that for a string with nodes at both ends, resonance frequencies proportional to the mode number appear.
For an effective string length of 1.20 m, the wavelengths from the first through fourth modes were 2.40, 1.20, 0.800, and 0.600 m, respectively. The wave speeds calculated from each resonance frequency and wavelength were 46.3–46.6 m/s and remained nearly constant despite the change in mode number.
This is because, for the same string with constant tension and linear density, the speed of waves traveling along the string is constant. As the mode number increases, the wavelength becomes shorter, but the resonance frequency increases correspondingly, so the product fλ remains nearly constant.
The theoretical wave speed calculated from a tension of 1.96 N and a linear density of 9.00 × 10-4 kg/m was 46.7 m/s. The relative error compared with the experimental value of 46.4 m/s obtained from the third mode was approximately 0.64%. The close agreement supports the theory that wave speed on a string is determined by tension and linear density.
Possible reasons why the experimental value did not exactly match the theoretical value include errors in reading the effective string length, pulley friction, stretching of the string, nonuniform linear density, and error in identifying the resonance frequency.
In particular, the theoretical calculation treats the string tension as the gravitational force Mg of the hanging mass. In practice, however, the pulley has friction, so the tensions on the two sides may not be exactly equal. In addition, the end at the vibration generator is not perfectly fixed and moves slightly, so the actual effective length may differ from the measured length.
For higher modes, the node spacing becomes shorter, so error in reading node positions represents a larger fraction of the wavelength. Small variations in string thickness or sag also become more significant at shorter wavelengths.
When tension was varied, the resonance frequency was approximately proportional not to the tension itself but to its square root. This is because wave speed is proportional to √(T/μ), while the wavelength remains constant when string length and mode number are fixed.
To improve measurement accuracy, the frequency should be varied in fine increments to locate the point of maximum amplitude, and measurements should be made when clear nodes and antinodes are visible along the entire string. It is also useful to check that the pulley rotates smoothly and to wait until string sag and oscillation of the weights have settled.
Overall, the experiment confirmed that a standing wave forms on a string fixed at both ends when an integer number of half-wavelengths fits within the string length, and that resonance frequency is proportional to both the mode number and the square root of the tension.
Discussion Example When the Experimental Wave Speed Is Lower Than the Theoretical Value
If the experimental wave speed is lower than the theoretical value, one possible cause is that pulley friction made the actual tension smaller than the gravitational force of the hanging mass. In addition, if the effective length was measured longer than its actual value or the resonance frequency was read too low, the wave speed calculated from v = fλ would also be smaller.
Discussion Example When Resonance Frequencies Deviate from Integer Multiples
If the resonance frequencies deviate from integer multiples of the fundamental frequency, the generator or pulley end may not have acted as a perfect node, causing the effective length to vary slightly with mode number. Variations in tension, nonuniformity of the string, and error in identifying resonance points can also cause deviations in the frequency ratios.
Summary
In a standing-wave-on-a-string experiment, the effective string length, tension, linear density, resonance frequency, and number of antinodes are measured to determine wavelength and wave speed.
For a string fixed at both ends, a standing wave forms when an integer number of half-wavelengths fits within the string length, and the resonance frequency is proportional to the mode number. The wave speed is given by v = √(T/μ), and the resonance frequency is proportional to the square root of the tension.
In the discussion, it is important to relate nodes and antinodes, wavelength, resonance frequency, tension, linear density, pulley friction, effective length, boundary conditions, and reading errors in higher modes.
