16 Waves
16.3 Wave Speed on a Stretched String
Learning Objectives
By the end of this section, you will be able to:
- Determine the factors that affect the speed of a wave on a string
- Write a mathematical expression for the speed of a wave on a string and generalize these concepts for other media
The speed of a wave depends on the characteristics of the medium. For example, in the case of a guitar, the strings vibrate to produce the sound. The speed of the waves on the strings, and the wavelength, determine the frequency of the sound produced. The strings on a guitar have different thickness but may be made of similar material. They have different linear densities, where the linear density is defined as the mass per length,
In this chapter, we consider only string with a constant linear density. If the linear density is constant, then the mass
Wave Speed on a String under Tension
To see how the speed of a wave on a string depends on the tension and the linear density, consider a pulse sent down a taut string (Figure). When the taut string is at rest at the equilibrium position, the tension in the string

If you pluck a string under tension, a transverse wave moves in the positive x-direction, as shown in Figure. The mass element is small but is enlarged in the figure to make it visible. The small mass element oscillates perpendicular to the wave motion as a result of the restoring force provided by the string and does not move in the x-direction. The tension

Assume that the inclination of the displaced string with respect to the horizontal axis is small. The net force on the element of the string, acting parallel to the string, is the sum of the tension in the string and the restoring force. The x-components of the force of tension cancel, so the net force is equal to the sum of the y-components of the force. The magnitude of the x-component of the force is equal to the horizontal force of tension of the string
The net force is on the small mass element can be written as
Using Newton’s second law, the net force is equal to the mass times the acceleration. The linear density of the string
Dividing by
Recall that the linear wave equation is
Therefore,
Solving for v, we see that the speed of the wave on a string depends on the tension and the linear density.
The speed of a pulse or wave on a string under tension can be found with the equation
where
Example
The Wave Speed of a Guitar Spring
On a six-string guitar, the high E string has a linear density of
Strategy
- The speed of the wave can be found from the linear density and the tension
- From the equation
if the linear density is increased by a factor of almost 20, the tension would need to be increased by a factor of 20. - Knowing the velocity and the linear density, the velocity equation can be solved for the force of tension
Solution
- Use the velocity equation to find the speed:
- The tension would need to be increased by a factor of approximately 20. The tension would be slightly less than 1128 N.
- Use the velocity equation to find the actual tension:
This solution is within
of the approximation.
Significance
The standard notes of the six string (high E, B, G, D, A, low E) are tuned to vibrate at the fundamental frequencies (329.63 Hz, 246.94Hz, 196.00Hz, 146.83Hz, 110.00Hz, and 82.41Hz) when plucked. The frequencies depend on the speed of the waves on the string and the wavelength of the waves. The six strings have different linear densities and are “tuned” by changing the tensions in the strings. We will see in Interference of Waves that the wavelength depends on the length of the strings and the boundary conditions. To play notes other than the fundamental notes, the lengths of the strings are changed by pressing down on the strings.
Check Your Understanding
The wave speed of a wave on a string depends on the tension and the linear mass density. If the tension is doubled, what happens to the speed of the waves on the string?
Show Solution
Since the speed of a wave on a taunt string is proportional to the square root of the tension divided by the linear density, the wave speed would increase by
Speed of Compression Waves in a Fluid
The speed of a wave on a string depends on the square root of the tension divided by the mass per length, the linear density. In general, the speed of a wave through a medium depends on the elastic property of the medium and the inertial property of the medium.
The elastic property describes the tendency of the particles of the medium to return to their initial position when perturbed. The inertial property describes the tendency of the particle to resist changes in velocity.
The speed of a longitudinal wave through a liquid or gas depends on the density of the fluid and the bulk modulus of the fluid,
Here the bulk modulus is defined as
Summary
- The speed of a wave on a string depends on the linear density of the string and the tension in the string. The linear density is mass per unit length of the string.
- In general, the speed of a wave depends on the square root of the ratio of the elastic property to the inertial property of the medium.
- The speed of a wave through a fluid is equal to the square root of the ratio of the bulk modulus of the fluid to the density of the fluid.
- The speed of sound through air at
is approximately
Conceptual Questions
If the tension in a string were increased by a factor of four, by what factor would the wave speed of a wave on the string increase?
Show Solution
The wave speed is proportional to the square root of the tension, so the speed is doubled.
Does a sound wave move faster in seawater or fresh water, if both the sea water and fresh water are at the same temperature and the sound wave moves near the surface?
Guitars have strings of different linear mass density. If the lowest density string and the highest density string are under the same tension, which string would support waves with the higher wave speed?
Show Solution
Since the speed of a wave on a string is inversely proportional to the square root of the linear mass density, the speed would be higher in the low linear mass density of the string.
Shown below are three waves that were sent down a string at different times. The tension in the string remains constant. (a) Rank the waves from the smallest wavelength to the largest wavelength. (b) Rank the waves from the lowest frequency to the highest frequency.
Electrical power lines connected by two utility poles are sometimes heard to hum when driven into oscillation by the wind. The speed of the waves on the power lines depend on the tension. What provides the tension in the power lines?
Show Solution
The tension in the wire is due to the weight of the electrical power cable.
Two strings, one with a low mass density and one with a high linear density are spliced together. The higher density end is tied to a lab post and a student holds the free end of the low-mass density string. The student gives the string a flip and sends a pulse down the strings. If the tension is the same in both strings, does the pulse travel at the same wave velocity in both strings? If not, where does it travel faster, in the low density string or the high density string?
Problems
A copper wire has a density of
Show Solution
a.
A piano wire has a linear mass density of
A string with a linear mass density of
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A cord has a linear mass density of
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Two strings are attached to poles, however the first string is twice as long as the second. If both strings have the same tension and mu, what is the ratio of the speed of the pulse of the wave from the first string to the second string?
Two strings are attached to poles, however the first string is twice the linear mass density mu of the second. If both strings have the same tension, what is the ratio of the speed of the pulse of the wave from the first string to the second string?
Show Solution
0.707
Transverse waves travel through a string where the tension equals 7.00 N with a speed of 20.00 m/s. What tension would be required for a wave speed of 25.00 m/s?
Two strings are attached between two poles separated by a distance of 2.00 m as shown below, both under the same tension of 600.00 N. String 1 has a linear density of
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Two strings are attached between two poles separated by a distance of 2.00 meters as shown in the preceding figure, both strings have a linear density of
The note
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Two transverse waves travel through a taut string. The speed of each wave is
A sinusoidal wave travels down a taut, horizontal string with a linear mass density of
Show Solution
a.
The speed of a transverse wave on a string is