How does increasing the tension of a string affect the speed of a wave travelling along it?

Increasing the tension increases the wave speed. According to the formula $v = \sqrt{T/\mu}$, the speed is directly proportional to the square root of the tension, as a higher tension provides a greater restoring force.

What is the difference between the total mass of a string and its linear mass density ($\mu$)?

Total mass is the absolute quantity of matter in the string ($kg$), while linear mass density is the mass per unit length ($\mu = m/L$). Wave speed depends on $\mu$ because it represents the inertia relative to the length the wave must travel.

What common error occurs when calculating $\mu$ from experimental data?

Students often forget to convert mass from grams to kilograms or length from centimeters to meters. Since the standard unit for $\mu$ is $kg\,m^{-1}$, failing to use SI units will result in a wave speed that is off by several orders of magnitude.

Define the 'Fundamental Frequency' of a stretched string.

The fundamental frequency ($f_0$) is the lowest frequency at which a stationary wave can form on a string. At this frequency, the string vibrates in a single loop with nodes at the fixed ends and one antinode in the center, where $\lambda = 2L$.

If the tension in a string is quadrupled, what happens to the frequency of the first harmonic?

The frequency doubles. Because $f \propto \sqrt{T}$, quadrupling the tension results in $\sqrt{4} = 2$ times the original frequency, assuming the length and linear density remain constant.

When comparing two strings of the same length and tension, why does the thicker string produce a lower pitch?

A thicker string has a higher mass per unit length ($\mu$). Since $v = \sqrt{T/\mu}$, a higher $\mu$ results in a slower wave speed, which leads to a lower frequency ($f = v/\lambda$) for the same wavelength.

What is the relationship between the string length ($L$) and wavelength ($\lambda$) for the third harmonic?

For the third harmonic, the string contains three half-wavelengths, so $L = 3(\lambda/2)$. Rearranging this gives $\lambda = 2L/3$.

Why is it incorrect to use the formula $v = f\lambda$ to say that speed increases with frequency on a specific string?

On a specific string under constant tension, the wave speed $v$ is fixed by $\sqrt{T/\mu}$. While $v = f\lambda$ holds true, an increase in frequency is always balanced by a proportional decrease in wavelength, leaving the speed unchanged.

How does the wave speed on a string differ from the speed of sound in air?

The speed of sound in air is determined by the air's temperature and pressure and is relatively constant. In contrast, wave speed on a string is highly adjustable by changing the mechanical tension or the physical thickness of the string used.

What happens to the wave speed if you use a string that is twice as long but has the same total mass and tension?

The wave speed increases by a factor of $\sqrt{2}$. Doubling the length while keeping total mass constant halves the linear mass density ($\mu$). Since $v \propto 1/\sqrt{\mu}$, halving $\mu$ increases $v$ by $\sqrt{2}$.

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2. Waves & Electricity

Wave Speed on a Stretched Spring

Summary

The speed of a transverse wave on a stretched string is determined by the physical properties of the medium, specifically its tension and linear mass density. This relationship defines how energy propagates through stringed instruments and forms the basis for calculating harmonic frequencies in stationary waves.

1. Definition & Core Concepts

The wave speed ( $v$ ) on a stretched string refers to the velocity at which a transverse pulse or continuous wave travels along the length of the medium. Unlike sound in air, this speed is not constant but depends entirely on the mechanical state of the string.

Tension ( $T$ ) is the pulling force exerted by each end of the string, measured in Newtons (N). Increasing the tension makes the string 'stiffer,' allowing it to return to its equilibrium position more rapidly after displacement, which increases wave speed.

Mass per unit length ( $\mu$ ), also known as linear mass density, represents the mass of the string divided by its total length ( $\mu = \frac{m}{L}$ ). It is measured in $kg\,m^{-1}$ and represents the inertia of the string; a heavier string is harder to accelerate, resulting in a slower wave speed.

Diagram of a vibrating string showing the first harmonic, tension force, and nodes/antinodes.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

Wave Speed on a Stretched Spring

Summary

1. Definition & Core Concepts

Diagram of a vibrating string showing the first harmonic, tension force, and nodes/antinodes.

2. Underlying Principles

The fundamental relationship for wave speed on a string is given by the formula $v = \sqrt{\frac{T}{\mu}}$ . This indicates that the speed is proportional to the square root of the tension and inversely proportional to the square root of the linear mass density.

When a string is fixed at both ends and vibrated, stationary waves are formed through the superposition of the incident wave and its reflection. The speed of these waves determines the frequency of the resulting sound according to the wave equation $v = f\lambda$ .

For the fundamental frequency ( $f_0$ ), the wavelength is exactly twice the length of the string ( $\lambda = 2L$ ). Substituting the speed formula into the wave equation yields the frequency formula: $f_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

3. Methods & Techniques

Calculating Linear Density: To find $\mu$ , you must measure the total mass of the string in kilograms and its total length in meters. It is a common mistake to use only the vibrating length; always use the mass of the specific length of string being weighed.
Determining Harmonics: To find the frequency of higher harmonics, multiply the fundamental frequency by the harmonic number ( $n$ ). For example, the second harmonic ( $n=2$ ) occurs at $f = 2f_0$ and has a wavelength $\lambda = L$ .
Experimental Verification: By varying the length $L$ and measuring the frequency $f$ required to maintain a stationary wave, one can plot $f$ against $1/L$ . The gradient of this linear graph is equal to $\frac{v}{2}$ , allowing for the experimental determination of wave speed.

4. Key Distinctions

Feature	Tension ( $T$ )	Linear Density ( $\mu$ )
Effect on Speed	Direct (Increasing $T$ increases $v$ )	Inverse (Increasing $\mu$ decreases $v$ )
Physical Cause	External stretching force	Material thickness and density
Mathematical Power	$v \propto \sqrt{T}$	$v \propto \frac{1}{\sqrt{\mu}}$

String Length vs. Wavelength: It is critical to distinguish between the physical length of the string ( $L$ ) and the wavelength ( $\lambda$ ). In the first harmonic, $\lambda = 2L$ , but in the second harmonic, $\lambda = L$ .
Wave Speed vs. Particle Speed: The wave speed $v$ is the constant horizontal speed of the energy transfer along the string, whereas the particle speed is the varying vertical velocity of the string fibers as they oscillate.

5. Exam Strategy & Tips

Unit Consistency: Always ensure mass is in $kg$ and length is in $m$ before calculating $\mu$ . Many problems provide mass in grams or length in centimeters to trap students.
The Square Root Trap: Remember that doubling the tension does NOT double the speed or frequency; it increases them by a factor of $\sqrt{2} \approx 1.41$ . To double the frequency, the tension must be quadrupled.
Identifying Harmonics: Count the number of 'loops' or antinodes in a diagram to identify the harmonic number. One loop is the 1st harmonic, two loops is the 2nd, and so on.
Sanity Check: Higher tension should always result in a higher pitch (frequency). If your calculation shows frequency decreasing as tension increases, check your algebraic rearrangement.

Calculating Linear Density: To find $\mu$ , you must measure the total mass of the string in kilograms and its total length in meters. It is a common mistake to use only the vibrating length; always use the mass of the specific length of string being weighed.
Determining Harmonics: To find the frequency of higher harmonics, multiply the fundamental frequency by the harmonic number ( $n$ ). For example, the second harmonic ( $n=2$ ) occurs at $f = 2f_0$ and has a wavelength $\lambda = L$ .
Experimental Verification: By varying the length $L$ and measuring the frequency $f$ required to maintain a stationary wave, one can plot $f$ against $1/L$ . The gradient of this linear graph is equal to $\frac{v}{2}$ , allowing for the experimental determination of wave speed.

Feature	Tension ( $T$ )	Linear Density ( $\mu$ )
Effect on Speed	Direct (Increasing $T$ increases $v$ )	Inverse (Increasing $\mu$ decreases $v$ )
Physical Cause	External stretching force	Material thickness and density
Mathematical Power	$v \propto \sqrt{T}$	$v \propto \frac{1}{\sqrt{\mu}}$

String Length vs. Wavelength: It is critical to distinguish between the physical length of the string ( $L$ ) and the wavelength ( $\lambda$ ). In the first harmonic, $\lambda = 2L$ , but in the second harmonic, $\lambda = L$ .
Wave Speed vs. Particle Speed: The wave speed $v$ is the constant horizontal speed of the energy transfer along the string, whereas the particle speed is the varying vertical velocity of the string fibers as they oscillate.

Unit Consistency: Always ensure mass is in $kg$ and length is in $m$ before calculating $\mu$ . Many problems provide mass in grams or length in centimeters to trap students.
The Square Root Trap: Remember that doubling the tension does NOT double the speed or frequency; it increases them by a factor of $\sqrt{2} \approx 1.41$ . To double the frequency, the tension must be quadrupled.
Identifying Harmonics: Count the number of 'loops' or antinodes in a diagram to identify the harmonic number. One loop is the 1st harmonic, two loops is the 2nd, and so on.
Sanity Check: Higher tension should always result in a higher pitch (frequency). If your calculation shows frequency decreasing as tension increases, check your algebraic rearrangement.