What is the difference between a node and an antinode in a stationary wave?

A node is a point of zero amplitude where destructive interference is total, while an antinode is a point of maximum amplitude where constructive interference occurs. Nodes remain stationary, whereas antinodes oscillate with the greatest displacement.

How does a stationary wave differ from a progressive wave regarding energy?

Progressive waves transfer energy from one location to another through the medium. In contrast, stationary waves store energy within the oscillation pattern, with no net transfer of energy along the string.

When comparing the first and second harmonics on the same string, how do their wavelengths relate?

The first harmonic has a wavelength $\lambda = 2L$, while the second harmonic has a wavelength $\lambda = L$. Therefore, the wavelength of the second harmonic is half that of the first harmonic.

What error occurs if the mass of the string itself is ignored when calculating tension?

Tension is usually assumed to be $mg$ from the hanging masses. If the string is very heavy, the tension will vary along its length due to its own weight, leading to inconsistent wave speeds and inaccurate frequency readings.

Why is it a mistake to measure the 'total length' of the string for the $L$ variable in the frequency formula?

The formula $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$ specifically refers to the vibrating length between the two fixed boundaries. Using the total length of the string (including the parts hanging over the pulley) will result in an incorrectly calculated wavelength.

What happens to the experimental results if the vibration generator is not placed exactly at a node?

The vibration generator acts as a driver; if it is not near a node, it may dampen the oscillations or fail to produce a clear resonance. This leads to poorly defined antinodes and difficulty in identifying the correct resonant frequency.

Define 'Mass per unit length' ($\mu$) and state its SI unit.

Mass per unit length is the ratio of the mass of a string to its length, representing the linear density of the material. Its SI unit is $kg\,m^{-1}$.

What is the formula for the speed of a transverse wave on a string?

The speed is given by $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in Newtons and $\mu$ is the mass per unit length in $kg\,m^{-1}$.

State the relationship between frequency ($f$) and length ($L$) for a fixed tension and string type.

Frequency is inversely proportional to length ($f \propto \frac{1}{L}$). If the length of the vibrating string is doubled, the fundamental frequency will be halved.

Why does increasing the tension in a string increase the frequency of the stationary wave?

Increasing tension increases the wave speed ($v = \sqrt{T/\mu}$). Since the wavelength is fixed by the string's length ($\lambda = 2L$), the frequency must increase to satisfy $v = f\lambda$.

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5. Waves & Particle Nature of Light

Core Practical 7: Investigating Stationary Waves

Summary

This practical investigation explores the formation of stationary waves on a stretched string and quantifies the relationship between frequency, length, tension, and mass per unit length. By observing harmonics, students verify the wave speed equation and the physical factors that govern wave propagation in a medium.

1. Definition & Core Concepts

A stationary wave (or standing wave) is the result of the superposition of two progressive waves with the same frequency and amplitude traveling in opposite directions. In this practical, a wave travels from a vibration generator to a fixed boundary (pulley) and reflects back, creating the interference pattern.

Nodes are points along the string where the displacement is always zero due to destructive interference. These occur at the fixed ends of the vibrating section of the string.

Antinodes are points of maximum displacement where the two waves interfere constructively. The distance between two adjacent nodes is exactly half a wavelength ( $\frac{\lambda}{2}$ ).

The First Harmonic (fundamental frequency) is the lowest frequency at which a stationary wave forms, characterized by a single 'loop' with a node at each end and one antinode in the center.

Diagram of a stationary wave in its first harmonic showing nodes at the ends and an antinode in the center.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

It is vital to distinguish between the physical properties of the string and the characteristics of the wave pattern observed.

Feature	Progressive Wave	Stationary Wave
Energy Transfer	Transmits energy in the direction of propagation	Stores energy; no net transfer of energy
Phase	All points within one wavelength have different phases	All points between two nodes are in phase
Amplitude	Same for all points (ignoring damping)	Varies from zero at nodes to maximum at antinodes

Vibrating Length vs. Total Length: When calculating $\mu$ , use the total length of the string. When using the wave formula, only use the length $L$ that is actually vibrating between the two fixed points.

5. Exam Strategy & Tips

Core Practical 7: Investigating Stationary Waves

Summary

1. Definition & Core Concepts

Nodes are points along the string where the displacement is always zero due to destructive interference. These occur at the fixed ends of the vibrating section of the string.

Antinodes are points of maximum displacement where the two waves interfere constructively. The distance between two adjacent nodes is exactly half a wavelength ( $\frac{\lambda}{2}$ ).

The First Harmonic (fundamental frequency) is the lowest frequency at which a stationary wave forms, characterized by a single 'loop' with a node at each end and one antinode in the center.

Diagram of a stationary wave in its first harmonic showing nodes at the ends and an antinode in the center.

2. Underlying Principles

The speed $v$ of a transverse wave on a stretched string is determined by the tension $T$ in the string and its mass per unit length $\mu$ . This is expressed by the formula: $v = \sqrt{\frac{T}{\mu}}$

Combining this with the wave equation $v = f\lambda$ , and noting that for the first harmonic the wavelength $\lambda = 2L$ , we derive the relationship for frequency: $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

This formula shows that the frequency is inversely proportional to the length ( $f \propto \frac{1}{L}$ ) and directly proportional to the square root of the tension ( $f \propto \sqrt{T}$ ). These relationships are the primary targets for experimental verification.

3. Methods & Techniques

Setup and Variable Control

Tension ( $T$ ): Controlled by hanging known masses ( $m$ ) over a pulley, where $T = mg$ . Ensure the pulley is low-friction to maintain accurate tension values.
Length ( $L$ ): Measured as the distance between the vibration generator and the bridge or pulley using a meter ruler.
Mass per unit length ( $\mu$ ): Calculated by measuring the total mass of a long section of the string and dividing by its total length before setting up the apparatus.

Procedure for Varying Length

Keep the tension constant by using a fixed mass. Adjust the position of the vibration generator to change the vibrating length $L$ .
Adjust the signal generator frequency until the first harmonic is clearly visible (maximum amplitude at the center). Record the frequency $f$ and length $L$ .

Procedure for Varying Tension

Keep the length $L$ constant. Add masses to the hanger to increase tension $T$ .
For each mass, adjust the frequency $f$ to find the first harmonic. Record $f$ and the total mass $m$ added.

4. Key Distinctions

It is vital to distinguish between the physical properties of the string and the characteristics of the wave pattern observed.

Feature	Progressive Wave	Stationary Wave
Energy Transfer	Transmits energy in the direction of propagation	Stores energy; no net transfer of energy
Phase	All points within one wavelength have different phases	All points between two nodes are in phase
Amplitude	Same for all points (ignoring damping)	Varies from zero at nodes to maximum at antinodes

Vibrating Length vs. Total Length: When calculating $\mu$ , use the total length of the string. When using the wave formula, only use the length $L$ that is actually vibrating between the two fixed points.

5. Exam Strategy & Tips

Linearizing Graphs: To verify $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$ , plot $f$ against $\frac{1}{L}$ (should be a straight line through the origin) or plot $f^2$ against $T$ . The gradient of an $f^2$ vs $T$ graph is $\frac{1}{4L^2\mu}$ .
Uncertainty in Nodes: Identifying the exact point of resonance can be difficult. To improve precision, find the frequency range over which the amplitude appears maximum and take the midpoint.
Mass per Unit Length: This is often a source of systematic error. Ensure the string is measured under slight tension to account for any stretching that might occur during the experiment.
Safety: Use a protective 'catch box' or soft padding under the hanging masses to prevent injury or floor damage if the string snaps.

Setup and Variable Control

Tension ( $T$ ): Controlled by hanging known masses ( $m$ ) over a pulley, where $T = mg$ . Ensure the pulley is low-friction to maintain accurate tension values.
Length ( $L$ ): Measured as the distance between the vibration generator and the bridge or pulley using a meter ruler.
Mass per unit length ( $\mu$ ): Calculated by measuring the total mass of a long section of the string and dividing by its total length before setting up the apparatus.

Procedure for Varying Length

Keep the tension constant by using a fixed mass. Adjust the position of the vibration generator to change the vibrating length $L$ .
Adjust the signal generator frequency until the first harmonic is clearly visible (maximum amplitude at the center). Record the frequency $f$ and length $L$ .

Procedure for Varying Tension

Keep the length $L$ constant. Add masses to the hanger to increase tension $T$ .
For each mass, adjust the frequency $f$ to find the first harmonic. Record $f$ and the total mass $m$ added.

Linearizing Graphs: To verify $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$ , plot $f$ against $\frac{1}{L}$ (should be a straight line through the origin) or plot $f^2$ against $T$ . The gradient of an $f^2$ vs $T$ graph is $\frac{1}{4L^2\mu}$ .
Uncertainty in Nodes: Identifying the exact point of resonance can be difficult. To improve precision, find the frequency range over which the amplitude appears maximum and take the midpoint.
Mass per Unit Length: This is often a source of systematic error. Ensure the string is measured under slight tension to account for any stretching that might occur during the experiment.
Safety: Use a protective 'catch box' or soft padding under the hanging masses to prevent injury or floor damage if the string snaps.