What is the primary difference between a travelling wave and a stationary wave?

A travelling wave transfers energy through a medium, with its crests and troughs propagating continuously. In contrast, a stationary wave does not transfer energy; instead, it features fixed points of zero displacement (nodes) and maximum displacement (antinodes) that oscillate in place.

How does varying the string's length affect the first harmonic frequency compared to varying its tension?

Increasing the string's length decreases the first harmonic frequency because a longer string supports a longer wavelength for the same wave speed. Conversely, increasing the tension increases the wave speed on the string, which in turn increases the first harmonic frequency for a given length.

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

A node is a point on a stationary wave where the amplitude of oscillation is always zero, resulting from complete destructive interference. An antinode, however, is a point where the amplitude of oscillation is maximum, occurring due to complete constructive interference.

What error might occur if the signal generator is not allowed to stabilize before taking readings?

If the signal generator is not stabilized, its output frequency might drift, leading to inaccurate frequency measurements. This introduces a systematic error, as the recorded frequency may not be the true resonant frequency, affecting the reliability of the experimental data.

What is a common difficulty in accurately identifying the first harmonic's resonant frequency?

A common difficulty is the 'sharpness of resonance,' where the amplitude of vibration can change rapidly with small frequency adjustments. This makes it challenging to pinpoint the exact frequency at which the maximum amplitude (and thus the clearest harmonic) occurs, potentially leading to random errors in frequency measurement.

What safety precaution is crucial when working with a string under high tension?

It is crucial to wear safety goggles, especially if using a metal wire, as there is a risk of the string snapping under high tension and causing injury. Additionally, standing clear of the hanging masses and placing a crash mat underneath them prevents injury if they fall.

Define the 'first harmonic' in the context of a vibrating string.

The first harmonic, also known as the fundamental frequency, is the lowest resonant frequency at which a string fixed at both ends can vibrate. It forms a stationary wave pattern with a single antinode in the middle and nodes at each end, corresponding to a wavelength twice the string's vibrating length.

What is 'mass per unit length' ($\mu$) and how is it measured in this experiment?

Mass per unit length ($\mu$) is a measure of how much mass a given length of string possesses, typically expressed in kilograms per meter (kg m$^{-1}$). It is measured by weighing a known length of the string (e.g., 1 meter) on a top-pan balance and then dividing the measured mass by that known length.

What is the role of the wooden bridge in the experimental setup?

The wooden bridge serves to define and adjust the precise vibrating length ($L$) of the string between the vibration generator and the bridge itself. By moving the bridge, the experimenter can systematically change the length of the string segment that is allowed to resonate, enabling investigation into its effect on frequency.

Why is it important to control variables in this experiment?

Controlling variables ensures that any observed changes in the dependent variable (frequency) are solely due to the manipulation of the independent variable (e.g., length, tension, or mass per unit length). This isolation of effects is fundamental to establishing a clear cause-and-effect relationship and validating scientific hypotheses.

Core Practical 5: Investigating Stationary Waves | Pearson Edexcel International AS Physics

1. Definition & Core Concepts

Stationary waves, also known as standing waves, are formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions. This typically occurs when a wave reflects off a boundary and interferes with the incident wave, creating a pattern where energy does not propagate.
A harmonic refers to a specific resonant frequency at which a stationary wave can form on a string or in an air column. The first harmonic, also called the fundamental frequency, is the simplest mode of vibration, characterized by a single antinode in the middle and nodes at both ends of the vibrating segment.
Nodes are points along a stationary wave where there is no displacement or vibration, resulting from complete destructive interference. Conversely, antinodes are points of maximum displacement or vibration, occurring where constructive interference is maximal.

2. Aims and Variables

The primary aim of this experiment is to investigate the relationships between the frequency of the first harmonic ( $f$ ) and three key physical parameters of the string: its vibrating length ( $L$ ), the tension ( $T$ ) applied to it, and its mass per unit length ( $\mu$ ). By systematically varying one parameter while keeping others constant, the individual influence of each can be determined.
The independent variable is the parameter intentionally changed by the experimenter, which can be the string's length, tension, or mass per unit length. The dependent variable is the frequency of the first harmonic, which is measured as a response to changes in the independent variable.
Control variables are factors that must be kept constant to ensure a fair test and isolate the effect of the independent variable. For instance, if string length is varied, both the tension and the mass per unit length of the string must remain unchanged to accurately assess the relationship between length and frequency.

3. Experimental Setup

The apparatus typically consists of a vibration generator (or mechanical oscillator) that creates transverse waves in the string, connected to a signal generator to control its frequency. One end of the string is attached to the vibration generator, while the other end passes over a pulley and is connected to a mass hanger.
Masses are added to the hanger to apply tension to the string, and a wooden bridge is used to define the vibrating length ( $L$ ) of the string between the generator and the bridge. A metre ruler is essential for accurately measuring this length, and a top-pan balance is used to determine the mass of the string for calculating mass per unit length.
The signal generator allows for precise adjustment of the frequency, which is crucial for identifying the resonant frequencies where stationary waves form. The resolution of measuring equipment, such as the metre ruler (1 mm) and top-pan balance (0.005 g), dictates the precision of the collected data.

Diagram showing the experimental setup for investigating stationary waves on a string. A vibration generator is on the left, connected to a string that passes over a wooden bridge. The string then goes over a pulley on the right, with masses hanging from it to provide tension. The vibrating length 'L' is indicated between the generator and the bridge, and the tension 'T' is shown by the hanging masses. A first harmonic stationary wave pattern is depicted on the string.

4. Methodology

The experiment begins by setting up the apparatus, ensuring the string is taut and passes smoothly over the pulley. The vibrating length ( $L$ ) is then precisely set using the wooden bridge and measured with a metre ruler, establishing the initial conditions for the investigation.
The signal generator is activated, and its frequency is gradually increased until the first harmonic (fundamental frequency) is observed on the string. This is identified by a clear, stable stationary wave pattern with a single antinode in the middle and nodes at both ends, indicating resonance.
The frequency at which the first harmonic occurs is recorded from the signal generator. This process is repeated for several different values of the chosen independent variable (e.g., varying $L$ by adjusting the bridge position), ensuring that other variables like tension and mass per unit length remain constant.
To enhance reliability, frequency readings for each setting should be taken multiple times (e.g., three repeats) and averaged. The tension ( $T$ ) in the string is calculated from the total mass ( $m$ ) on the hanger using the formula $T = mg$ , where $g$ is the gravitational field strength. The mass per unit length ( $\mu$ ) is determined by weighing a known length of the string on a balance.

5. Data Analysis

For the first harmonic, the wavelength ( $\lambda$ ) of the stationary wave is twice the vibrating length of the string, i.e., $\lambda = 2L$ . Using the fundamental wave equation, the wave speed ( $v$ ) can be expressed as $v = f\lambda = f \times 2L$ .
To analyze the relationship between frequency and length, the equation can be rearranged to $f = \frac{v}{2L}$ . This linear relationship suggests that plotting frequency ( $f$ ) on the y-axis against the inverse of length ( $\frac{1}{L}$ ) on the x-axis should yield a straight line passing through the origin.
The gradient of this $f$ vs. $\frac{1}{L}$ graph will be equal to $\frac{v}{2}$ . Therefore, the experimental wave speed ( $v_{exp}$ ) can be determined by multiplying the calculated gradient by two. This graphical method provides a robust way to extract the wave speed from the collected data.
The experimentally determined wave speed ( $v_{exp}$ ) can then be compared with the theoretically predicted wave speed ( $v_{theory}$ ) calculated using the formula $v_{theory} = \sqrt{\frac{T}{\mu}}$ . This comparison helps validate the experimental results and the underlying physical principles. Uncertainties in measurements of length and frequency should also be assessed to determine the overall uncertainty in the calculated wave speed.

6. Evaluation & Error Management

Systematic errors can arise from equipment calibration or environmental factors. For instance, the signal generator should be allowed to stabilize for about 20 minutes before taking readings, and an oscilloscope can be used to verify its frequency output. To improve resolution, measurements should span a wide range of lengths (e.g., 20 cm intervals over at least 1.0 m).
Random errors are often associated with the precision of observation. A significant challenge is accurately identifying the exact point of resonance (the 'sharpness of resonance'), as the string's amplitude can vary rapidly near the resonant frequency. A technique to minimize this involves carefully adjusting the frequency while observing a node, aiming for the largest response.
To reduce random errors in frequency measurement, a specific repeat procedure is recommended: first, find the frequency for the largest vibration, then increase the frequency and gradually reduce it to find the first harmonic again, recording both values. Averaging these readings, possibly with a third repeat, provides a more reliable frequency value.
Safety considerations are paramount. If using a metal wire, safety goggles should be worn in case it snaps under tension. Using a rubber string can mitigate this risk. Additionally, it is crucial to stand clear of the hanging masses and place a crash mat underneath them to prevent injury or damage if they fall.

7. Underlying Physics Principles

The formation of stationary waves is a direct consequence of the principle of superposition, where two or more waves combine to form a resultant wave. In this practical, the incident wave from the vibration generator and its reflection from the bridge superpose to create the stationary wave pattern.
The relationship between wave speed ( $v$ ), frequency ( $f$ ), and wavelength ( $\lambda$ ) is given by the fundamental wave equation: $v = f\lambda$ . This equation is central to understanding how the observed frequency relates to the physical dimensions of the vibrating string.
The speed of a transverse wave on a stretched string is determined by the string's physical properties: its tension ( $T$ ) and its mass per unit length ( $\mu$ ). The theoretical formula for wave speed is $v = \sqrt{\frac{T}{\mu}}$ , which highlights that higher tension increases wave speed, while greater inertia (mass per unit length) decreases it.

Core Practical 5: Investigating Stationary Waves

Summary

1. Definition & Core Concepts

2. Aims and Variables

3. Experimental Setup

4. Methodology

5. Data Analysis

6. Evaluation & Error Management

7. Underlying Physics Principles

Core Practical 5: Investigating Stationary Waves

Summary

1. Definition & Core Concepts

2. Aims and Variables

3. Experimental Setup

4. Methodology

5. Data Analysis

6. Evaluation & Error Management

7. Underlying Physics Principles