Standing Waves on a String
Two identical waves travelling in opposite directions make a pattern that stays in place. The standing wave \(y = [2A\sin kx]\cos\omega t\), its nodes, antinodes and loops, resonance on a string of fixed length, the harmonics of strings with fixed and free ends, and the laws of a vibrating string with the sonometer and Melde's experiment.
Why this chapter
A guitar, a sitar, a violin and a veena all make their notes the same way. A string tied at both ends is set vibrating, and its waves run to the ends and back. The chapters before this one showed that waves add when they meet, and that an end sends a wave back. This chapter puts the two ideas together. A wave and its reflection add to a standing wave, a pattern that does not travel. On a string of fixed length only certain standing waves can last, and their frequencies are the notes the string can play. The chapter ends with the three laws that let a musician tune a string, and with two classic experiments that test them.
You will learn
- Add two identical waves travelling in opposite directions to get \(y = [2A\sin kx]\cos\omega t\). Read its space part and its time part.
- Find the positions of the nodes and antinodes, and state the phase rules for particles in one loop and in neighbouring loops.
- Explain why a standing wave carries no energy along the string, and how the energy of a loop changes between kinetic and potential.
- Explain resonance on a string through the round trip of a wave, and derive \(2L = n\lambda\) from it and from the boundary conditions.
- Find the harmonics of a string fixed at both ends and of a string with one free end. Say why a free end allows only odd harmonics.
- Use \(f = (1/2L)\sqrt{T/\mu}\) and the three laws of a vibrating string, and explain the sonometer and the two arrangements of Melde's experiment.
Before you begin
From Introduction to Waves you should know the harmonic wave \(y = A\sin(kx - \omega t)\), with \(k = 2\pi/\lambda\) and \(\omega = 2\pi f\). From Propagating Waves on a String you need the speed \(v = \sqrt{T/\mu}\). From Superposition and Reflection you need the principle of superposition and the two reflections. A fixed end returns a wave inverted, and a free end, where the slope is zero, returns it upright. You should be able to use the identity for the sum of two sines.
Lessons
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%Glossary
The key words of this chapter, with the meaning this chapter gives them.
- antinode
- A point of a standing wave where the amplitude is largest, \(2A\). Antinodes lie midway between nodes, a quarter of a wavelength from each.
- boundary condition
- A rule that a wave must obey at the edge of its medium. A fixed end of a string must be a node, and a free end must be an antinode.
- envelope
- The curve \(\pm 2A\sin(kx)\) that bounds a standing wave. At each position it gives the largest displacement the medium reaches there.
- fundamental frequency
- The lowest resonant frequency of a system, \(f_1\). For a string fixed at both ends, \(f_1 = v/2L\). It is also called the first harmonic.
- harmonic
- A resonant frequency that is a whole-number multiple of the fundamental. The \(n\)th harmonic is \(f_n = nf_1\).
- harmonic series
- The full set of resonant frequencies of a system. For a string fixed at both ends it is \(f_1, 2f_1, 3f_1, \dots\), and with one free end only \(f_1, 3f_1, 5f_1, \dots\).
- loop
- The part of a standing wave between two consecutive nodes, half a wavelength long. All its particles oscillate in phase.
- mass per unit length
- The mass of one metre of a string or a wire, \(\mu\), in kg/m. Also called the linear mass density. For a wire of radius \(r\) and density \(\rho\), \(\mu = \pi r^2\rho\).
- Melde's experiment
- An experiment in which a string tied to a prong of a vibrating tuning fork forms standing waves. With the prongs across the string it vibrates at the fork's frequency, and with the prongs along it at half that frequency.
- node
- A point of a standing wave where the amplitude is zero, so the medium there never moves. Adjacent nodes are half a wavelength apart.
- normal mode
- A pattern in which every part of a system oscillates at one frequency. On a string, each normal mode is a standing wave with its own number of loops.
- octave
- The step between two notes whose frequencies are in the ratio \(2 : 1\). The ear hears the higher note as the same note, raised.
- overtone
- A resonant frequency above the fundamental. The first overtone is the next resonant frequency after \(f_1\).
- paper rider
- A small V-shaped piece of paper placed on a sonometer wire. It jumps off when the wire resonates, because the middle of the wire is then an antinode swinging widely.
- resonance
- The large, steady vibration of a system driven at one of its natural frequencies. On a string, each reflected wave then returns in phase with the source and adds to the new wave.
- resonant frequency
- A frequency at which a system resonates. A string of fixed length has a whole series of them.
- sonometer
- A hollow wooden box carrying a wire stretched over two bridges, with its tension set by hanging masses. It is used to study the laws of a vibrating string.
- standing wave
- The pattern formed by two identical waves travelling in opposite directions along the same medium. Its profile oscillates in place and does not travel. Also called a stationary wave.
- tuning fork
- A two-pronged steel fork that vibrates at one fixed frequency when struck. It is used as a standard of frequency.