LECTURE 20 — Stationary Waves At the end

LECTURE 20 - Stationary Waves At the end of this lecture you should be able to: • describe in English what a stationary wave is and how it is formed • Know what is meant by normal modes of vibration for a string attached at both ends • be able to derive expressions for the fundamental and harmonic series for such a string • understand derive the resonant modes for columns of air closed at both ends, open at both ends, and closed at one end but open the other. • know what is meant by a node and an antinode • be able to show mathematically how a stationary wave arises from the superposition of two waves of equal amplitude and frequency, travelling in opposite directions.

Stationary (standing) waves • • • Many examples in engineering and science: Acoustics (e. g. waves on a violin string or in wind instruments) Atomic physics (e. g. ‘electron waves’ in an atom) Engineering (e. g. waves in mechanical structures such as bridges) Optics - lasers Microwaves Etc.

Nodes and antinodes

Transverse standing waves on a stretched string or spring wavelength nodes antinodes frequency

Resonant modes for sound waves in a tube or cavity • Tube closed at both ends: use exactly the same diagrams and frequencies as for a string fixed at both ends (modes and nodes are not the same thing)

Sound waves in a pipe Fundamental mode or first harmonic

Resonant frequencies for pipe closed at one end and open at the other length wavelength frequency

Resonant frequencies for pipe open at both ends wavelength frequency

Modes of vibration of a metal rod If suspended freely : the modes of vibration are similar to a pipe open at both ends applet for flexible metal rod: http: //www. falstad. com/barwaves/index. html

Mathematical expression for a stationary wave Consider two progressive waves moving towards each other. Assume they have the same amplitude, frequency and wavelength.

Superposition of two waves travelling in opposite directions (graphical approach)

http: //science. sbcc. edu/physics/f lash/oscillationswaves/standing waves. html (there are many other examples on the internet)

Superposition of two waves travelling in opposite directions (mathematical approach) y = 2 Asinkx cosωt At each value of x the particle oscillates up and down with frequency ω. The amplitude of the oscillation varies with x. Energy is not transported along the wave.

Nodes and antinodes y = 2 Asinkx cosωt If x = 0, λ/2, λ, 3λ/2, 2λ, etc, y = 0 for all values of t. (At these points kx = nπ) Such a point is called a node. If sinkx = 1, the wave oscillates between y = -2 A through y = 0 to y = 2 A, its maximum value. Such a point is called an antinode.

Progressive and stationary waves We can thus classify waves as a) progressive (travelling), in which the energy travels in the direction of the wave b) stationary, in which energy is stored in the vibration of the wave.

Comparison between a progressive wave and a stationary wave Progressive Energy Amplitude of vibrating particles Phase of vibrating particles Stationary

Example 1 A 10 m long string, fixed at both ends is under a tension of 96. 7 N. The linear mass density of the string is 0. 122 kg/m. Find the third harmonic frequency of the string. .

Example 2 A piano wire of length 1. 5 m is fixed at both ends. The frequency of the 3 rd harmonic is 440 Hz and the maximum amplitude is 4. 0 x 10 -4 m. This stationary wave can be considered to be the superposition of two travelling waves. What are the amplitudes and speeds of these waves? State the equation of this third harmonic stationary wave

Example 3 A pipe 90 cm long is open at both ends. How long must a second pipe, closed at one end, be if it is to have the same fundamental frequency as the open pipe?

LECTURE READING 20 Stationary Waves Adams and Allday: 6. 15, 6. 16, 6. 17 At the end of this lecture you should • be able to describe in English what a stationary wave is and how it is formed • understand what is meant by the normal modes of vibration for a string attached at both ends • be able to derive expressions for the fundamental and harmonic series for such a string • understand derive the resonant modes for columns of air – closed at both ends, open at both ends, and closed at one end but open the other. • know what is meant by a node and an antinode • be able to show mathematically how a stationary wave arises from the superposition of two waves of equal amplitude and frequency, travelling in opposite directions.

Addendum: Answers to Examples • Example 1 : f 3 = 4. 22 Hz • Example 2 : A = 2 x 10 -4 m; v = 440 m/s y (x, t) = (4 x 10 -4 )(sin 2πx) (cos 880πt) • Example 3 : L 2 = ½ L 1