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# Wavelength: Λ (Lambda) Example: Wave on a String Review L 22 – and -3 •A mechanical is a disturbance that • travels through a medium – , liquids or • clocks – pendulum gases – it is a that propagates • springs • The disturbance moves because of the • elastic nature of the material • mechanical waves • waves • As the disturbance moves, the parts of the • The periodic wave relation material (segment of string, air molecules) • execute harmonic motion (move up and – standing waves down or back and forth) – beats – --- waves on strings • musical instruments – --- sound

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Wavelength: lambda) Period (T) or ( f ) Snapshot of the string at some time – freezes the motion Wave crests • An observer at a fixed position observes the wave moving by Time (s) • The time between

Displacement (m) T T   successive crests Wave troughs passing by (or troughs) is the PERIOD T • each segment of the string undergoes simple Period and frequency harmonic motion as the wave passes by are inversely related • The number of crests • between successive peaks (wave crests) 1 passing by per unit time is called the  (lambda), it is T  is the FREQUENCY f measured in meters or centimeters f 3 4

The periodic wave relation Example: wave on a string

•The wavelength, period (or frequency) and wave speed are related

•In one period the wave moves one 2 cm 2 cm 2 cm wavelength, so the wave speed v = /T •Since f = 1/T, this can be written as • A wave moves on a string at a speed of 4 cm/s f = v • A snapshot of the motion shows that the wavelength,  = 2 cm, what is the frequency,  ? which is the periodic wave relation. •v = , so  = v /  = (4 cm/s ) / (2 cm) = 2 Hz •T = 1 / f = 1 / (2 Hz) = 0.5 s 5 6

1 Making sound waves Tuning forks make sound waves  longitudinal disturbances • When the diaphragm in the speaker moves out it • The vibration of the fork causes the air near it to compresses the layer of vibrate air in front of it. • The length of the fork • This compressed air determines the frequency layer then expands and – longer fork  lower f pushes on another layer – shorter fork  higher f of air adjacent to it • It produces a pure pitch single frequency • A propagating sound wave is produced 7 8

Stringed instruments Bowed instruments • Three types – Plucked: guitar, bass, harp, harpsichord – Bowed: , viola, cello, bass – Struck: piano • In , violas, cellos and basses, a bow made of horse hair • All use strings that are fixed at both ends is used to excite the strings into vibration • The speed of the wave on the string depends on: • Each of these instruments are successively bigger (longer and – The in the string which is adjustable (tuning) heavier strings). – The thickness of the string (instruments have some • The shorter strings make the high and the long thin and some thicker strings) strings make the low frequencies • The periodic wave relation applies: f = v • Bowing excites many vibration modes simultaneously  includes a mixture of tones (richness) 9 10

Wind instruments: organs, flutes… St. Vincent’s Episcopal Church in Bedford, Texas Gravissima 8.2 Hz • The air pressure inside the pipe can vibrate, in some 4400 Hz places it is high and in other places low • Depending on the length of the pipe, various resonant modes are excited, just like blowing across a pop bottle • The long pipes make the low notes, the short pipes make the high notes

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2 Constructive Destructive Wave interference Interference Interference • If there are 2 waves on a string, they can A A combine together to make another type of wave called a • Standing waves are produced by an effect called wave interference, and there are two B B types of interference: – Constructive interference – the combination wave is bigger than the 2 waves – Destructive interference- the combination wave is smaller than the 2 waves A+B A+B 13 14

Wave interference effects Standing waves • standing waves are produced by wave interference • when a transverse wave is launched on a string, a • Waves can interfere with each other in reflected wave is produced at the other end space or in time • Wave interference in space gives rise to standing waves • Wave interference in time gives rise to beats • the primary and reflected waves interfere with each other to produce a standing wave • In some places along the string, the waves interfere

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Modes of vibration Standing waves • Nodes N  the string does not move • Antinodes A string has maximum • At the NODES, the string does not move

NNFundamental mode • At the ANTINODES the string moves up A Wavelength = 2 L and down harmonically Frequency = f L o • Since the ends are fixed, only an integer N N First harmonic mode number of half can fit A N A Wavelength = L • e. g., 2 L, L, 2/3 L, ½ L, etc. Frequency = 2 fo • The frequency is determined by the N N Second harmonic mode velocity and mode number (wavelength) A Wavelength = 2/3 L N A N A Frequency = 3 fo 17 18

3 Mode vibration frequencies Beats – sound wave interference • In general, f = v / , where v is the Beats occur when 2 waves of slightly propagation speed of the string different frequencies are combined. • The propagation speed depends on the diameter and tension of the string • Modes

– Fundamental: fo = v / 2L

– First harmonic: f1 = v / L = 2 fo

– Second harmonic: f2 = v / (2/3)L = 3 fo • The effective length can be changed by the musician “fingering” the strings 19 20

Wave interference can be used to eliminate Room noise – anti-noise technology