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Waves and Sound Chapter 14

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Wave Motion A wave is a moving self-sustained disturbance of a medium – either a field or a substance. Mechanical waves are waves in a material medium. Mechanical waves require Some source of disturbance A medium that can be disturbed Some physical connection between or mechanism though which adjacent portions of the medium influence each other All waves carry energy and momentum

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Wave Characteristics The state of being displaced moves through the medium as a wave. A progressive or travelling wave is a self-sustaining disturbance of a medium that propagates from one region to another, carrying energy and momentum. Examples: waves on a string, surface waves on liquids, sound waves in air, and compression waves in solids or liquids. In all cases the disturbance advances and not the medium.

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Traveling Waves Flip one end of a long rope that is under tension and fixed at one end The pulse travels to the right with a definite speed A disturbance of this type is called a traveling wave

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Description of a Wave A steady stream of pulses on a very long string produces a continuous wave The blade oscillates in simple harmonic motion Each small segment of the string, such as P, oscillates with simple harmonic motion

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**Amplitude and Wavelength**

Amplitude (A) is the maximum displacement of string above the equilibrium position Wavelength (λ), is the distance between two successive points that behave identically

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Longitudinal Waves In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave A longitudinal wave is also called a compression wave

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**Longitudinal Wave Represented as a Sine Curve**

A longitudinal wave can also be represented as a sine curve Compressions correspond to crests and stretches correspond to troughs Also called density waves or pressure waves

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Transverse Waves In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion

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Waveforms Wavepulse in taut rope. Shape of pulse is determined by motion of driver. If driver (hand) oscillates up and down in a regular way, it generates a wave train – a constant frequency carrier whose amplitude is modulated (varies with time.)

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**Waveform – The shape of a Wave**

The high points are crests of the wave The low points are troughs of the wave As a 2-D or 3-D wave propagates, it creates a wavefront

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Velocity of Waves Period (T) of a periodic wave - time it takes for a single profile to pass a point in space - the number of seconds per cycles. The inverse of the period (1 /T) is the frequency f, the number of profiles passing per second, the number of cycles per second. The distance in space over which the wave executes one cycle of its basic repeated form is the wavelength, l – the length of the profile.

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Velocity of Waves The speed of the wave — the rate (in m/s) at which the wave advances Is derived from the basic speed equation of distance/time Since a length of wave l passes by in a time T, its speed must equal l /T = f l The speed of any progressive periodic wave: v = fl

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Example 1 A youngster in a boat watches waves on a lake that seem to be an endless succession of identical crests passing, with a half-second between them. If one wave takes 1.5 s to sweep straight down the length of her 4.5 m-long boat, what are the frequency, period, and wavelength of the waves? Given: The waves are periodic; 0.5 s between crests; L = 4.5 m; t = 1.5 s Find: T, f, v, and l

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**Transverse Waves: Strings**

The speed of a mechanical wave is determined by the inertial and elastic properties of the medium and not in any way by the motion of the source Pulse traveling with a speed v along a lightweight, flexible string under constant tension FT v (11.3) When m/L is large, there is a lot of inertia and the speed is low. When FT is large, the string tends to spring back rapidly, and the speed is high

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Example 2 A 2.0 m-long horizontal string having a mass of 40 g is slung over a light frictionless pulley, and its end is attached to a hanging 2.0 kg mass. Compute the speed of the wavepulse on the string. Ignore the weight of the overhanging length of rope. Given: A string of length l = 2.0 m, m = 40 g supporting a 2.0 kg load Find: v

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**Reflection, Refraction, Diffraction and Absorption**

End of rope is held stationary; energy pumped in at the other end, the reflected wave ideally carries away all the original energy It is inverted – 180° out-of-phase with the incident wave End of the rope is free; it will rise up as the pulse arrives until all the energy is stored elastically. The rope then snaps back down, producing a reflected wavepulse that is right side up.

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**Reflection of Waves – Fixed Boundary**

Whenever a traveling wave reaches a boundary, some or all of the wave is reflected When it is reflected from a fixed end, the wave is inverted The shape remains the same

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**Reflected Wave – Open Boundary**

When a traveling wave reaches an open boundary, all or part of it is reflected When reflected from an open boundary, the pulse is not inverted

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**Reflection, Refraction, Diffraction and Absorption**

When a wave passes from one medium to another having different physical characteristics, there will be a redistribution of energy. Medium is also displaced, and a portion of the incident energy appears as a refracted wave. If the incident wave is periodic, the transmitted wave has the same frequency but a different speed and therefore a different wavelength: the larger the density of the refracting medium, the smaller the length of the wave.

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**Reflection, Refraction, Diffraction and Absorption**

When a wave meets a hole or another obstacle, it can be bent around it or through it—Diffraction A wave can lose part or all of its energy when it meets a boundary – Absorption.

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**Reflection, Refraction, Diffraction and Absorption**

A wave passing through a “lens” will be both reflected AND refracted. Examples include light (of course) and also sound (through the balloon of different gas) Absorption can either SUBTRACT (beach sand) or ADD (wind) energy to a wave, depending on which way the energy is being transferred.

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**Superposition of Waves**

Superposition Principle: In the region where two or more waves overlap, the resultant is the algebraic sum of the various contributions at each point. Superimposing two harmonic waves of the same frequency and amplitude: at every value of x, add the heights of the two sine curves – above the axis as positive and below it as negative. The sum of any number of harmonic waves of the same frequency traveling in the same direction is also a harmonic wave of that frequency.

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Interference of Waves Two traveling waves can meet and pass through each other without being destroyed or even altered Waves obey the Superposition Principle If two or more traveling waves are moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point Actually only true for waves with small amplitudes

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**Constructive Interference**

Two waves, a and b, have the same frequency and amplitude Are in phase The combined wave, c, has the same frequency and a greater amplitude

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**Destructive Interference**

Two waves, a and b, have the same amplitude and frequency They are 180° out of phase When they combine, the waveforms cancel

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Superposition When two or more waves interact, their amplitudes are added (superimposed) one upon the other, creating interference. Constructive interference occurs when the superposition increases amplitude. Destructive interference decreases the amplitude.

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**Natural Frequency/Harmonics**

If a periodic force occurs at the appropriate frequency, a standing wave will be produced in the medium. The lowest natural frequency in a medium is its fundamental harmonic. Double this frequency to produce the 2nd harmonic. Triple this frequency to produce the 3rd harmonic

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**Natural Frequency/Harmonics**

REQUIRES FIXED BOUNDARIES

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Frequency and Period w0 - the natural angular frequency, the specific frequency at which a physical system oscillates all by itself once set in motion natural angular frequency and since w0 = 2pf0 natural linear frequency Since T= 1/f0 Period

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Waves and Energy As waves propagate, their energy alternates between two froms: Transverse Waves – Potential <> Kinetic Longitudinal Waves – Pressure <> Kinetic Light Waves – Electric <> Magnetic

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**Waves and Energy Generally – HIGHER FREQUENCY = HIGHER ENERGY**

HIGHER AMPLITUDE = HIGHER ENERGY

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Nodes and Modes Nodes occur/are located at points of equilibrium within a wave. Anitnodes occur/are located at points of greatest displacement (amplitude) within a wave.

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**Nodes and Modes One-dimensional modes:**

Transverse – guitar or piano strings Rotational – jump rope, lasso Two- and Three-dimensional modes: Radial – concentric circular nodes and anti-nodes Angular – linear nodes and anti-nodes radiating outward from center.

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Nodes and Modes

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Nodes and Modes

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Chapter 14 Vibrations and Waves. Hooke’s Law F s = - k x F s is the spring force k is the spring constant It is a measure of the stiffness of the spring.

Chapter 14 Vibrations and Waves. Hooke’s Law F s = - k x F s is the spring force k is the spring constant It is a measure of the stiffness of the spring.

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